Finite Group Databasebeta

← #2678×C211 #41010298

Subgroups — #2678×C211 #41010298

Order 27008.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#123×C211C211yesmaximal, characteristic, complemented
2C2×#121×C211C211yesmaximal, characteristic, complemented
3C2×#120×C211C211yesmaximal, characteristic, complemented
4C2×#120×C211C211yesmaximal, characteristic, complemented
5#397×C211C211yesmaximal, characteristic, complemented
6#396×C211C211yesmaximal, characteristic, complemented
7#394×C211C211yesmaximal, characteristic, complemented
8#394×C211C211yesmaximal, characteristic, complemented
9#392×C211C211yesmaximal, characteristic, complemented
10#392×C211C211yesmaximal, characteristic, complemented
11#390×C211C211yesmaximal, characteristic, complemented
12#390×C211C211yesmaximal, characteristic, complemented
13#388×C211C211yesmaximal, characteristic, complemented
14#386×C211C211yesmaximal, characteristic, complemented
15#379×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, complemented
25#123×C211—24no—
26#121×C211—24no—
27#120×C211—24no—
28#120×C211—24no—
29C2×C4⋊C4×C211C2211yescharacteristic, complemented
30C2×C4⋊C4×C211C2211yescharacteristic, complemented
31C2×C4⋊C4×C211C2211yescharacteristic, complemented
32C2×C4⋊C4×C211C2211yescharacteristic, complemented
33C2×C22⋊C4×C211C2211yescharacteristic
34C2×C22⋊C4×C211C2211yescharacteristic
35C2×C22⋊C4×C211C2211yescharacteristic
36C2×C22⋊C4×C211C2211yescharacteristic
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, complemented
46C2×C22⋊C4×C211C2211yescharacteristic, complemented
47C2×C22⋊C4×C211C2211yescharacteristic, complemented
48C2×C22⋊C4×C211C2211yescharacteristic, complemented
49C2×C22⋊C4×C211C2211yescharacteristic, complemented
50C2×C42×C211C2211yescharacteristic, complemented
51#95×C211C2211yescharacteristic, complemented
52#95×C211C2211yescharacteristic, complemented
53#95×C211C2211yescharacteristic, complemented
54#95×C211C2211yescharacteristic, complemented
55C24×C211—22no—
56C24×C211—22no—
57C24×C211—22no—
58C24×C211—41no—
59C24×C211—41no—
60C24×C211—41no—
61C24×C211—41no—
62C24×C211C2311yescharacteristic
63C24×C211C2311yescharacteristic
64C24×C211C2311yescharacteristic
65C24×C211C2311yescharacteristic
66C24×C211C2311yescharacteristic
67C2×D4×C211—22no—
68C2×D4×C211—22no—
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—41no—
78C2×D4×C211—41no—
79C2×D4×C211—41no—
80C2×D4×C211—41no—
81C2×D4×C211—41no—
82C2×D4×C211—41no—
83C2×D4×C211—41no—
84C2×D4×C211—41no—
85C2×D4×C211—41no—
86C2×D4×C211—41no—
87C2×D4×C211D414yescomplemented
88C22×C4×C211—22no—
89C22×C4×C211—22no—
90C22×C4×C211—41no—
91C22×C4×C211C2311yescharacteristic
92C22×C4×C211C2311yescharacteristic
93C22×C4×C211C2311yescharacteristic
94C22×C4×C211C2311yescharacteristic
95C22×C4×C211C2311yescharacteristic
96C22×C4×C211C2311yescharacteristic
97C22×C4×C211C2311yescharacteristic
98C22×C4×C211C2311yescharacteristic
99C22×C4×C211C2311yescharacteristic
100C22×C4×C211C2311yescharacteristic
101C4⋊C4×C211—22no—
102C4⋊C4×C211—22no—
103C4⋊C4×C211—22no—
104C4⋊C4×C211—41no—
105C22⋊C4×C211—22no—
106C22⋊C4×C211—22no—
107C22⋊C4×C211—22no—
108C22⋊C4×C211—22no—
109C22⋊C4×C211—22no—
110C22⋊C4×C211—22no—
111C22⋊C4×C211—22no—
112C22⋊C4×C211—22no—
113C22⋊C4×C211—22no—
114C22⋊C4×C211—41no—
115C22⋊C4×C211—41no—
116C22⋊C4×C211—41no—
117C22⋊C4×C211—41no—
118C22⋊C4×C211—41no—
119C22⋊C4×C211—41no—
120C22⋊C4×C211—41no—
121C22⋊C4×C211D414yescomplemented
122C42×C211—41no—
123C23×C211—21no—
124C23×C211—21no—
125C23×C211—21no—
126C23×C211—21no—
127C23×C211—21no—
128C23×C211—21no—
129C23×C211—21no—
130C23×C211—21no—
131C23×C211—21no—
132C23×C211—21no—
133C23×C211—21no—
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—44no—
155C23×C211—44no—
156C23×C211—44no—
157C23×C211—44no—
158C23×C211—44no—
159C23×C211—44no—
160C23×C211—44no—
161C23×C211C2×D412yes—
162C23×C211C2×D412yes—
163C23×C211C2×D412yes—
164C23×C211#4112yes—
165C23×C211C2411yescharacteristic
166D4×C211—44no—
167D4×C211—44no—
168D4×C211—44no—
169D4×C211—44no—
170D4×C211—44no—
171D4×C211—44no—
172D4×C211—44no—
173C2×C4×C211—21no—
174C2×C4×C211—21no—
175C2×C4×C211—21no—
176C2×C4×C211—21no—
177C2×C4×C211—21no—
178C2×C4×C211—21no—
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—44no—
201C2×C4×C211C2×D412yes—
202C2×C4×C211C2×D412yes—
203C2×C4×C211C2×D412yes—
204C22×C211—22no—
205C22×C211—22no—
206C22×C211—22no—
207C22×C211—22no—
208C22×C211—22no—
209C22×C211—22no—
210C22×C211—22no—
211C22×C211—22no—
212C22×C211—22no—
213C22×C211—22no—
214C22×C211—22no—
215C22×C211—22no—
216C22×C211—416no—
217C22×C211—41no—
218C22×C211—41no—
219C22×C211—41no—
220C22×C211—41no—
221C22×C211—41no—
222C22×C211—41no—
223C22×C211—41no—
224C22×C211—41no—
225C22×C211—41no—
226C22×C211—41no—
227C22×C211—41no—
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×C211C22×D411yescharacteristic
241C22×C211C22×D411yescharacteristic
242C22×C211C2×#4111yescharacteristic
243C22×C211H(5,𝔽2)11yescharacteristic
244C22×C211H(5,𝔽2)11yescharacteristic
245C22×C211H(5,𝔽2)11yescharacteristic
246C22×C211H(5,𝔽2)11yescharacteristic
247C4×C211—22no—
248C4×C211—22no—
249C4×C211—22no—
250C4×C211—41no—
251C4×C211—41no—
252C4×C211—41no—
253C4×C211—41no—
254C4×C211—41no—
255C4×C211—41no—
256C4×C211—41no—
257C2×C211—42no—
258C2×C211—42no—
259C2×C211—42no—
260C2×C211—42no—
261C2×C211—81no—
262C2×C211#53411yescharacteristic
263C2×C211#53511yescharacteristic
264C2×C211#53811yescharacteristic
265C2×C211D4211yescharacteristic
266C2×C211#54611yescharacteristic
267C2×C211#54611yescharacteristic
268C2×C211#56011yescharacteristic
269C211#267811yescharacteristic, direct product factor, Sylow
270#2678C21111yesmaximal, characteristic, direct product factor, Sylow
271C2×#123C2×C21111yescharacteristic, complemented
272C2×#121C2×C21111yescharacteristic, complemented
273C2×#120C2×C21111yescharacteristic, complemented
274C2×#120C2×C21111yescharacteristic, complemented
275#397C2×C21111yescharacteristic, complemented
276#396C2×C21111yescharacteristic, complemented
277#394C2×C21111yescharacteristic, complemented
278#394C2×C21111yescharacteristic, complemented
279#392C2×C21111yescharacteristic, complemented
280#392C2×C21111yescharacteristic, complemented
281#390C2×C21111yescharacteristic, complemented
282#390C2×C21111yescharacteristic, complemented
283#388C2×C21111yescharacteristic, complemented
284#386C2×C21111yescharacteristic, complemented
285#379C2×C21111yescharacteristic, complemented
286C25C22×C21111yescharacteristic
287C22×D4C22×C21111yescharacteristic
288C22×D4C22×C21111yescharacteristic
289C22×D4C22×C21111yescharacteristic
290C22×D4C22×C21111yescharacteristic
291C22×D4C22×C21111yescharacteristic
292C22×D4C22×C21111yescharacteristic
293C22×D4C22×C21111yescharacteristic, complemented
294C23×C4C22×C21111yescharacteristic, complemented
295#123—24no—
296#121—24no—
297#120—24no—
298#120—24no—
299C2×C4⋊C4C22×C21111yescharacteristic, complemented
300C2×C4⋊C4C22×C21111yescharacteristic, complemented
301C2×C4⋊C4C22×C21111yescharacteristic, complemented
302C2×C4⋊C4C22×C21111yescharacteristic, complemented
303C2×C22⋊C4C22×C21111yescharacteristic
304C2×C22⋊C4C22×C21111yescharacteristic
305C2×C22⋊C4C22×C21111yescharacteristic
306C2×C22⋊C4C22×C21111yescharacteristic
307C2×C22⋊C4C22×C21111yescharacteristic
308C2×C22⋊C4C22×C21111yescharacteristic
309C2×C22⋊C4C22×C21111yescharacteristic
310C2×C22⋊C4C22×C21111yescharacteristic
311C2×C22⋊C4C22×C21111yescharacteristic
312C2×C22⋊C4C22×C21111yescharacteristic
313C2×C22⋊C4C22×C21111yescharacteristic
314C2×C22⋊C4C22×C21111yescharacteristic
315C2×C22⋊C4C22×C21111yescharacteristic, complemented
316C2×C22⋊C4C22×C21111yescharacteristic, complemented
317C2×C22⋊C4C22×C21111yescharacteristic, complemented
318C2×C22⋊C4C22×C21111yescharacteristic, complemented
319C2×C22⋊C4C22×C21111yescharacteristic, complemented
320C2×C42C22×C21111yescharacteristic, complemented
321#95C22×C21111yescharacteristic, complemented
322#95C22×C21111yescharacteristic, complemented
323#95C22×C21111yescharacteristic, complemented
324#95C22×C21111yescharacteristic, complemented
325C24—22no—
326C24—22no—
327C24—22no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24C23×C21111yescharacteristic
333C24C23×C21111yescharacteristic
334C24C23×C21111yescharacteristic
335C24C23×C21111yescharacteristic
336C24C23×C21111yescharacteristic
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4D4×C21114yescomplemented
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4C23×C21111yescharacteristic
362C22×C4C23×C21111yescharacteristic
363C22×C4C23×C21111yescharacteristic
364C22×C4C23×C21111yescharacteristic
365C22×C4C23×C21111yescharacteristic
366C22×C4C23×C21111yescharacteristic
367C22×C4C23×C21111yescharacteristic
368C22×C4C23×C21111yescharacteristic
369C22×C4C23×C21111yescharacteristic
370C22×C4C23×C21111yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—41no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4D4×C21114yescomplemented
392C42—41no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
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—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23C2×D4×C21112yes—
432C23C2×D4×C21112yes—
433C23C2×D4×C21112yes—
434C23#41×C21112yes—
435C23C24×C21111yescharacteristic
436D4—44no—
437D4—44no—
438D4—44no—
439D4—44no—
440D4—44no—
441D4—44no—
442D4—44no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
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—44no—
471C2×C4C2×D4×C21112yes—
472C2×C4C2×D4×C21112yes—
473C2×C4C2×D4×C21112yes—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—416no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22C22×D4×C21111yescharacteristic
511C22C22×D4×C21111yescharacteristic
512C22C2×#41×C21111yescharacteristic
513C22H(5,𝔽2)×C21111yescharacteristic
514C22H(5,𝔽2)×C21111yescharacteristic
515C22H(5,𝔽2)×C21111yescharacteristic
516C22H(5,𝔽2)×C21111yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—42no—
531C2—81no—
532C2#534×C21111yescharacteristic
533C2#535×C21111yescharacteristic
534C2#538×C21111yescharacteristic
535C2D42×C21111yescharacteristic
536C2#546×C21111yescharacteristic
537C2#546×C21111yescharacteristic
538C2#560×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×#123×C211C2semidirect
2. C2×#121×C211C2semidirect
3. C2×#120×C211C2semidirect
4. C2×#120×C211C2semidirect
5. #397×C211C2semidirect
6. #396×C211C2semidirect
7. #394×C211C2semidirect
8. #394×C211C2semidirect
9. #392×C211C2semidirect
10. #392×C211C2semidirect
11. #390×C211C2semidirect
12. #390×C211C2semidirect
13. #388×C211C2semidirect
14. #386×C211C2semidirect
15. #379×C211C2semidirect
23. C22×D4×C211C22semidirect
24. C23×C4×C211C22semidirect
29. C2×C4⋊C4×C211C22semidirect
30. C2×C4⋊C4×C211C22semidirect
31. C2×C4⋊C4×C211C22semidirect
32. C2×C4⋊C4×C211C22semidirect
45. C2×C22⋊C4×C211C22semidirect
46. C2×C22⋊C4×C211C22semidirect
47. C2×C22⋊C4×C211C22semidirect
48. C2×C22⋊C4×C211C22semidirect
49. C2×C22⋊C4×C211C22semidirect
50. C2×C42×C211C22semidirect
51. #95×C211C22semidirect
52. #95×C211C22semidirect
53. #95×C211C22semidirect
54. #95×C211C22semidirect
87. C2×D4×C211D4semidirect
121. C22⋊C4×C211D4semidirect
269. C211#2678direct (N × H)
270. #2678C211direct (N × H)
271. C2×#123C2×C211semidirect
272. C2×#121C2×C211semidirect
273. C2×#120C2×C211semidirect
274. C2×#120C2×C211semidirect
275. #397C2×C211semidirect
276. #396C2×C211semidirect
277. #394C2×C211semidirect
278. #394C2×C211semidirect
279. #392C2×C211semidirect
280. #392C2×C211semidirect
281. #390C2×C211semidirect
282. #390C2×C211semidirect
283. #388C2×C211semidirect
284. #386C2×C211semidirect
285. #379C2×C211semidirect
293. C22×D4C22×C211semidirect
294. C23×C4C22×C211semidirect
299. C2×C4⋊C4C22×C211semidirect
300. C2×C4⋊C4C22×C211semidirect
301. C2×C4⋊C4C22×C211semidirect
302. C2×C4⋊C4C22×C211semidirect
315. C2×C22⋊C4C22×C211semidirect
316. C2×C22⋊C4C22×C211semidirect
317. C2×C22⋊C4C22×C211semidirect
318. C2×C22⋊C4C22×C211semidirect
319. C2×C22⋊C4C22×C211semidirect
320. C2×C42C22×C211semidirect
321. #95C22×C211semidirect
322. #95C22×C211semidirect
323. #95C22×C211semidirect
324. #95C22×C211semidirect
357. C2×D4D4×C211semidirect
391. C22⋊C4D4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C2112
C2×#121×C2112
C2×#120×C2112
#397×C2112
#396×C2112
#394×C2112
#392×C2112
#390×C2112
#388×C2112
#386×C2112
#379×C2112
#2678211