Finite Group Databasebeta

← #63033

Subgroups — #63033

Order 256.

3255 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3565C216yesmaximal, complemented
2#3559C216yesmaximal, complemented
3#3557C216yesmaximal, complemented
4#3555C213yesmaximal, complemented
5#3554C213yesmaximal, complemented
6#3552C213yesmaximal, complemented
7#3524C212yesmaximal, complemented
8#3522C211yesmaximal, characteristic, complemented
9#3522C213yesmaximal, complemented
10#3517C211yesmaximal, characteristic, complemented
11#3516C212yesmaximal, complemented
12#3515C213yesmaximal, complemented
13#3515C216yesmaximal, complemented
14#3511C213yesmaximal, complemented
15#3507C213yesmaximal, complemented
16#3504C212yesmaximal, complemented
17#3502C211yesmaximal, characteristic, complemented
18#3482C213yesmaximal, complemented
19#3482C211yesmaximal, characteristic, complemented
20#3480C212yesmaximal, complemented
21#3480C211yesmaximal, characteristic, complemented
22Q8×#41C211yesmaximal, characteristic, complemented
23C2×#557C211yesmaximal, characteristic, complemented
24#563C22112yes—
25#563C22112yes—
26#562C22112yes—
27#559C2213yes—
28#559C2216yes—
29#559C2213yes—
30#559C2216yes—
31Q82C2211yescharacteristic
32Q82C2213yes—
33#557C2211yescharacteristic
34#557C2211yescharacteristic
35#557C2212yes—
36#557C2213yes—
37#557C2213yes—
38#557C2216yes—
39#557C2212yes—
40#557C2212yes—
41#557C2211yescharacteristic
42#557C2213yes—
43#557C2213yes—
44#557C2216yes—
45#556C2213yes—
46#556C2216yes—
47#556C2213yes—
48#556C2213yes—
49#556C2216yes—
50#556C2216yes—
51#556C2216yes—
52#555C2216yes—
53#554C2213yes—
54#554C2216yes—
55#554C2216yes—
56#554C2216yes—
57#554C2216yes—
58#554C2216yes—
59#554C2216yes—
60#554C2213yes—
61#553C2216yes—
62#553C2216yes—
63#553C2216yes—
64#553C2216yes—
65#553C2216yes—
66#553C2216yes—
67#552C2211yescharacteristic
68#552C2213yes—
69#552C2216yes—
70#552C2213yes—
71#552C2216yes—
72#552C2213yes—
73#552C22112yes—
74#552C22112yes—
75#552C2216yes—
76#552C2216yes—
77#550C2211yescharacteristic
78#550C2212yes—
79#550C2213yes—
80D4×Q8C2213yes—
81D4×Q8C2211yescharacteristic
82D4×Q8C2213yes—
83#548C2211yescharacteristic
84#548C2212yes—
85#548C2212yes—
86#548C2213yes—
87#548C2211yescharacteristic
88#548C2213yes—
89#548C2216yes—
90#547C2213yes—
91#547C2216yes—
92#547C2213yes—
93#547C2216yes—
94#547C2213yes—
95#547C22112yes—
96#547C2216yes—
97#544C2216yes—
98#544C2216yes—
99#544C2216yes—
100#542C2216yes—
101#542C2216yes—
102#542C2216yes—
103#542C2216yes—
104#542C2216yes—
105#542C22112yes—
106#542C2216yes—
107#541C2216yes—
108#541C22112yes—
109#541C2216yes—
110#540C2216yes—
111#540C2216yes—
112#539C22112yes—
113#539C22112yes—
114#539C22112yes—
115#539C22112yes—
116#537C2216yes—
117#536C2216yes—
118#533C2213yes—
119#533C2216yes—
120#533C2213yes—
121#533C2216yes—
122#533C2213yes—
123#533C2213yes—
124#532C2213yes—
125#532C2216yes—
126#532C2213yes—
127#532C2213yes—
128C2×C4⋊Q8C2213yes—
129#529C2216yes—
130#529C2216yes—
131#529C2216yes—
132#529C2213yes—
133#529C2216yes—
134#529C2213yes—
135#529C2213yes—
136#529C2213yes—
137#529C2216yes—
138#529C2216yes—
139#529C2216yes—
140C2×#125C2216yes—
141C2×#125C2216yes—
142#520C2212yes—
143#520C2212yes—
144#520C2216yes—
145#520C2216yes—
146#519C2212yes—
147C4×#41C2212yes—
148C4×#41C2211yescharacteristic
149C4×#41C2212yes—
150C4×#41C2216yes—
151C4×#41C2213yes—
152C4×#41C2213yes—
153C2×C4×Q8C2212yes—
154C2×C4×Q8C2213yes—
155C2×C4×Q8C2211yescharacteristic
156C2×C4×Q8C2213yes—
157C2×C4×Q8C2211yescharacteristic
158C2×#41C2311yescharacteristic
159C2×#41C2312yes—
160C2×#41C2316yes—
161C2×#41C2313yes—
162C2×#41C2313yes—
163C2×#41C2316yes—
164C2×#41C2316yes—
165C22×Q8C2311yescharacteristic
166C22×Q8C2311yescharacteristic
167C22×Q8C2313yes—
168C22×Q8C2313yes—
169C4⋊Q8C2313yes—
170C4⋊Q8C2313yes—
171C4⋊Q8C2316yes—
172C4⋊Q8C2316yes—
173C4⋊Q8C2316yes—
174C4⋊Q8C2313yes—
175C4⋊Q8C2316yes—
176C4⋊Q8C2313yes—
177C4⋊Q8C2313yes—
178C4⋊Q8C2313yes—
179C4⋊Q8C2316yes—
180C4⋊Q8C2316yes—
181C4⋊Q8C2313yes—
182C4⋊Q8C2316yes—
183C4⋊Q8C2316yes—
184C4⋊Q8C23112yes—
185#127C2313yes—
186#127C2316yes—
187#126C2316yes—
188#126C2312yes—
189#126C2316yes—
190#126C2316yes—
191#126C2316yes—
192#126C23112yes—
193#126C23112yes—
194#126C23112yes—
195#126C2316yes—
196#126C23112yes—
197#126C2316yes—
198#126C2316yes—
199#126C23112yes—
200#126C23112yes—
201#125C2313yes—
202#125C2316yes—
203#125C2316yes—
204#125C2316yes—
205#125C2316yes—
206#125C2316yes—
207#125C23112yes—
208#125C23112yes—
209#125C23112yes—
210#125C2316yes—
211#125C2316yes—
212#125C2311yescharacteristic
213#125C2316yes—
214#125C2313yes—
215#125C2316yes—
216#125C2313yes—
217#125C2316yes—
218#125C2312yes—
219#125C2316yes—
220#125C2316yes—
221#125C2316yes—
222#125C23112yes—
223#125C23112yes—
224#124C2313yes—
225#124C2316yes—
226#124C2313yes—
227#124C2313yes—
228#124C2316yes—
229#124C2313yes—
230#124C2316yes—
231#124C2316yes—
232#124C2316yes—
233#124C2316yes—
234#124C23112yes—
235#124C23112yes—
236#123C2316yes—
237#123C2312yes—
238#123C23112yes—
239#123C23112yes—
240#123C2316yes—
241#123C2316yes—
242#123C2316yes—
243#123C2316yes—
244#123C23112yes—
245#123C23112yes—
246#123C2316yes—
247#123C2316yes—
248#123C23112yes—
249#123C23112yes—
250#123C23112yes—
251#123C23112yes—
252C22⋊Q8C2311yescharacteristic
253C22⋊Q8C2313yes—
254C22⋊Q8C2313yes—
255C22⋊Q8C2313yes—
256C22⋊Q8C2316yes—
257C22⋊Q8C2313yes—
258C22⋊Q8C2316yes—
259C22⋊Q8C2316yes—
260C22⋊Q8C2313yes—
261C22⋊Q8C2316yes—
262C22⋊Q8C2313yes—
263C22⋊Q8C2311yescharacteristic
264C22⋊Q8C2316yes—
265C22⋊Q8C2316yes—
266C22⋊Q8C2316yes—
267C22⋊Q8C2316yes—
268C22⋊Q8C2316yes—
269C22⋊Q8C23112yes—
270C22⋊Q8C23112yes—
271C22⋊Q8C2316yes—
272C22⋊Q8C2316yes—
273C22⋊Q8C2316yes—
274C22⋊Q8C2316yes—
275C22⋊Q8C23112yes—
276C22⋊Q8C23112yes—
277C22⋊Q8C23112yes—
278C22⋊Q8C23112yes—
279C22⋊Q8C23112yes—
280C22⋊Q8C23112yes—
281#121C2316yes—
282#121C2313yes—
283#121C2313yes—
284#121C2316yes—
285#121C2316yes—
286#121C2313yes—
287#121C2316yes—
288#121C2316yes—
289#121C2316yes—
290#121C2313yes—
291#121C2316yes—
292#121C23112yes—
293#121C2316yes—
294#121C2316yes—
295#121C23112yes—
296#121C23112yes—
297C4×Q8C2311yescharacteristic
298C4×Q8C2311yescharacteristic
299C4×Q8C2313yes—
300C4×Q8C2312yes—
301C4×Q8C2311yescharacteristic
302C4×Q8C2312yes—
303C4×Q8C2311yescharacteristic
304C4×Q8C2313yes—
305C4×Q8C2316yes—
306C4×Q8C2313yes—
307C4×Q8C2313yes—
308C4×Q8C2313yes—
309C4×Q8C2316yes—
310C4×Q8C2316yes—
311C4×Q8C2313yes—
312C4×Q8C2313yes—
313C4×Q8C2313yes—
314C4×Q8C2316yes—
315C4×Q8C2316yes—
316C4×Q8C2313yes—
317C4×Q8C2316yes—
318C4×Q8C2313yes—
319C4×Q8C2312yes—
320C4×Q8C2311yescharacteristic
321C4×Q8C2313yes—
322C4×Q8C2316yes—
323C4×Q8C2313yes—
324C4×Q8C2313yes—
325C4×Q8C2316yes—
326C4×Q8C2316yes—
327C4×Q8C2316yes—
328C4×Q8C2316yes—
329C4×Q8C2316yes—
330C4×Q8C2316yes—
331C4×D4C2311yescharacteristic
332C4×D4C2313yes—
333C4×D4C2313yes—
334C4×D4C2316yes—
335C4×D4C2313yes—
336C4×D4C2313yes—
337C4×D4C2316yes—
338C4×D4C2313yes—
339C4×D4C2313yes—
340C4×D4C2313yes—
341C4×D4C2316yes—
342C4×D4C2313yes—
343C4×D4C2316yes—
344C4×D4C2316yes—
345C4×D4C2316yes—
346C4×D4C2316yes—
347C4×D4C2316yes—
348C4×D4C2316yes—
349C4×D4C2316yes—
350C4×D4C2313yes—
351C4×D4C2316yes—
352C4×D4C23112yes—
353C4×D4C2316yes—
354C4×D4C23112yes—
355C4×D4C23112yes—
356C4×D4C2316yes—
357C4×D4C2316yes—
358C4×D4C2316yes—
359C4×D4C2316yes—
360C4×D4C2316yes—
361C4×D4C2316yes—
362#117C2311yescharacteristic
363#117C2312yes—
364#117C2313yes—
365#117C2316yes—
366#117C2316yes—
367#117C2316yes—
368#117C2313yes—
369#117C2316yes—
370#117C2313yes—
371#117C2313yes—
372#117C2316yes—
373#117C2316yes—
374#117C2316yes—
375#117C2316yes—
376#117C2316yes—
377#117C2316yes—
378#117C23112yes—
379#117C23112yes—
380C2×C4⋊C4C2311yescharacteristic
381C2×C4⋊C4C2313yes—
382C2×C4⋊C4C2313yes—
383C2×C4⋊C4C2316yes—
384C2×C4⋊C4C2316yes—
385C2×C4⋊C4C2316yes—
386C2×C4⋊C4C2313yes—
387C2×C4⋊C4C2316yes—
388C2×C4⋊C4C23112yes—
389C2×C4⋊C4C23112yes—
390C2×C4⋊C4C2316yes—
391C2×C4⋊C4C2316yes—
392C2×C42C2316yes—
393C2×C42C2313yes—
394C2×C42C2316yes—
395C2×C42C2313yes—
396C2×C42C2313yes—
397C2×C42C2316yes—
398#41—24no—
399#41—28no—
400#41—412no—
401#41—46no—
402#41—412no—
403#41—412no—
404#41—46no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—46no—
408C2×Q8—46no—
409C2×Q8C2413yes—
410C2×Q8C2413yes—
411C2×Q8C2411yescharacteristic
412C2×Q8C2413yes—
413C2×Q8C2413yes—
414C2×Q8C2416yes—
415C2×Q8C2413yes—
416C2×Q8C2416yes—
417C2×Q8C2416yes—
418C2×Q8C2416yes—
419C2×Q8C2411yescharacteristic
420C2×D4C2413yes—
421C2×D4C2416yes—
422C2×D4C2411yescharacteristic
423C2×D4C2413yes—
424C2×D4C2416yes—
425C2×D4C2413yes—
426C2×D4C2413yes—
427C2×D4C2416yes—
428C2×D4C24112yes—
429C2×D4C2416yes—
430C2×D4C2416yes—
431C22×C4C2412yes—
432C22×C4C2413yes—
433C22×C4C2411yescharacteristic
434C22×C4C2413yes—
435C22×C4C2411yescharacteristic
436C22×C4C2413yes—
437C22×C4C2416yes—
438C22×C4C2413yes—
439C22×C4C2413yes—
440C22×C4C2416yes—
441C22×C4C2416yes—
442C22×C4C2416yes—
443C22×C4C2416yes—
444C22×C4C2413yes—
445C22×C4C2413yes—
446C22×C4C2416yes—
447C22×C4C24112yes—
448C22×C4C24112yes—
449C4⋊C4C2411yescharacteristic
450C4⋊C4C2413yes—
451C4⋊C4C2416yes—
452C4⋊C4C2413yes—
453C4⋊C4C2416yes—
454C4⋊C4C2412yes—
455C4⋊C4C2413yes—
456C4⋊C4C2413yes—
457C4⋊C4C2416yes—
458C4⋊C4C2416yes—
459C4⋊C4C2416yes—
460C4⋊C4C24112yes—
461C4⋊C4C2416yes—
462C4⋊C4C2416yes—
463C4⋊C4C2416yes—
464C4⋊C4C2416yes—
465C4⋊C4C2413yes—
466C4⋊C4C2416yes—
467C4⋊C4C2413yes—
468C4⋊C4C2416yes—
469C4⋊C4C24112yes—
470C4⋊C4C2416yes—
471C4⋊C4C24112yes—
472C4⋊C4C2411yescharacteristic
473C4⋊C4C2412yes—
474C4⋊C4C2416yes—
475C4⋊C4C2416yes—
476C4⋊C4C2416yes—
477C4⋊C4C2416yes—
478C4⋊C4C2416yes—
479C4⋊C4C24112yes—
480C4⋊C4C2416yes—
481C4⋊C4C2413yes—
482C4⋊C4C2411yescharacteristic
483C4⋊C4C2413yes—
484C4⋊C4C2416yes—
485C4⋊C4C2412yes—
486C4⋊C4C2412yes—
487C4⋊C4C2412yes—
488C4⋊C4C2411yescharacteristic
489C4⋊C4C2412yes—
490C4⋊C4C2413yes—
491C4⋊C4C2416yes—
492C4⋊C4C2416yes—
493C4⋊C4C2416yes—
494C4⋊C4C2413yes—
495C4⋊C4C2416yes—
496C4⋊C4C2416yes—
497C22⋊C4C2411yescharacteristic
498C22⋊C4C2416yes—
499C22⋊C4C2413yes—
500C22⋊C4C2416yes—
501C22⋊C4C2411yescharacteristic
502C22⋊C4C2416yes—
503C22⋊C4C2412yes—
504C22⋊C4C2413yes—
505C22⋊C4C2412yes—
506C22⋊C4C2412yes—
507C22⋊C4C2416yes—
508C22⋊C4C2416yes—
509C22⋊C4C2416yes—
510C22⋊C4C24112yes—
511C22⋊C4C24112yes—
512C22⋊C4C2416yes—
513C22⋊C4C2416yes—
514C22⋊C4C2416yes—
515C22⋊C4C2416yes—
516C22⋊C4C2416yes—
517C22⋊C4C2416yes—
518C22⋊C4C24112yes—
519C22⋊C4C24112yes—
520C22⋊C4C2413yes—
521C22⋊C4C2416yes—
522C22⋊C4C2413yes—
523C42C2412yes—
524C42C2412yes—
525C42C2416yes—
526C42C2413yes—
527C42C2413yes—
528C42C2411yescharacteristic
529C42C2413yes—
530C42C2416yes—
531C42C2413yes—
532C42C2416yes—
533C42C2413yes—
534C42C2413yes—
535C42C2413yes—
536C42C2416yes—
537C42C2413yes—
538C42C2416yes—
539C42C2416yes—
540C42C2413yes—
541C42C2416yes—
542C42C24112yes—
543C23C2511yescharacteristic
544C23C2513yes—
545C23C2516yes—
546C23C2511yescharacteristic
547Q8—26no—
548Q8—26no—
549Q8—22no—
550Q8—26no—
551Q8—26no—
552Q8—46no—
553Q8—46no—
554Q8—43no—
555Q8—46no—
556Q8—46no—
557Q82−1+414yes—
558D4—22no—
559D4—26no—
560D4—212no—
561D4—26no—
562D4—212no—
563D4—26no—
564D4—43no—
565D4—46no—
566D4—46no—
567D4—412no—
568D4—46no—
569C2×C4—24no—
570C2×C4—26no—
571C2×C4—26no—
572C2×C4—26no—
573C2×C4—26no—
574C2×C4—212no—
575C2×C4—22no—
576C2×C4—26no—
577C2×C4—22no—
578C2×C4—46no—
579C2×C4—46no—
580C2×C4—46no—
581C2×C4—46no—
582C2×C4—46no—
583C2×C4—412no—
584C2×C4—412no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4C2511yescharacteristic
588C2×C4C2511yescharacteristic
589C2×C4C2512yes—
590C2×C4C2513yes—
591C2×C4C2513yes—
592C2×C4C2511yescharacteristic
593C2×C4C2512yes—
594C2×C4C2512yes—
595C2×C4C2512yes—
596C2×C4C2513yes—
597C2×C4C2511yescharacteristic
598C2×C4C2516yes—
599C2×C4C2516yes—
600C2×C4C2513yes—
601C2×C4C2516yes—
602C2×C4C2513yes—
603C2×C4C2513yes—
604C2×C4C2513yes—
605C2×C4C2511yescharacteristic
606C22—21no—
607C22—21no—
608C22—21no—
609C22—23no—
610C22—26no—
611C22—23no—
612C22—21no—
613C22—21no—
614C22—23no—
615C22—26no—
616C22—26no—
617C22#58512yes—
618C22C2611yescharacteristic
619C4—21no—
620C4—22no—
621C4—23no—
622C4—23no—
623C4—22no—
624C4—21no—
625C4—23no—
626C4—23no—
627C4—23no—
628C4—22no—
629C4—21no—
630C4—23no—
631C4—26no—
632C4—21no—
633C4—26no—
634C4—26no—
635C4—22no—
636C4C2×2−1+416yes—
637C4#58512yes—
638C2—22no—
639C2—41no—
640C2—43no—
641C2—46no—
642C2C22×2−1+411yescharacteristic
643C2C2×#58511yescharacteristic
644C2C2×#58511yescharacteristic
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. #3559C2semidirect
3. #3557C2semidirect
4. #3555C2semidirect
5. #3554C2semidirect
6. #3552C2semidirect
7. #3524C2semidirect
8. #3522C2semidirect
9. #3522C2semidirect
10. #3517C2semidirect
11. #3516C2semidirect
12. #3515C2semidirect
13. #3515C2semidirect
14. #3511C2semidirect
15. #3507C2semidirect
16. #3504C2semidirect
17. #3502C2semidirect
18. #3482C2semidirect
19. #3482C2semidirect
20. #3480C2semidirect
21. #3480C2semidirect
22. Q8×#41C2semidirect
23. C2×#557C2semidirect

Maximal subgroups

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

subgroupindex
#35652
#35592
#35572
#35552
#35542
#35522
#35242
#35222
#35172
#35162
#35152
#35112
#35072
#35042
#35022
#34822
#34802
Q8×#412
C2×#5572