Finite Group Databasebeta

← #2424×C211 #41010044

Subgroups — #2424×C211 #41010044

Order 27008.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×C211C211yesmaximal, characteristic, complemented
2C2×#121×C211C211yesmaximal, characteristic, complemented
3C2×#120×C211C211yesmaximal, characteristic, complemented
4C2×#120×C211C211yesmaximal, characteristic, complemented
5C2×C4×D4×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊C4×C211C211yesmaximal, characteristic, complemented
7#396×C211C211yesmaximal, characteristic, complemented
8#394×C211C211yesmaximal, characteristic, complemented
9#394×C211C211yesmaximal, characteristic, complemented
10#392×C211C211yesmaximal, characteristic, complemented
11#390×C211C211yesmaximal, characteristic, complemented
12#388×C211C211yesmaximal, characteristic, complemented
13#386×C211C211yesmaximal, characteristic, complemented
14#386×C211C211yesmaximal, characteristic, complemented
15#385×C211C211yesmaximal, characteristic, complemented
16C25×C211C2211yescharacteristic
17C22×D4×C211C2211yescharacteristic
18C22×D4×C211C2211yescharacteristic
19C22×D4×C211C2211yescharacteristic
20C22×D4×C211C2211yescharacteristic
21C22×D4×C211C2211yescharacteristic
22C22×D4×C211C2211yescharacteristic
23C22×D4×C211C2211yescharacteristic, complemented
24C23×C4×C211C2211yescharacteristic
25C23×C4×C211C2211yescharacteristic
26C23×C4×C211C2211yescharacteristic, complemented
27#121×C211—24no—
28#121×C211—24no—
29#120×C211—24no—
30#120×C211—24no—
31C4×D4×C211—24no—
32C2×C4⋊C4×C211C2211yescharacteristic, complemented
33C2×C4⋊C4×C211C2211yescharacteristic, complemented
34C2×C4⋊C4×C211C2211yescharacteristic, complemented
35C2×C4⋊C4×C211C2211yescharacteristic, complemented
36C2×C22⋊C4×C211—24no—
37C2×C22⋊C4×C211C2211yescharacteristic
38C2×C22⋊C4×C211C2211yescharacteristic
39C2×C22⋊C4×C211C2211yescharacteristic
40C2×C22⋊C4×C211C2211yescharacteristic
41C2×C22⋊C4×C211C2211yescharacteristic
42C2×C22⋊C4×C211C2211yescharacteristic
43C2×C22⋊C4×C211C2211yescharacteristic
44C2×C22⋊C4×C211C2211yescharacteristic
45C2×C22⋊C4×C211C2211yescharacteristic
46C2×C22⋊C4×C211C2211yescharacteristic
47C2×C22⋊C4×C211C2211yescharacteristic, complemented
48C2×C22⋊C4×C211C2211yescharacteristic, complemented
49C2×C22⋊C4×C211C2211yescharacteristic, complemented
50C2×C22⋊C4×C211C2211yescharacteristic, complemented
51C2×C22⋊C4×C211C2211yescharacteristic, complemented
52C2×C42×C211C2211yescharacteristic, complemented
53#95×C211C2211yescharacteristic, complemented
54#95×C211C2211yescharacteristic, complemented
55#95×C211C2211yescharacteristic, complemented
56#95×C211C2211yescharacteristic, complemented
57C24×C211—22no—
58C24×C211—22no—
59C24×C211—22no—
60C24×C211—22no—
61C24×C211—22no—
62C24×C211—41no—
63C24×C211—41no—
64C24×C211C2311yescharacteristic
65C24×C211C2311yescharacteristic
66C24×C211C2311yescharacteristic
67C24×C211C2311yescharacteristic
68C24×C211C2311yescharacteristic
69C2×D4×C211—22no—
70C2×D4×C211—22no—
71C2×D4×C211—22no—
72C2×D4×C211—22no—
73C2×D4×C211—22no—
74C2×D4×C211—22no—
75C2×D4×C211—22no—
76C2×D4×C211—22no—
77C2×D4×C211—22no—
78C2×D4×C211—22no—
79C2×D4×C211—22no—
80C2×D4×C211—22no—
81C2×D4×C211—22no—
82C2×D4×C211—22no—
83C2×D4×C211—22no—
84C2×D4×C211—41no—
85C2×D4×C211—41no—
86C2×D4×C211—41no—
87C2×D4×C211—41no—
88C2×D4×C211—41no—
89C2×D4×C211D414yescomplemented
90C22×C4×C211—22no—
91C22×C4×C211—22no—
92C22×C4×C211—22no—
93C22×C4×C211—22no—
94C22×C4×C211—22no—
95C22×C4×C211—22no—
96C22×C4×C211—41no—
97C22×C4×C211—41no—
98C22×C4×C211—41no—
99C22×C4×C211C2311yescharacteristic
100C22×C4×C211C2311yescharacteristic
101C22×C4×C211C2311yescharacteristic
102C22×C4×C211C2311yescharacteristic
103C22×C4×C211C2311yescharacteristic
104C22×C4×C211C2311yescharacteristic
105C22×C4×C211C2311yescharacteristic
106C22×C4×C211C2311yescharacteristic
107C22×C4×C211C2311yescharacteristic
108C22×C4×C211C2311yescharacteristic
109C4⋊C4×C211—22no—
110C4⋊C4×C211—22no—
111C4⋊C4×C211—22no—
112C4⋊C4×C211—41no—
113C22⋊C4×C211—22no—
114C22⋊C4×C211—22no—
115C22⋊C4×C211—22no—
116C22⋊C4×C211—22no—
117C22⋊C4×C211—22no—
118C22⋊C4×C211—22no—
119C22⋊C4×C211—22no—
120C22⋊C4×C211—22no—
121C22⋊C4×C211—22no—
122C22⋊C4×C211—22no—
123C22⋊C4×C211—22no—
124C22⋊C4×C211—22no—
125C22⋊C4×C211—41no—
126C22⋊C4×C211D414yescomplemented
127C22⋊C4×C211D414yescomplemented
128C42×C211—22no—
129C23×C211—21no—
130C23×C211—21no—
131C23×C211—21no—
132C23×C211—21no—
133C23×C211—28no—
134C23×C211—21no—
135C23×C211—21no—
136C23×C211—21no—
137C23×C211—21no—
138C23×C211—21no—
139C23×C211—21no—
140C23×C211—21no—
141C23×C211—21no—
142C23×C211—21no—
143C23×C211—21no—
144C23×C211—21no—
145C23×C211—21no—
146C23×C211—21no—
147C23×C211—21no—
148C23×C211—21no—
149C23×C211—21no—
150C23×C211—21no—
151C23×C211—21no—
152C23×C211—21no—
153C23×C211—21no—
154C23×C211—21no—
155C23×C211—21no—
156C23×C211—21no—
157C23×C211—21no—
158C23×C211—21no—
159C23×C211—44no—
160C23×C211—44no—
161C23×C211—44no—
162C23×C211—44no—
163C23×C211—44no—
164C23×C211—44no—
165C23×C211C2×D412yes—
166C23×C211C2×D412yes—
167C23×C211C2×D412yes—
168C23×C211C2×D412yes—
169C23×C211#4112yes—
170C23×C211#4112yes—
171C23×C211C2411yescharacteristic
172D4×C211—28no—
173D4×C211—44no—
174D4×C211—44no—
175D4×C211—44no—
176D4×C211—44no—
177D4×C211—44no—
178D4×C211—44no—
179C2×C4×C211—21no—
180C2×C4×C211—21no—
181C2×C4×C211—21no—
182C2×C4×C211—21no—
183C2×C4×C211—21no—
184C2×C4×C211—21no—
185C2×C4×C211—21no—
186C2×C4×C211—21no—
187C2×C4×C211—21no—
188C2×C4×C211—21no—
189C2×C4×C211—21no—
190C2×C4×C211—21no—
191C2×C4×C211—21no—
192C2×C4×C211—21no—
193C2×C4×C211—21no—
194C2×C4×C211—21no—
195C2×C4×C211—21no—
196C2×C4×C211—21no—
197C2×C4×C211—21no—
198C2×C4×C211—21no—
199C2×C4×C211—21no—
200C2×C4×C211—21no—
201C2×C4×C211—21no—
202C2×C4×C211—21no—
203C2×C4×C211—21no—
204C2×C4×C211—44no—
205C2×C4×C211—44no—
206C2×C4×C211—44no—
207C2×C4×C211C2×D412yes—
208C2×C4×C211C2×D412yes—
209C2×C4×C211C2×D412yes—
210C2×C4×C211C2×D412yes—
211C2×C4×C211C2×D412yes—
212C22×C211—22no—
213C22×C211—22no—
214C22×C211—22no—
215C22×C211—22no—
216C22×C211—22no—
217C22×C211—22no—
218C22×C211—22no—
219C22×C211—22no—
220C22×C211—22no—
221C22×C211—22no—
222C22×C211—22no—
223C22×C211—22no—
224C22×C211—22no—
225C22×C211—22no—
226C22×C211—22no—
227C22×C211—416no—
228C22×C211—41no—
229C22×C211—41no—
230C22×C211—41no—
231C22×C211—41no—
232C22×C211—41no—
233C22×C211—41no—
234C22×C211—41no—
235C22×C211—41no—
236C22×C211—41no—
237C22×C211—41no—
238C22×C211—41no—
239C22×C211—41no—
240C22×C211—41no—
241C22×C211—41no—
242C22×C211—41no—
243C22×C211—41no—
244C22×C211—41no—
245C22×C211—41no—
246C22×C211—41no—
247C22×C211#12114yescomplemented
248C22×C211C22×D411yescharacteristic
249C22×C211C22×D411yescharacteristic
250C22×C211C22×D411yescharacteristic
251C22×C211C2×#4111yescharacteristic
252C22×C211C2×#4111yescharacteristic
253C22×C211H(5,𝔽2)11yescharacteristic
254C22×C211H(5,𝔽2)11yescharacteristic
255C4×C211—22no—
256C4×C211—22no—
257C4×C211—22no—
258C4×C211—22no—
259C4×C211—22no—
260C4×C211—41no—
261C4×C211—41no—
262C4×C211—41no—
263C4×C211—41no—
264C4×C211—41no—
265C2×C211—24no—
266C2×C211—42no—
267C2×C211—42no—
268C2×C211—42no—
269C2×C211—81no—
270C2×C211C2×#12111yescharacteristic
271C2×C211#52511yescharacteristic
272C2×C211#53811yescharacteristic
273C2×C211D4211yescharacteristic
274C2×C211D4211yescharacteristic
275C2×C211#54611yescharacteristic
276C2×C211#54611yescharacteristic
277C211#242411yescharacteristic, direct product factor, Sylow
278#2424C21111yesmaximal, characteristic, direct product factor, Sylow
279C2×#121C2×C21111yescharacteristic, complemented
280C2×#121C2×C21111yescharacteristic, complemented
281C2×#120C2×C21111yescharacteristic, complemented
282C2×#120C2×C21111yescharacteristic, complemented
283C2×C4×D4C2×C21111yescharacteristic, complemented
284C22×C22⋊C4C2×C21111yescharacteristic, complemented
285#396C2×C21111yescharacteristic, complemented
286#394C2×C21111yescharacteristic, complemented
287#394C2×C21111yescharacteristic, complemented
288#392C2×C21111yescharacteristic, complemented
289#390C2×C21111yescharacteristic, complemented
290#388C2×C21111yescharacteristic, complemented
291#386C2×C21111yescharacteristic, complemented
292#386C2×C21111yescharacteristic, complemented
293#385C2×C21111yescharacteristic, complemented
294C25C22×C21111yescharacteristic
295C22×D4C22×C21111yescharacteristic
296C22×D4C22×C21111yescharacteristic
297C22×D4C22×C21111yescharacteristic
298C22×D4C22×C21111yescharacteristic
299C22×D4C22×C21111yescharacteristic
300C22×D4C22×C21111yescharacteristic
301C22×D4C22×C21111yescharacteristic, complemented
302C23×C4C22×C21111yescharacteristic
303C23×C4C22×C21111yescharacteristic
304C23×C4C22×C21111yescharacteristic, complemented
305#121—24no—
306#121—24no—
307#120—24no—
308#120—24no—
309C4×D4—24no—
310C2×C4⋊C4C22×C21111yescharacteristic, complemented
311C2×C4⋊C4C22×C21111yescharacteristic, complemented
312C2×C4⋊C4C22×C21111yescharacteristic, complemented
313C2×C4⋊C4C22×C21111yescharacteristic, complemented
314C2×C22⋊C4—24no—
315C2×C22⋊C4C22×C21111yescharacteristic
316C2×C22⋊C4C22×C21111yescharacteristic
317C2×C22⋊C4C22×C21111yescharacteristic
318C2×C22⋊C4C22×C21111yescharacteristic
319C2×C22⋊C4C22×C21111yescharacteristic
320C2×C22⋊C4C22×C21111yescharacteristic
321C2×C22⋊C4C22×C21111yescharacteristic
322C2×C22⋊C4C22×C21111yescharacteristic
323C2×C22⋊C4C22×C21111yescharacteristic
324C2×C22⋊C4C22×C21111yescharacteristic
325C2×C22⋊C4C22×C21111yescharacteristic, complemented
326C2×C22⋊C4C22×C21111yescharacteristic, complemented
327C2×C22⋊C4C22×C21111yescharacteristic, complemented
328C2×C22⋊C4C22×C21111yescharacteristic, complemented
329C2×C22⋊C4C22×C21111yescharacteristic, complemented
330C2×C42C22×C21111yescharacteristic, complemented
331#95C22×C21111yescharacteristic, complemented
332#95C22×C21111yescharacteristic, complemented
333#95C22×C21111yescharacteristic, complemented
334#95C22×C21111yescharacteristic, complemented
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24C23×C21111yescharacteristic
343C24C23×C21111yescharacteristic
344C24C23×C21111yescharacteristic
345C24C23×C21111yescharacteristic
346C24C23×C21111yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4D4×C21114yescomplemented
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4C23×C21111yescharacteristic
378C22×C4C23×C21111yescharacteristic
379C22×C4C23×C21111yescharacteristic
380C22×C4C23×C21111yescharacteristic
381C22×C4C23×C21111yescharacteristic
382C22×C4C23×C21111yescharacteristic
383C22×C4C23×C21111yescharacteristic
384C22×C4C23×C21111yescharacteristic
385C22×C4C23×C21111yescharacteristic
386C22×C4C23×C21111yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—41no—
404C22⋊C4D4×C21114yescomplemented
405C22⋊C4D4×C21114yescomplemented
406C42—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23C2×D4×C21112yes—
444C23C2×D4×C21112yes—
445C23C2×D4×C21112yes—
446C23C2×D4×C21112yes—
447C23#41×C21112yes—
448C23#41×C21112yes—
449C23C24×C21111yescharacteristic
450D4—28no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—44no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4C2×D4×C21112yes—
486C2×C4C2×D4×C21112yes—
487C2×C4C2×D4×C21112yes—
488C2×C4C2×D4×C21112yes—
489C2×C4C2×D4×C21112yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—416no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22#121×C21114yescomplemented
526C22C22×D4×C21111yescharacteristic
527C22C22×D4×C21111yescharacteristic
528C22C22×D4×C21111yescharacteristic
529C22C2×#41×C21111yescharacteristic
530C22C2×#41×C21111yescharacteristic
531C22H(5,𝔽2)×C21111yescharacteristic
532C22H(5,𝔽2)×C21111yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—42no—
547C2—81no—
548C2C2×#121×C21111yescharacteristic
549C2#525×C21111yescharacteristic
550C2#538×C21111yescharacteristic
551C2D42×C21111yescharacteristic
552C2D42×C21111yescharacteristic
553C2#546×C21111yescharacteristic
554C2#546×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C24
commutator, Frattini, cosocleC23C24×C211
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C211 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#121×C211C2semidirect
2. C2×#121×C211C2semidirect
3. C2×#120×C211C2semidirect
4. C2×#120×C211C2semidirect
5. C2×C4×D4×C211C2semidirect
6. C22×C22⋊C4×C211C2semidirect
7. #396×C211C2semidirect
8. #394×C211C2semidirect
9. #394×C211C2semidirect
10. #392×C211C2semidirect
11. #390×C211C2semidirect
12. #388×C211C2semidirect
13. #386×C211C2semidirect
14. #386×C211C2semidirect
15. #385×C211C2semidirect
23. C22×D4×C211C22semidirect
26. C23×C4×C211C22semidirect
32. C2×C4⋊C4×C211C22semidirect
33. C2×C4⋊C4×C211C22semidirect
34. C2×C4⋊C4×C211C22semidirect
35. C2×C4⋊C4×C211C22semidirect
47. C2×C22⋊C4×C211C22semidirect
48. C2×C22⋊C4×C211C22semidirect
49. C2×C22⋊C4×C211C22semidirect
50. C2×C22⋊C4×C211C22semidirect
51. C2×C22⋊C4×C211C22semidirect
52. C2×C42×C211C22semidirect
53. #95×C211C22semidirect
54. #95×C211C22semidirect
55. #95×C211C22semidirect
56. #95×C211C22semidirect
89. C2×D4×C211D4semidirect
126. C22⋊C4×C211D4semidirect
127. C22⋊C4×C211D4semidirect
247. C22×C211#121semidirect
277. C211#2424direct (N × H)
278. #2424C211direct (N × H)
279. C2×#121C2×C211semidirect
280. C2×#121C2×C211semidirect
281. C2×#120C2×C211semidirect
282. C2×#120C2×C211semidirect
283. C2×C4×D4C2×C211semidirect
284. C22×C22⋊C4C2×C211semidirect
285. #396C2×C211semidirect
286. #394C2×C211semidirect
287. #394C2×C211semidirect
288. #392C2×C211semidirect
289. #390C2×C211semidirect
290. #388C2×C211semidirect
291. #386C2×C211semidirect
292. #386C2×C211semidirect
293. #385C2×C211semidirect
301. C22×D4C22×C211semidirect
304. C23×C4C22×C211semidirect
310. C2×C4⋊C4C22×C211semidirect
311. C2×C4⋊C4C22×C211semidirect
312. C2×C4⋊C4C22×C211semidirect
313. C2×C4⋊C4C22×C211semidirect
325. C2×C22⋊C4C22×C211semidirect
326. C2×C22⋊C4C22×C211semidirect
327. C2×C22⋊C4C22×C211semidirect
328. C2×C22⋊C4C22×C211semidirect
329. C2×C22⋊C4C22×C211semidirect
330. C2×C42C22×C211semidirect
331. #95C22×C211semidirect
332. #95C22×C211semidirect
333. #95C22×C211semidirect
334. #95C22×C211semidirect
367. C2×D4D4×C211semidirect
404. C22⋊C4D4×C211semidirect
405. C22⋊C4D4×C211semidirect
525. C22#121×C211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#121×C2112
C2×#120×C2112
C2×C4×D4×C2112
C22×C22⋊C4×C2112
#396×C2112
#394×C2112
#392×C2112
#390×C2112
#388×C2112
#386×C2112
#385×C2112
#2424211