Finite Group Databasebeta

← #14373

Subgroups — #14373

Order 256.

987 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#385C211yesmaximal, characteristic, complemented
4C2×#381C211yesmaximal, characteristic, complemented
5C2×#380C211yesmaximal, characteristic, complemented
6#1443C211yesmaximal, characteristic, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1442C211yesmaximal, characteristic, complemented
9#1439C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic, complemented
11#1438C211yesmaximal, characteristic
12#1437C211yesmaximal, characteristic, complemented
13#1437C211yesmaximal, characteristic, complemented
14#1436C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C42C2211yescharacteristic
33#386—24no—
34#386C418yescomplemented
35#385—24no—
36#381—24no—
37#380—24no—
38C2×#95C2211yescharacteristic
39C2×#95C2211yescharacteristic
40C2×#95C2211yescharacteristic
41C2×#95C2211yescharacteristic
42C2×#95C2211yescharacteristic
43C2×#95C2211yescharacteristic
44C2×#95C2211yescharacteristic
45C2×#95C2211yescharacteristic
46C2×#95C2211yescharacteristic
47C2×#95C2211yescharacteristic
48C2×#95C2211yescharacteristic
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C25C2311yescharacteristic
57C25C2311yescharacteristic
58C22×D4—22no—
59C22×D4—22no—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4C2×C414yes—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4C2×C414yescomplemented
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4C2×C414yes—
128C2×C22⋊C4C2×C414yes—
129C2×C22⋊C4C2×C414yescomplemented
130C2×C22⋊C4D414yes—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134#95—22no—
135#95—22no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95C2×C414yescomplemented
150#95C2×C414yescomplemented
151#95D414yescomplemented
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24C22×C412yes—
177C24C22×C412yes—
178C24C2×D412yes—
179C24#4112yes—
180C24#4112yes—
181C24#4112yes—
182C24C2411yescharacteristic
183C2×D4—28no—
184C2×D4—28no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—44no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—28no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—28no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4C22×C412yes—
285C22×C4C22×C412yes—
286C22×C4C22×C412yes—
287C22×C4C22×C412yes—
288C22×C4C22×C412yes—
289C22×C4C2×D412yes—
290C22×C4C2×D412yes—
291C22×C4C2×D412yes—
292C22×C4C2×D412yes—
293C22×C4C2×D412yes—
294C22×C4#4112yes—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C22⋊C4—28no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C42—44no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23#11714yes—
378C23C4×D414yes—
379C23C23×C411yescharacteristic
380C23C22×D411yescharacteristic
381C23C22×D411yescharacteristic
382C23C2×#4111yescharacteristic
383C23C2×#4111yescharacteristic
384C23C2×#4111yescharacteristic
385C23C2×#4111yescharacteristic
386C23H(5,𝔽2)11yescharacteristic
387C23H(5,𝔽2)11yescharacteristic
388C23H(5,𝔽2)11yescharacteristic
389C23H(5,𝔽2)11yescharacteristic
390C23H(5,𝔽2)11yescharacteristic
391C23H(5,𝔽2)11yescharacteristic
392C232−1+411yescharacteristic
393C232−1+411yescharacteristic
394D4—416no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—416no—
426C2×C4—416no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4C4×D414yes—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22C2×#11711yescharacteristic
519C22C2×C4×D411yescharacteristic
520C22C2×C4×D411yescharacteristic
521C22#51811yescharacteristic
522C22#51811yescharacteristic
523C22#52011yescharacteristic
524C22#52011yescharacteristic
525C22#53411yescharacteristic
526C22#53511yescharacteristic
527C22#53611yescharacteristic
528C22#53711yescharacteristic
529C22#53811yescharacteristic
530C22#53811yescharacteristic
531C22#53911yescharacteristic
532C22#53911yescharacteristic
533C22#53911yescharacteristic
534C22#53911yescharacteristic
535C22#54011yescharacteristic
536C22#54211yescharacteristic
537C22D4211yescharacteristic
538C22#54611yescharacteristic
539C22#54611yescharacteristic
540C22#54611yescharacteristic
541C22#54611yescharacteristic
542C22#54611yescharacteristic
543C22#54811yescharacteristic
544C22#55111yescharacteristic
545C22#55111yescharacteristic
546C22#55111yescharacteristic
547C22#55211yescharacteristic
548C22#55311yescharacteristic
549C22#56011yescharacteristic
550C22#56011yescharacteristic
551C22#56211yescharacteristic
552C22#56211yescharacteristic
553C4—24no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C2—44no—
567C2—44no—
568C2#231811yescharacteristic
569C2#232211yescharacteristic
570C2#232411yescharacteristic
571C2#234211yescharacteristic
572C2#235211yescharacteristic
573C2#235811yescharacteristic
574C2#237711yescharacteristic
575C2#267811yescharacteristic
576C2#268111yescharacteristic
577C2#268711yescharacteristic
578C2#268811yescharacteristic
579C2#269311yescharacteristic
580C2#269611yescharacteristic
581C2#274311yescharacteristic
582C2#274611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#385C2semidirect
4. C2×#381C2semidirect
5. C2×#380C2semidirect
6. #1443C2semidirect
7. #1442C2semidirect
8. #1442C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
12. #1437C2semidirect
13. #1437C2semidirect
14. #1436C2semidirect
15. C22⋊#95C2semidirect
34. #386C4semidirect
78. C23×C4C2×C4semidirect
129. C2×C22⋊C4C2×C4semidirect
149. #95C2×C4semidirect
150. #95C2×C4semidirect
151. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3862
C2×#3852
C2×#3812
C2×#3802
#14432
#14422
#14392
#14382
#14372
#14362
C22⋊#952