Finite Group Databasebeta

← #62913

Subgroups — #62913

Order 256.

3487 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3554C216yesmaximal, complemented
2#3552C216yesmaximal, complemented
3#3550C216yesmaximal, complemented
4#3547C216yesmaximal, complemented
5#3541C2112yesmaximal, complemented
6#3523C212yesmaximal, complemented
7#3518C212yesmaximal, complemented
8#3511C213yesmaximal, complemented
9#3506C213yesmaximal, complemented
10#3503C213yesmaximal, complemented
11#3499C212yesmaximal, complemented
12#3497C212yesmaximal, complemented
13#3486C214yesmaximal, complemented
14#3480C211yesmaximal, characteristic, complemented
15Q8×#41C211yesmaximal, characteristic, complemented
16C2×D4×Q8C211yesmaximal, characteristic, complemented
17#3460C213yesmaximal, complemented
18C2×2−1+4C2214yescomplemented
19C23×Q8C2212yes—
20#563C22124yescomplemented
21#562C22124yescomplemented
22#559C22112yes—
23#559C2213yescomplemented
24Q82C2211yescharacteristic, complemented
25Q82C2216yescomplemented
26#557C2213yescomplemented
27#557C2211yescharacteristic, complemented
28#557C2216yescomplemented
29#556C2216yes—
30#556C2213yescomplemented
31#556C2216yescomplemented
32#556C22112yescomplemented
33#555C2216yes—
34#555C2216yescomplemented
35#554C2216yes—
36#554C22112yes—
37#554C2213yescomplemented
38#554C2216yescomplemented
39#554C2216yescomplemented
40#553C2216yes—
41#553C22112yes—
42#553C22112yes—
43#552C22112yescomplemented
44#550C2216yes—
45#550C2211yescharacteristic, complemented
46D4×Q8C2212yes—
47D4×Q8C2212yes—
48D4×Q8C2216yes—
49D4×Q8C2216yes—
50D4×Q8C2212yes—
51D4×Q8C22112yes—
52D4×Q8C2211yescharacteristic, complemented
53D4×Q8C2214yescomplemented
54D4×Q8C22112yescomplemented
55D4×Q8C2216yescomplemented
56#548C2212yes—
57#548C22112yes—
58#548C2214yescomplemented
59#548C22112yescomplemented
60#548C22112yescomplemented
61#547C2216yes—
62#547C2216yescomplemented
63#547C22112yescomplemented
64#544C2213yes—
65#544C22112yescomplemented
66#542C2216yes—
67#542C2216yes—
68#542C22112yescomplemented
69#542C22112yescomplemented
70#542C22124yescomplemented
71#541C22112yescomplemented
72#540C2216yes—
73#540C2216yes—
74#539C22124yescomplemented
75#538C22112yes—
76#538C22112yes—
77#537C2213yes—
78#537C22112yescomplemented
79#536C22112yescomplemented
80#536C22112yescomplemented
81#536C22112yescomplemented
82#535C2211yescharacteristic
83#535C2213yes—
84#533C2213yes—
85#533C2216yescomplemented
86#533C2216yescomplemented
87#533C2216yescomplemented
88#532C2216yes—
89#532C2216yes—
90#532C2213yes—
91#532C2216yescomplemented
92C2×C4⋊Q8C2213yes—
93C2×C4⋊Q8C2216yescomplemented
94#529C2216yes—
95#529C2216yes—
96#529C2216yescomplemented
97#525C22112yes—
98#525C22112yes—
99C2×C22⋊Q8C2216yes—
100#520C2216yes—
101#519C2214yescomplemented
102C4×#41C2213yes—
103C4×#41C2213yes—
104C4×#41C2212yescomplemented
105C2×C4×Q8C2211yescharacteristic
106C2×C4×Q8C2212yescomplemented
107C2×C4×D4C2213yes—
1082−1+4—416no—
109C2×#41C2311yescharacteristic
110C2×#41C2313yes—
111C2×#41C2312yes—
112C2×#41C23112yes—
113C2×#41C23112yes—
114C2×#41C23112yes—
115C22×Q8—48no—
116C22×Q8C2311yescharacteristic
117C22×Q8C2312yes—
118C22×Q8C2314yes—
119C22×Q8C2316yes—
120C22×Q8C23112yes—
121C22×Q8C23112yes—
122C22×D4C2311yescharacteristic
123C23×C4C2316yes—
124C4⋊Q8C2316yes—
125C4⋊Q8C2316yes—
126C4⋊Q8C2316yes—
127C4⋊Q8C2316yes—
128C4⋊Q8C2316yes—
129C4⋊Q8C2313yes—
130C4⋊Q8C23112yes—
131C4⋊Q8C23112yes—
132C4⋊Q8C2313yes—
133C4⋊Q8C2316yes—
134C4⋊Q8C2313yes—
135C4⋊Q8C2316yes—
136C4⋊Q8C2316yes—
137C4⋊Q8C23124yes—
138#127C2312yes—
139#127C2313yes—
140#127C2316yes—
141#126C23112yes—
142#126C23112yes—
143#126C23112yes—
144#126C23124yes—
145#126C23124yes—
146#125C2316yes—
147#125C2316yes—
148#125C2316yes—
149#125C2316yes—
150#125C23112yes—
151#125C23112yes—
152#125C23124yes—
153#124C2312yes—
154#124C2316yes—
155#124C2316yes—
156#124C23112yes—
157#124C23112yes—
158#124C2316yes—
159#124C23112yes—
160#124C23124yes—
161#124C23124yes—
162#123C23112yes—
163#123C23112yes—
164#123C23124yes—
165#123C23112yes—
166#123C23112yes—
167#123C23112yes—
168#123C23124yes—
169#123C23124yes—
170C22⋊Q8C23112yes—
171C22⋊Q8C23112yes—
172C22⋊Q8C23124yes—
173C22⋊Q8C2316yes—
174C22⋊Q8C23112yes—
175C22⋊Q8C23112yes—
176C22⋊Q8C23112yes—
177C22⋊Q8C2316yes—
178C22⋊Q8C2316yes—
179C22⋊Q8C2316yes—
180C22⋊Q8C2316yes—
181C22⋊Q8C2316yes—
182C22⋊Q8C23112yes—
183C22⋊Q8C2316yes—
184C22⋊Q8C23112yes—
185C22⋊Q8C23112yes—
186C22⋊Q8C23112yes—
187C22⋊Q8C23124yes—
188C22⋊Q8C23124yes—
189C22⋊Q8C23124yes—
190#121C2316yes—
191#121C23112yes—
192#121C23112yes—
193#121C23112yes—
194#121C2312yes—
195#121C2316yes—
196#121C2316yes—
197#121C2312yes—
198#121C2316yes—
199#121C2316yes—
200#121C2316yes—
201#121C23112yes—
202#121C2316yes—
203#121C23112yes—
204#121C23112yes—
205#121C23124yes—
206#120C2314yes—
207#120C23112yes—
208C4×Q8C2311yescharacteristic
209C4×Q8C2311yescharacteristic
210C4×Q8C2313yes—
211C4×Q8C2313yes—
212C4×Q8C2312yes—
213C4×Q8C2316yes—
214C4×Q8C2316yes—
215C4×Q8C2313yes—
216C4×Q8C2313yes—
217C4×Q8C2313yes—
218C4×Q8C2313yes—
219C4×Q8C2312yes—
220C4×Q8C23112yes—
221C4×Q8C23112yes—
222C4×Q8C23112yes—
223C4×Q8C2316yes—
224C4×Q8C23112yes—
225C4×D4C2316yes—
226C4×D4C2316yes—
227C4×D4C2316yes—
228C4×D4C2316yes—
229C4×D4C2316yes—
230C4×D4C2316yes—
231C4×D4C2313yes—
232C4×D4C2313yes—
233C4×D4C2316yes—
234C4×D4C2316yes—
235C4×D4C2313yes—
236C4×D4C2313yes—
237C4×D4C2316yes—
238C4×D4C2316yes—
239C4×D4C23124yes—
240C4×D4C23112yes—
241C4×D4C23112yes—
242C4×D4C23112yes—
243C4×D4C23124yes—
244C4×D4C2316yes—
245C4×D4C23112yes—
246#117C2311yescharacteristic
247#117C2313yes—
248#117C2316yes—
249#117C2316yes—
250#117C2316yes—
251#117C23112yes—
252#117C23112yes—
253#117C23112yes—
254#117C23112yes—
255C2×C4⋊C4C2313yes—
256C2×C4⋊C4C2313yes—
257C2×C4⋊C4C2316yes—
258C2×C4⋊C4C23112yes—
259C2×C4⋊C4C2316yes—
260C2×C4⋊C4C23112yes—
261C2×C22⋊C4C2316yes—
262C2×C42C2313yes—
263C2×C42C2316yes—
264C2×C42C2316yes—
265C24C2412yes—
266#41—28no—
267#41—24no—
268#41—212no—
269#41—424no—
270#41—424no—
271#41—424no—
272C2×Q8—24no—
273C2×Q8—216no—
274C2×Q8—28no—
275C2×Q8—412no—
276C2×Q8—424no—
277C2×Q8—424no—
278C2×Q8C2411yescharacteristic
279C2×Q8C2413yes—
280C2×Q8C2413yes—
281C2×Q8C24112yes—
282C2×Q8C2416yes—
283C2×Q8C2416yes—
284C2×Q8C24112yes—
285C2×Q8C24112yes—
286C2×Q8C2411yescharacteristic
287C2×D4—24no—
288C2×D4C2412yes—
289C2×D4C2416yes—
290C2×D4C2412yes—
291C2×D4C2411yescharacteristic
292C2×D4C2412yes—
293C2×D4C2413yes—
294C2×D4C2416yes—
295C2×D4C2414yes—
296C2×D4C24112yes—
297C2×D4C2416yes—
298C2×D4C24112yes—
299C2×D4C2414yes—
300C2×D4C24112yes—
301C22×C4—412no—
302C22×C4C2412yes—
303C22×C4C2416yes—
304C22×C4C2413yes—
305C22×C4C2416yes—
306C22×C4C2416yes—
307C22×C4C24112yes—
308C22×C4C2413yes—
309C22×C4C2416yes—
310C22×C4C2416yes—
311C22×C4C2416yes—
312C22×C4C24124yes—
313C22×C4C24112yes—
314C22×C4C2411yescharacteristic
315C4⋊C4C2413yes—
316C4⋊C4C2416yes—
317C4⋊C4C2413yes—
318C4⋊C4C24112yes—
319C4⋊C4C24112yes—
320C4⋊C4C2416yes—
321C4⋊C4C2413yes—
322C4⋊C4C2416yes—
323C4⋊C4C24112yes—
324C4⋊C4C2416yes—
325C4⋊C4C24124yes—
326C4⋊C4C24112yes—
327C4⋊C4C2416yes—
328C4⋊C4C2413yes—
329C4⋊C4C2416yes—
330C4⋊C4C2416yes—
331C4⋊C4C2411yescharacteristic
332C4⋊C4C2411yescharacteristic
333C4⋊C4C2416yes—
334C4⋊C4C2416yes—
335C4⋊C4C2416yes—
336C4⋊C4C2416yes—
337C4⋊C4C24112yes—
338C4⋊C4C24112yes—
339C4⋊C4C24112yes—
340C4⋊C4C24112yes—
341C22⋊C4C24112yes—
342C22⋊C4C24112yes—
343C22⋊C4C2414yes—
344C22⋊C4C24112yes—
345C22⋊C4C24112yes—
346C22⋊C4C2416yes—
347C22⋊C4C2412yes—
348C22⋊C4C2416yes—
349C22⋊C4C24124yes—
350C22⋊C4C2416yes—
351C22⋊C4C2416yes—
352C22⋊C4C2416yes—
353C22⋊C4C24112yes—
354C22⋊C4C24112yes—
355C22⋊C4C2414yes—
356C22⋊C4C24112yes—
357C22⋊C4C24112yes—
358C42C2412yes—
359C42C2413yes—
360C42C2413yes—
361C42C2413yes—
362C42C2416yes—
363C42C2416yes—
364C42C2413yes—
365C42C2416yes—
366C42C2416yes—
367C42C2416yes—
368C42C24124yes—
369C23—24no—
370C23—42no—
371C23—42no—
372C23C2514yes—
373C23C2512yes—
374C23C2516yes—
375C23C2511yescharacteristic
376Q8—26no—
377Q8—224no—
378Q8—212no—
379Q8—22no—
380Q8—26no—
381Q8—412no—
382Q8—412no—
383Q8—46no—
384Q8H(5,𝔽2)14yes—
385D4—24no—
386D4—212no—
387D4—212no—
388D4—24no—
389D4—24no—
390D4—22no—
391D4—26no—
392D4—44no—
393D4—46no—
394D4—412no—
395D4—44no—
396D4—412no—
397D4—412no—
398C2×C4—26no—
399C2×C4—224no—
400C2×C4—212no—
401C2×C4—212no—
402C2×C4—212no—
403C2×C4—24no—
404C2×C4—46no—
405C2×C4—412no—
406C2×C4—43no—
407C2×C4—46no—
408C2×C4—46no—
409C2×C4—424no—
410C2×C42−1+414yescomplemented
411C2×C4C2516yes—
412C2×C4C2512yes—
413C2×C4C2516yes—
414C2×C4C2513yes—
415C2×C4C2512yes—
416C2×C4C2513yes—
417C2×C4C2512yes—
418C2×C4C2513yes—
419C2×C4C25112yes—
420C2×C4C2516yes—
421C2×C4C2513yes—
422C2×C4C2511yescharacteristic
423C2×C4C2511yescharacteristic
424C22—21no—
425C22—21no—
426C22—24no—
427C22—22no—
428C22—26no—
429C22—22no—
430C22—26no—
431C22—24no—
432C22—22no—
433C22—26no—
434C22—24no—
435C22—48no—
436C22C2×2−1+412yes—
437C22C2611yescharacteristic
438C4—22no—
439C4—26no—
440C4—212no—
441C4—23no—
442C4—26no—
443C4—22no—
444C4—23no—
445C4—22no—
446C4—23no—
447C4—26no—
448C4C2×H(5,𝔽2)16yes—
449C4C2×2−1+412yes—
450C4C2×2−1+412yes—
451C2—22no—
452C2—42no—
453C2—46no—
454C2—44no—
455C2C22×H(5,𝔽2)11yescharacteristic
456C2C22×2−1+411yescharacteristic
457C22−1+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. #3554C2semidirect
2. #3552C2semidirect
3. #3550C2semidirect
4. #3547C2semidirect
5. #3541C2semidirect
6. #3523C2semidirect
7. #3518C2semidirect
8. #3511C2semidirect
9. #3506C2semidirect
10. #3503C2semidirect
11. #3499C2semidirect
12. #3497C2semidirect
13. #3486C2semidirect
14. #3480C2semidirect
15. Q8×#41C2semidirect
16. C2×D4×Q8C2semidirect
17. #3460C2semidirect
18. C2×2−1+4C22semidirect
20. #563C22semidirect
21. #562C22semidirect
23. #559C22semidirect
24. Q82C22semidirect
25. Q82C22semidirect
26. #557C22semidirect
27. #557C22semidirect
28. #557C22semidirect
30. #556C22semidirect
31. #556C22semidirect
32. #556C22semidirect
34. #555C22semidirect
37. #554C22semidirect
38. #554C22semidirect
39. #554C22semidirect
43. #552C22semidirect
45. #550C22semidirect
52. D4×Q8C22semidirect
53. D4×Q8C22semidirect
54. D4×Q8C22semidirect
55. D4×Q8C22semidirect
58. #548C22semidirect
59. #548C22semidirect
60. #548C22semidirect
62. #547C22semidirect
63. #547C22semidirect
65. #544C22semidirect
68. #542C22semidirect
69. #542C22semidirect
70. #542C22semidirect
71. #541C22semidirect
74. #539C22semidirect
78. #537C22semidirect
79. #536C22semidirect
80. #536C22semidirect
81. #536C22semidirect
85. #533C22semidirect
86. #533C22semidirect
87. #533C22semidirect
91. #532C22semidirect
93. C2×C4⋊Q8C22semidirect
96. #529C22semidirect
101. #519C22semidirect
104. C4×#41C22semidirect
106. C2×C4×Q8C22semidirect
410. C2×C42−1+4semidirect

Maximal subgroups

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

subgroupindex
#35542
#35522
#35502
#35472
#35412
#35232
#35182
#35112
#35062
#35032
#34992
#34972
#34862
#34802
Q8×#412
C2×D4×Q82
#34602