Finite Group Databasebeta

← #62127

Subgroups — #62127

Order 256.

905 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503C211yesmaximal, characteristic, complemented
2#3481C211yesmaximal, characteristic
3#3385C212yesmaximal, complemented
4#3375C212yesmaximal, complemented
5#3346C214yesmaximal, complemented
6#3338C212yesmaximal, complemented
7#3314C214yesmaximal, complemented
8#3295C212yesmaximal, complemented
9#3254C211yesmaximal, characteristic, complemented
10#3248C212yesmaximal, complemented
11#3239C212yesmaximal, complemented
12#3226C211yesmaximal, characteristic, complemented
13#3110C212yesmaximal, complemented
14#3107C211yesmaximal, characteristic, complemented
15#3079C211yesmaximal, characteristic, complemented
16C2×#474C212yesmaximal, complemented
17#2990C211yesmaximal, characteristic, complemented
18#559C2211yescharacteristic
19Q82C2212yescomplemented
20#557—21no—
21#554C2211yescharacteristic
22#554C2211yescharacteristic
23#554C2212yes—
24#554C2214yescomplemented
25#553—21no—
26#553C2212yes—
27#551—22no—
28D4×Q8—21no—
29D4×Q8—22no—
30#546—21no—
31#546C2212yes—
32D42C2211yescharacteristic
33#544—22no—
34#544—21no—
35#533C2211yescharacteristic, complemented
36#533C2212yescomplemented
37#532C2211yescharacteristic
38C2×C4⋊Q8C2211yescharacteristic, complemented
39C2×C4⋊Q8C2212yescomplemented
40#529—21no—
41#529C2212yes—
42#525—22no—
43#520—21no—
44C4×#41C2211yescharacteristic
45C4×#41C2211yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2212yes—
48#501C2214yescomplemented
49#500C2212yescomplemented
50#489C2214yescomplemented
51#486C2212yes—
52#485C2212yescomplemented
53#484C2214yescomplemented
54#484C2214yescomplemented
55#481C2214yes—
56#480C2214yescomplemented
57#479C2214yescomplemented
58#477C2212yescomplemented
59#477C2212yescomplemented
60#476C2212yes—
61#475C2212yescomplemented
62#475C2211yescharacteristic, complemented
63#474C2212yes—
64#474C2211yescharacteristic
65#474C2214yes—
66#474C2214yescomplemented
67#474C2212yescomplemented
68#474C2212yescomplemented
69#474C2212yescomplemented
70#468C2214yescomplemented
71#467C2214yescomplemented
72#465C2214yes—
73#464C2214yescomplemented
74#462C2211yescharacteristic, complemented
75#461C2211yescharacteristic
76#461C2212yescomplemented
77#460C2212yes—
78#452C2212yes—
79#450C2214yes—
80#446C2212yescomplemented
81#440C2214yes—
82#440C2214yescomplemented
83C4×Q16C2212yescomplemented
84C4×SD16C2212yes—
85#435C2214yescomplemented
86C8×D4C2212yes—
87#432C2212yescomplemented
88#431C2211yescharacteristic, complemented
89#428C2212yescomplemented
90C2×#107C2214yescomplemented
91#423C2211yescharacteristic, complemented
92C2×C4⋊C8C2212yescomplemented
93#417C2212yes—
94C2×#102C2214yes—
95C2×#41—21no—
96C2×#41D411yescharacteristic
97C2×#41C2311yescharacteristic
98C22×Q8—22no—
99C22×Q8—21no—
100C22×D4—21no—
101C22×D4C2312yes—
102C23×C4—22no—
103C2×Q16C2312yes—
104C2×SD16C2312yes—
105C2×SD16C2314yes—
106C2×M16C2312yes—
107C22×C8C2314yes—
108C4⋊Q8—22no—
109C4⋊Q8—24no—
110C4⋊Q8—21no—
111C4⋊Q8—22no—
112C4⋊Q8—21no—
113C4⋊Q8—22no—
114C4⋊Q8C2311yescharacteristic
115C4⋊Q8C2312yes—
116C4⋊Q8C2311yescharacteristic
117C4⋊Q8C2312yes—
118C4⋊Q8C2314yes—
119C4⋊Q8C2311yescharacteristic
120#127C2311yescharacteristic
121#126—22no—
122#126—22no—
123#125—21no—
124#125—22no—
125#125—24no—
126#125—21no—
127#125—22no—
128#125C2312yes—
129#124—21no—
130#124—21no—
131#124C2312yes—
132#123—22no—
133#123—22no—
134#123—22no—
135#123—22no—
136C22⋊Q8—21no—
137C22⋊Q8—22no—
138C22⋊Q8—21no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8C2312yes—
145C22⋊Q8C2314yes—
146C22⋊Q8C2312yes—
147#121—22no—
148#121—21no—
149#121—21no—
150#121—22no—
151#121C2312yes—
152#121C2312yes—
153#120—22no—
154#120—22no—
155C4×Q8—22no—
156C4×Q8—21no—
157C4×Q8—22no—
158C4×Q8D411yescharacteristic
159C4×Q8C2311yescharacteristic
160C4×Q8C2311yescharacteristic
161C4×Q8C2312yes—
162C4×Q8C2314yes—
163C4×Q8C2311yescharacteristic
164C4×D4—21no—
165C4×D4—22no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4D411yescharacteristic
176C4×D4D412yes—
177C4×D4C2311yescharacteristic
178C4×D4C2312yes—
179C4×D4C2312yes—
180C4×D4C2312yes—
181C4×D4C2311yescharacteristic
182C4×D4C2311yescharacteristic
183C4×D4C2312yes—
184#117—22no—
185#117—21no—
186#117—21no—
187#117—22no—
188#117D412yes—
189#117D411yescharacteristic
190#117C2311yescharacteristic
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2312yes—
195C2×C4⋊C4C2314yes—
196C2×C4⋊C4C2312yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—21no—
200C2×C42C2312yes—
201C2×C42C2311yescharacteristic
202#107C2312yes—
203#107C2314yes—
204#107C2314yes—
205#107C2314yes—
206#107C2314yes—
207#106C2314yes—
208#106C2312yes—
209C4⋊C8C2311yescharacteristic
210C4⋊C8C2312yes—
211C4⋊C8C2314yes—
212C4⋊C8C2312yes—
213C4⋊C8C2311yescharacteristic
214#103C2314yes—
215#103C2312yes—
216#103C2312yes—
217#103C2312yes—
218#103C2314yes—
219#102C2312yes—
220#102C2314yes—
221#102C2314yes—
222#102C2314yes—
223#102C2314yes—
224C22⋊C8C2312yes—
225C22⋊C8C2314yes—
226#97C2314yes—
227C4×C8C2312yes—
228C24—22no—
229#41—24no—
230#41—22no—
231#41—41no—
232#41—82no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—84no—
239C2×Q8—82no—
240C2×Q8C2×D411yescharacteristic
241C2×Q8C2411yescharacteristic
242C2×Q8C2412yes—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4C2×D412yes—
253C2×D4C2×D411yescharacteristic
254C2×D4C2412yes—
255C2×D4C2411yescharacteristic
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—84no—
266C22×C4C2×D412yes—
267C22×C4C2×D411yescharacteristic
268C22×C4C2×D412yes—
269C22×C4C2×D411yescharacteristic
270C22×C4C2412yes—
271C22×C4C2411yescharacteristic
272Q16—42no—
273SD16—24no—
274SD16—44no—
275M16—42no—
276C2×C8—44no—
277C2×C8C2412yes—
278C2×C8C2412yes—
279C2×C8C2414yes—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4C2×D411yescharacteristic
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D411yescharacteristic
302C4⋊C4C2×D412yes—
303C4⋊C4C2412yes—
304C4⋊C4C2414yes—
305C4⋊C4C2411yescharacteristic
306C4⋊C4C2411yescharacteristic
307C4⋊C4C2412yes—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—24no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—21no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4C2×D412yes—
322C22⋊C4C2×D412yes—
323C22⋊C4C2×D412yes—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—24no—
328C42—22no—
329C42C2×D411yescharacteristic
330C42C2×D411yescharacteristic
331C42C2×D412yes—
332C42C2×D412yes—
333C42C2411yescharacteristic
334C42C2412yes—
335C42C2411yescharacteristic
336C23—21no—
337C23—22no—
338C23—44no—
339C23—44no—
340C23—82no—
341C23C22×D411yescharacteristic
342C23C22×D412yes—
343Q8—22no—
344Q8—22no—
345Q8—21no—
346Q8—22no—
347Q8—21no—
348Q8—42no—
349Q8—44no—
350Q8—84no—
351Q8—82no—
352Q8—81no—
353D4—24no—
354D4—22no—
355D4—21no—
356D4—22no—
357D4—44no—
358D4—42no—
359D4—44no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4#13714yes—
364D42−1+412yes—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D412yes—
393C2×C4C22×D412yes—
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C2511yescharacteristic
398C8—22no—
399C8—22no—
400C8—24no—
401C22—22no—
402C22—22no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—88no—
413C22C2×#13714yes—
414C22C23×D411yescharacteristic
415C4—21no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—41no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—42no—
426C4—42no—
427C4—44no—
428C4C2×#13712yes—
429C4#57612yes—
430C4C2×2−1+411yescharacteristic
431C4#58511yescharacteristic
432C2—24no—
433C2—41no—
434C2—42no—
435C2—82no—
436C2#348911yescharacteristic
437C2C22×#13711yescharacteristic
438C2C2×#57611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3481C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3503C2semidirect
3. #3385C2semidirect
4. #3375C2semidirect
5. #3346C2semidirect
6. #3338C2semidirect
7. #3314C2semidirect
8. #3295C2semidirect
9. #3254C2semidirect
10. #3248C2semidirect
11. #3239C2semidirect
12. #3226C2semidirect
13. #3110C2semidirect
14. #3107C2semidirect
15. #3079C2semidirect
16. C2×#474C2semidirect
17. #2990C2semidirect
19. Q82C22semidirect
24. #554C22semidirect
35. #533C22semidirect
36. #533C22semidirect
38. C2×C4⋊Q8C22semidirect
39. C2×C4⋊Q8C22semidirect
48. #501C22semidirect
49. #500C22semidirect
50. #489C22semidirect
52. #485C22semidirect
53. #484C22semidirect
54. #484C22semidirect
56. #480C22semidirect
57. #479C22semidirect
58. #477C22semidirect
59. #477C22semidirect
61. #475C22semidirect
62. #475C22semidirect
66. #474C22semidirect
67. #474C22semidirect
68. #474C22semidirect
69. #474C22semidirect
70. #468C22semidirect
71. #467C22semidirect
73. #464C22semidirect
74. #462C22semidirect
76. #461C22semidirect
80. #446C22semidirect
82. #440C22semidirect
83. C4×Q16C22semidirect
85. #435C22semidirect
87. #432C22semidirect
88. #431C22semidirect
89. #428C22semidirect
90. C2×#107C22semidirect
91. #423C22semidirect
92. C2×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35032
#34812
#33852
#33752
#33462
#33382
#33142
#32952
#32542
#32482
#32392
#32262
#31102
#31072
#30792
C2×#4742
#29902