Finite Group Databasebeta

← #62953

Subgroups — #62953

Order 256.

3172 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3565C216yesmaximal, complemented
2#3565C216yesmaximal, complemented
3#3557C216yesmaximal, complemented
4#3555C216yesmaximal, complemented
5#3555C213yesmaximal, complemented
6#3552C213yesmaximal, complemented
7#3550C213yesmaximal, complemented
8#3547C213yesmaximal, complemented
9#3524C212yesmaximal, complemented
10#3523C213yesmaximal, complemented
11#3523C211yesmaximal, characteristic, complemented
12#3518C213yesmaximal, complemented
13#3518C212yesmaximal, complemented
14#3518C211yesmaximal, characteristic, complemented
15#3516C212yesmaximal, complemented
16#3509C213yesmaximal, complemented
17#3503C211yesmaximal, characteristic, complemented
18#3502C211yesmaximal, characteristic, complemented
19#3502C212yesmaximal, complemented
20#3482C211yesmaximal, characteristic, complemented
21#3480C211yesmaximal, characteristic, complemented
22Q8×#41C211yesmaximal, characteristic, complemented
23#3461C213yesmaximal, complemented
24#563C22112yes—
25#563C22112yes—
26#563C22112yes—
27#562C22112yes—
28#559C2213yes—
29#559C2216yes—
30Q82C2211yescharacteristic
31Q82C2212yes—
32Q82C2213yes—
33Q82C2216yes—
34#557C2212yes—
35#557C2211yescharacteristic
36#557C2211yescharacteristic
37#557C2211yescharacteristic
38#557C2212yes—
39#557C2216yes—
40#557C2213yes—
41#557C2216yes—
42#557C2216yes—
43#557C2216yes—
44#557C2216yes—
45#557C2213yes—
46#556C2213yes—
47#556C2213yes—
48#556C2216yes—
49#556C2216yes—
50#556C2216yes—
51#556C2216yes—
52#556C2213yes—
53#556C2213yes—
54#556C2216yes—
55#556C2216yes—
56#556C22112yes—
57#556C2216yes—
58#555C2213yes—
59#555C2216yes—
60#555C2216yes—
61#554C2213yes—
62#554C2216yes—
63#554C2216yes—
64#554C2216yes—
65#553C2213yes—
66#553C2213yes—
67#553C2216yes—
68#553C2216yes—
69#552C2216yes—
70#552C2216yes—
71#552C2216yes—
72#552C2216yes—
73#552C2216yes—
74#552C2216yes—
75#552C22112yes—
76#552C2216yes—
77#550C2211yescharacteristic
78D4×Q8C2213yes—
79D4×Q8C2211yescharacteristic
80D4×Q8C2213yes—
81D4×Q8C2213yes—
82D4×Q8C2212yes—
83D4×Q8C2216yes—
84#548C2211yescharacteristic
85#548C2211yescharacteristic
86#548C2212yes—
87#548C2212yes—
88#548C2216yes—
89#547C2213yes—
90#544C2213yes—
91#544C2216yes—
92#544C2216yes—
93#544C2216yes—
94#544C2216yes—
95#542C2211yescharacteristic
96#542C2213yes—
97#542C2216yes—
98#542C2216yes—
99#542C2216yes—
100#542C2216yes—
101#542C2216yes—
102#542C22112yes—
103#542C2216yes—
104#542C22112yes—
105#542C22112yes—
106#541C2213yes—
107#541C2213yes—
108#541C2216yes—
109#541C2216yes—
110#541C2216yes—
111#541C2216yes—
112#541C22112yes—
113#541C22112yes—
114#540C2216yes—
115#539C2216yes—
116#539C2216yes—
117#539C22112yes—
118#537C2213yes—
119#536C2213yes—
120#536C2213yes—
121#536C2216yes—
122#533C2213yes—
123#533C2216yes—
124#533C2213yes—
125#533C2213yes—
126#533C2216yes—
127#533C2213yes—
128#533C2213yes—
129#533C2216yes—
130#533C2213yes—
131#533C2213yes—
132#533C2216yes—
133#532C2213yes—
134C2×C4⋊Q8C2213yes—
135#529C2216yes—
136#529C2213yes—
137#529C2216yes—
138#529C2213yes—
139#529C2216yes—
140#529C2213yes—
141#529C2213yes—
142#529C2216yes—
143#529C2216yes—
144#529C2216yes—
145#520C2212yes—
146#519C2212yes—
147#519C2212yes—
148#519C2212yes—
149#519C2216yes—
150C4×#41C2213yes—
151C4×#41C2213yes—
152C4×#41C2211yescharacteristic
153C4×#41C2212yes—
154C4×#41C2211yescharacteristic
155C2×C4×Q8C2213yes—
156C2×C4×Q8C2211yescharacteristic
157C2×C4×Q8C2212yes—
158C2×#41C2311yescharacteristic
159C2×#41C2311yescharacteristic
160C2×#41C2313yes—
161C2×#41C2313yes—
162C22×Q8C2311yescharacteristic
163C22×Q8C2312yes—
164C22×Q8C2313yes—
165C22×Q8C2316yes—
166C4⋊Q8C2313yes—
167C4⋊Q8C2313yes—
168C4⋊Q8C2313yes—
169C4⋊Q8C2316yes—
170C4⋊Q8C2316yes—
171C4⋊Q8C2313yes—
172C4⋊Q8C2311yescharacteristic
173C4⋊Q8C2313yes—
174C4⋊Q8C2313yes—
175C4⋊Q8C2313yes—
176C4⋊Q8C2316yes—
177C4⋊Q8C2316yes—
178C4⋊Q8C2316yes—
179C4⋊Q8C2316yes—
180C4⋊Q8C2316yes—
181C4⋊Q8C2316yes—
182C4⋊Q8C2313yes—
183C4⋊Q8C2316yes—
184C4⋊Q8C2316yes—
185C4⋊Q8C23112yes—
186C4⋊Q8C23112yes—
187C4⋊Q8C23112yes—
188#127C2313yes—
189#126C2316yes—
190#126C2312yes—
191#126C2316yes—
192#126C2316yes—
193#126C2316yes—
194#126C2316yes—
195#126C2316yes—
196#126C2316yes—
197#126C2316yes—
198#126C23112yes—
199#126C2316yes—
200#126C23112yes—
201#126C23112yes—
202#126C23112yes—
203#126C23112yes—
204#126C23112yes—
205#126C23112yes—
206#126C23112yes—
207#125C2316yes—
208#125C2313yes—
209#125C2316yes—
210#125C2313yes—
211#125C2316yes—
212#125C2316yes—
213#125C2313yes—
214#125C2316yes—
215#125C2313yes—
216#125C2316yes—
217#125C2316yes—
218#125C2316yes—
219#125C2316yes—
220#125C2316yes—
221#125C2316yes—
222#125C2316yes—
223#125C2316yes—
224#125C2316yes—
225#125C2316yes—
226#125C23112yes—
227#125C23112yes—
228#125C23112yes—
229#125C23112yes—
230#125C23112yes—
231#124C2311yescharacteristic
232#124C2312yes—
233#124C2313yes—
234#124C2316yes—
235#124C2313yes—
236#124C2316yes—
237#124C2316yes—
238#124C2313yes—
239#124C2316yes—
240#124C2316yes—
241#124C23112yes—
242#124C23112yes—
243#124C23112yes—
244#124C23112yes—
245#123C2312yes—
246#123C2316yes—
247#123C2316yes—
248#123C2316yes—
249#123C2316yes—
250#123C2316yes—
251#123C2316yes—
252#123C2316yes—
253#123C23112yes—
254#123C2316yes—
255#123C23112yes—
256#123C2316yes—
257#123C23112yes—
258#123C23112yes—
259C22⋊Q8C2313yes—
260C22⋊Q8C2316yes—
261C22⋊Q8C2313yes—
262C22⋊Q8C2313yes—
263C22⋊Q8C2316yes—
264C22⋊Q8C2316yes—
265C22⋊Q8C2316yes—
266C22⋊Q8C2313yes—
267C22⋊Q8C2316yes—
268C22⋊Q8C2316yes—
269C22⋊Q8C2311yescharacteristic
270C22⋊Q8C2313yes—
271C22⋊Q8C2312yes—
272C22⋊Q8C2316yes—
273C22⋊Q8C2316yes—
274C22⋊Q8C2316yes—
275C22⋊Q8C2316yes—
276C22⋊Q8C2313yes—
277C22⋊Q8C2316yes—
278C22⋊Q8C2313yes—
279C22⋊Q8C2316yes—
280C22⋊Q8C23112yes—
281C22⋊Q8C2316yes—
282C22⋊Q8C2316yes—
283C22⋊Q8C23112yes—
284C22⋊Q8C2316yes—
285C22⋊Q8C23112yes—
286C22⋊Q8C2316yes—
287C22⋊Q8C2316yes—
288C22⋊Q8C23112yes—
289C22⋊Q8C2316yes—
290C22⋊Q8C23112yes—
291C22⋊Q8C23112yes—
292C22⋊Q8C23112yes—
293#121C2311yescharacteristic
294#121C2313yes—
295#121C2316yes—
296#121C2316yes—
297#121C2313yes—
298#121C2316yes—
299#121C2316yes—
300#121C2316yes—
301#121C2313yes—
302#121C2313yes—
303#121C2313yes—
304#121C2316yes—
305#121C23112yes—
306C4×Q8C2311yescharacteristic
307C4×Q8C2311yescharacteristic
308C4×Q8C2311yescharacteristic
309C4×Q8C2313yes—
310C4×Q8C2311yescharacteristic
311C4×Q8C2312yes—
312C4×Q8C2313yes—
313C4×Q8C2313yes—
314C4×Q8C2313yes—
315C4×Q8C2316yes—
316C4×Q8C2316yes—
317C4×Q8C2316yes—
318C4×Q8C2313yes—
319C4×Q8C2316yes—
320C4×Q8C2313yes—
321C4×Q8C2312yes—
322C4×Q8C2311yescharacteristic
323C4×Q8C2312yes—
324C4×Q8C2313yes—
325C4×Q8C2313yes—
326C4×Q8C2313yes—
327C4×Q8C2316yes—
328C4×Q8C2316yes—
329C4×Q8C2316yes—
330C4×Q8C2316yes—
331C4×Q8C2316yes—
332C4×Q8C23112yes—
333C4×Q8C23112yes—
334C4×Q8C2316yes—
335C4×Q8C2316yes—
336C4×Q8C2316yes—
337C4×Q8C2316yes—
338C4×Q8C2316yes—
339C4×Q8C2316yes—
340C4×Q8C2316yes—
341C4×Q8C23112yes—
342C4×Q8C2316yes—
343C4×D4C2311yescharacteristic
344C4×D4C2316yes—
345C4×D4C2313yes—
346C4×D4C2313yes—
347C4×D4C2316yes—
348C4×D4C2313yes—
349C4×D4C2313yes—
350C4×D4C2313yes—
351C4×D4C2313yes—
352C4×D4C2313yes—
353C4×D4C2313yes—
354C4×D4C2313yes—
355C4×D4C2316yes—
356C4×D4C2316yes—
357C4×D4C2313yes—
358C4×D4C2313yes—
359C4×D4C2316yes—
360C4×D4C2316yes—
361C4×D4C2316yes—
362C4×D4C2316yes—
363C4×D4C2316yes—
364C4×D4C23112yes—
365C4×D4C23112yes—
366#117C2311yescharacteristic
367#117C2313yes—
368#117C2313yes—
369#117C2313yes—
370#117C2316yes—
371#117C2313yes—
372#117C2316yes—
373#117C2313yes—
374#117C2316yes—
375#117C2316yes—
376#117C2316yes—
377#117C23112yes—
378#117C23112yes—
379#117C2316yes—
380#117C2316yes—
381#117C2316yes—
382#117C2316yes—
383#117C2316yes—
384#117C2316yes—
385#117C23112yes—
386#117C2316yes—
387C2×C4⋊C4C2313yes—
388C2×C4⋊C4C2316yes—
389C2×C4⋊C4C2316yes—
390C2×C4⋊C4C2313yes—
391C2×C4⋊C4C2316yes—
392C2×C4⋊C4C2316yes—
393C2×C42C2313yes—
394C2×C42C2316yes—
395C2×C42C2316yes—
396C2×C42C2313yes—
397C2×C42C2313yes—
398#41—24no—
399#41—24no—
400#41—46no—
401#41—46no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—412no—
405C2×Q8—46no—
406C2×Q8C2413yes—
407C2×Q8C2416yes—
408C2×Q8C2411yescharacteristic
409C2×Q8C2413yes—
410C2×Q8C2413yes—
411C2×Q8C2412yes—
412C2×Q8C2416yes—
413C2×Q8C2416yes—
414C2×Q8C2416yes—
415C2×Q8C24112yes—
416C2×Q8C2413yes—
417C2×Q8C2411yescharacteristic
418C2×D4C2413yes—
419C2×D4C2413yes—
420C2×D4C2411yescharacteristic
421C2×D4C2412yes—
422C2×D4C2416yes—
423C2×D4C2416yes—
424C2×D4C2416yes—
425C2×D4C2413yes—
426C2×D4C2413yes—
427C2×D4C2413yes—
428C22×C4C2413yes—
429C22×C4C2411yescharacteristic
430C22×C4C2413yes—
431C22×C4C2416yes—
432C22×C4C2413yes—
433C22×C4C2413yes—
434C22×C4C2413yes—
435C22×C4C2412yes—
436C22×C4C2413yes—
437C22×C4C2416yes—
438C22×C4C2413yes—
439C22×C4C2416yes—
440C22×C4C2413yes—
441C22×C4C2416yes—
442C22×C4C24112yes—
443C4⋊C4C2411yescharacteristic
444C4⋊C4C2412yes—
445C4⋊C4C2413yes—
446C4⋊C4C2416yes—
447C4⋊C4C2416yes—
448C4⋊C4C2416yes—
449C4⋊C4C2413yes—
450C4⋊C4C2416yes—
451C4⋊C4C2413yes—
452C4⋊C4C2413yes—
453C4⋊C4C2416yes—
454C4⋊C4C2416yes—
455C4⋊C4C2416yes—
456C4⋊C4C2416yes—
457C4⋊C4C2416yes—
458C4⋊C4C2416yes—
459C4⋊C4C24112yes—
460C4⋊C4C2416yes—
461C4⋊C4C2416yes—
462C4⋊C4C2413yes—
463C4⋊C4C2413yes—
464C4⋊C4C2411yescharacteristic
465C4⋊C4C2416yes—
466C4⋊C4C2412yes—
467C4⋊C4C2416yes—
468C4⋊C4C2416yes—
469C4⋊C4C2416yes—
470C4⋊C4C2411yescharacteristic
471C4⋊C4C2411yescharacteristic
472C4⋊C4C2413yes—
473C4⋊C4C2413yes—
474C4⋊C4C2413yes—
475C4⋊C4C2416yes—
476C4⋊C4C2413yes—
477C4⋊C4C24112yes—
478C4⋊C4C2416yes—
479C4⋊C4C24112yes—
480C4⋊C4C2416yes—
481C4⋊C4C2416yes—
482C4⋊C4C2416yes—
483C4⋊C4C2416yes—
484C4⋊C4C2416yes—
485C4⋊C4C2416yes—
486C4⋊C4C2416yes—
487C4⋊C4C24112yes—
488C4⋊C4C2416yes—
489C4⋊C4C2416yes—
490C4⋊C4C2416yes—
491C4⋊C4C2412yes—
492C22⋊C4C2413yes—
493C22⋊C4C2413yes—
494C22⋊C4C2412yes—
495C22⋊C4C2412yes—
496C22⋊C4C2416yes—
497C22⋊C4C2416yes—
498C22⋊C4C2416yes—
499C22⋊C4C24112yes—
500C22⋊C4C2412yes—
501C22⋊C4C2412yes—
502C22⋊C4C24112yes—
503C22⋊C4C2416yes—
504C22⋊C4C2416yes—
505C22⋊C4C2411yescharacteristic
506C22⋊C4C2411yescharacteristic
507C22⋊C4C2416yes—
508C22⋊C4C2413yes—
509C22⋊C4C2416yes—
510C22⋊C4C24112yes—
511C22⋊C4C2416yes—
512C22⋊C4C2416yes—
513C22⋊C4C2413yes—
514C22⋊C4C2416yes—
515C22⋊C4C2416yes—
516C22⋊C4C2416yes—
517C22⋊C4C2412yes—
518C22⋊C4C2416yes—
519C22⋊C4C2416yes—
520C42C2411yescharacteristic
521C42C2411yescharacteristic
522C42C2413yes—
523C42C2413yes—
524C42C2412yes—
525C42C2413yes—
526C42C2416yes—
527C42C2416yes—
528C42C2416yes—
529C42C2413yes—
530C42C2416yes—
531C42C2416yes—
532C42C2413yes—
533C42C2413yes—
534C42C2413yes—
535C42C2416yes—
536C42C2413yes—
537C42C24112yes—
538C42C24112yes—
539C42C24112yes—
540C23C2511yescharacteristic
541C23C2512yes—
542C23C2513yes—
543C23C2513yes—
544Q8—26no—
545Q8—212no—
546Q8—26no—
547Q8—22no—
548Q8—26no—
549Q8—46no—
550Q8—412no—
551Q8—42no—
552Q8—46no—
553Q8—46no—
554Q8—43no—
555Q82−1+414yes—
556D4—26no—
557D4—26no—
558D4—22no—
559D4—42no—
560D4—46no—
561D4—46no—
562D4—43no—
563D4—43no—
564D4—43no—
565D4—46no—
566C2×C4—212no—
567C2×C4—26no—
568C2×C4—26no—
569C2×C4—26no—
570C2×C4—26no—
571C2×C4—26no—
572C2×C4—22no—
573C2×C4—43no—
574C2×C4—42no—
575C2×C4—46no—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—412no—
579C2×C4—46no—
580C2×C4—46no—
581C2×C4C2511yescharacteristic
582C2×C4C2511yescharacteristic
583C2×C4C2516yes—
584C2×C4C2513yes—
585C2×C4C2512yes—
586C2×C4C2516yes—
587C2×C4C2513yes—
588C2×C4C2513yes—
589C2×C4C2511yescharacteristic
590C2×C4C2511yescharacteristic
591C2×C4C2511yescharacteristic
592C2×C4C2513yes—
593C2×C4C2512yes—
594C2×C4C2516yes—
595C2×C4C2512yes—
596C2×C4C2513yes—
597C2×C4C2516yes—
598C2×C4C2513yes—
599C2×C4C2511yescharacteristic
600C22—21no—
601C22—22no—
602C22—23no—
603C22—23no—
604C22—23no—
605C22—23no—
606C22—21no—
607C22—22no—
608C22—21no—
609C22—23no—
610C22—22no—
611C22—23no—
612C22C2611yescharacteristic
613C4—21no—
614C4—21no—
615C4—26no—
616C4—23no—
617C4—23no—
618C4—21no—
619C4—23no—
620C4—26no—
621C4—26no—
622C4—22no—
623C4—21no—
624C4—22no—
625C4—23no—
626C4—21no—
627C4—23no—
628C4—26no—
629C4—22no—
630C4C2×2−1+416yes—
631C4#58512yes—
632C2—41no—
633C2—42no—
634C2—43no—
635C2—43no—
636C2C22×2−1+411yescharacteristic
637C2C2×#58511yescharacteristic
638C22−1+611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3565C2semidirect
2. #3565C2semidirect
3. #3557C2semidirect
4. #3555C2semidirect
5. #3555C2semidirect
6. #3552C2semidirect
7. #3550C2semidirect
8. #3547C2semidirect
9. #3524C2semidirect
10. #3523C2semidirect
11. #3523C2semidirect
12. #3518C2semidirect
13. #3518C2semidirect
14. #3518C2semidirect
15. #3516C2semidirect
16. #3509C2semidirect
17. #3503C2semidirect
18. #3502C2semidirect
19. #3502C2semidirect
20. #3482C2semidirect
21. #3480C2semidirect
22. Q8×#41C2semidirect
23. #3461C2semidirect

Maximal subgroups

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

subgroupindex
#35652
#35572
#35552
#35522
#35502
#35472
#35242
#35232
#35182
#35162
#35092
#35032
#35022
#34822
#34802
Q8×#412
#34612