Finite Group Databasebeta

← #60888

Subgroups — #60888

Order 256.

868 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503C211yesmaximal, characteristic, complemented
2#3460C211yesmaximal, characteristic
3#3243C214yesmaximal, complemented
4#3241C214yesmaximal, complemented
5#3239C212yesmaximal, complemented
6#3235C212yesmaximal, complemented
7#3233C212yesmaximal, complemented
8#3226C212yesmaximal, complemented
9#3164C212yesmaximal, complemented
10#3136C212yesmaximal, complemented
11#3098C212yesmaximal, complemented
12#3096C212yesmaximal, complemented
13#3079C211yesmaximal, characteristic, complemented
14#3077C211yesmaximal, characteristic, complemented
15#3076C211yesmaximal, characteristic, complemented
16C2×#474C211yesmaximal, characteristic, complemented
17#2925C211yesmaximal, characteristic, complemented
18Q82—21no—
19#557—21no—
20#554—22no—
21#554—21no—
22D4×Q8—22no—
23D4×Q8—21no—
24#544C2212yes—
25#544C2211yescharacteristic
26#544C2214yescomplemented
27#542—21no—
28#542C2212yes—
29#540—21no—
30#540C2212yes—
31#538—22no—
32#537C2211yescharacteristic
33#535C2211yescharacteristic
34#535C2211yescharacteristic
35#533C2211yescharacteristic
36#533C2212yescomplemented
37#532—21no—
38#532C2212yes—
39C2×C4⋊Q8C2211yescharacteristic
40C2×C4⋊Q8C2212yescomplemented
41#529—21no—
42#529C2212yes—
43#525—22no—
44#520C2212yes—
45C4×#41C2211yescharacteristic
46C4×#41C2211yescharacteristic
47C2×C4×D4C2211yescharacteristic
48#501C2214yescomplemented
49#501C2214yescomplemented
50#500C2214yescomplemented
51#499C2214yescomplemented
52#490C2214yes—
53#489C2214yes—
54#488C2214yes—
55#486C2214yes—
56#485C2214yescomplemented
57#484C2214yescomplemented
58#482C2214yes—
59#482C2214yes—
60#482C2214yescomplemented
61#481C2214yes—
62#480C2214yes—
63#480C2214yescomplemented
64#479C2212yescomplemented
65#478C2214yes—
66#478C2212yescomplemented
67#477C2211yescharacteristic, complemented
68#477C2212yescomplemented
69#476C2212yes—
70#476C2211yescharacteristic, complemented
71#476C2212yescomplemented
72#475C2211yescharacteristic, complemented
73#475C2212yescomplemented
74#474C2212yes—
75#474C2212yes—
76#474C2212yes—
77#474C2214yes—
78#474C2211yescharacteristic, complemented
79#474C2212yescomplemented
80#433C2214yescomplemented
81#432C2212yescomplemented
82#431C2212yescomplemented
83#428C2212yes—
84#428C2212yes—
85#427C2212yes—
86C2×#107C2212yes—
87#423C2211yescharacteristic
88C2×C4⋊C8C2211yescharacteristic
89#418C2212yes—
90#417C2212yes—
91#416C2212yes—
92C2×#102C2212yes—
93#408C2212yes—
94#405C2212yes—
95C2×#41—22no—
96C2×#41C2311yescharacteristic
97C2×#41C2311yescharacteristic
98C22×Q8—21no—
99C22×Q8—22no—
100C22×D4C2311yescharacteristic
101C23×C4—21no—
102C2×M16C2312yes—
103C22×C8C2312yes—
104C4⋊Q8—21no—
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8C2312yes—
112C4⋊Q8C2312yes—
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2314yes—
115C4⋊Q8C2312yes—
116C4⋊Q8C2311yescharacteristic
117#127—21no—
118#127C2312yes—
119#126—22no—
120#126—22no—
121#125—21no—
122#125—22no—
123#125—24no—
124#125C2312yes—
125#125C2312yes—
126#125C2314yes—
127#124—21no—
128#124—22no—
129#124—22no—
130#124C2312yes—
131#123—22no—
132#123—22no—
133#123—22no—
134#123—22no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—21no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8C2312yes—
146#121—22no—
147#121—21no—
148#121—22no—
149#121—21no—
150#121—21no—
151#121—22no—
152#121—22no—
153#121C2312yes—
154#121C2312yes—
155#121C2312yes—
156#120—22no—
157#120—22no—
158C4×Q8—22no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8D411yescharacteristic
164C4×Q8D411yescharacteristic
165C4×Q8C2311yescharacteristic
166C4×Q8C2311yescharacteristic
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—22no—
175C4×D4D412yes—
176C4×D4D411yescharacteristic
177C4×D4D412yes—
178C4×D4D411yescharacteristic
179C4×D4C2312yes—
180C4×D4C2312yes—
181C4×D4C2311yescharacteristic
182C4×D4C2311yescharacteristic
183#117—21no—
184#117—22no—
185#117C2311yescharacteristic
186#117C2312yes—
187#117C2314yes—
188#117C2311yescharacteristic
189#117C2312yes—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C2311yescharacteristic
193C2×C4⋊C4C2312yes—
194C2×C4⋊C4C2314yes—
195C2×C22⋊C4—21no—
196C2×C42—21no—
197C2×C42C2311yescharacteristic
198C2×C42C2312yes—
199#107C2314yes—
200#107C2314yes—
201#107C2312yes—
202#107C2312yes—
203#107C2314yes—
204#107C2314yes—
205#106C2312yes—
206#106C2312yes—
207#106C2314yes—
208#106C2314yes—
209C4⋊C8C2311yescharacteristic
210C4⋊C8C2311yescharacteristic
211C4⋊C8C2312yes—
212C4⋊C8C2312yes—
213C4⋊C8C2312yes—
214#103C2312yes—
215#103C2312yes—
216#103C2314yes—
217#102C2314yes—
218#102C2314yes—
219#102C2312yes—
220#102C2312yes—
221#102C2314yes—
222#102C2314yes—
223#102C2314yes—
224C22⋊C8C2314yes—
225C22⋊C8C2314yes—
226#97C2314yes—
227C4×C8C2314yes—
228C24—21no—
229#41—22no—
230#41—22no—
231#41—22no—
232#41—22no—
233#41—84no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8C2×D411yescharacteristic
243C2×Q8C2411yescharacteristic
244C2×D4—21no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4C2×D412yes—
254C2×D4C2×D411yescharacteristic
255C2×D4C2412yes—
256C2×D4C2411yescharacteristic
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—82no—
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D412yes—
268C22×C4C2×D411yescharacteristic
269C22×C4C2×D412yes—
270C22×C4C2412yes—
271C22×C4C2411yescharacteristic
272M16—24no—
273C2×C8—24no—
274C2×C8C2412yes—
275C2×C8C2414yes—
276C2×C8C2412yes—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4C2×D411yescharacteristic
296C4⋊C4C2×D411yescharacteristic
297C4⋊C4C2×D412yes—
298C4⋊C4C2×D412yes—
299C4⋊C4C2412yes—
300C4⋊C4C2414yes—
301C4⋊C4C2412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2412yes—
304C4⋊C4C2411yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4C2×D412yes—
319C22⋊C4C2×D412yes—
320C22⋊C4C2×D412yes—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—24no—
325C42—22no—
326C42C2×D411yescharacteristic
327C42C2×D412yes—
328C42C2×D411yescharacteristic
329C42C2×D412yes—
330C42C2412yes—
331C42C2411yescharacteristic
332C42C2411yescharacteristic
333C23—22no—
334C23—21no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23C22×D412yes—
339C23C22×D411yescharacteristic
340Q8—22no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—42no—
345Q8—44no—
346Q8—82no—
347Q8—84no—
348Q8—82no—
349D4—24no—
350D4—22no—
351D4—22no—
352D4—22no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—42no—
357D4—82no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4C22×D412yes—
384C2×C4C22×D411yescharacteristic
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D412yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D412yes—
389C2×C4C22×D411yescharacteristic
390C2×C4C22×D411yescharacteristic
391C2×C42−1+412yes—
392C2×C42−1+412yes—
393C2×C4C2511yescharacteristic
394C8—22no—
395C8—22no—
396C8—24no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—42no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—84no—
409C22#53612yes—
410C22C23×D411yescharacteristic
411C4—21no—
412C4—21no—
413C4—22no—
414C4—22no—
415C4—21no—
416C4—22no—
417C4—41no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—41no—
422C4—44no—
423C4—42no—
424C4#53612yes—
425C4#57612yes—
426C4C2×2−1+411yescharacteristic
427C4C2×2−1+411yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—82no—
432C2C2×#53611yescharacteristic
433C2C2×#57611yescharacteristic
434C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3460C2
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. #3243C2semidirect
4. #3241C2semidirect
5. #3239C2semidirect
6. #3235C2semidirect
7. #3233C2semidirect
8. #3226C2semidirect
9. #3164C2semidirect
10. #3136C2semidirect
11. #3098C2semidirect
12. #3096C2semidirect
13. #3079C2semidirect
14. #3077C2semidirect
15. #3076C2semidirect
16. C2×#474C2semidirect
17. #2925C2semidirect
26. #544C22semidirect
36. #533C22semidirect
40. C2×C4⋊Q8C22semidirect
48. #501C22semidirect
49. #501C22semidirect
50. #500C22semidirect
51. #499C22semidirect
56. #485C22semidirect
57. #484C22semidirect
60. #482C22semidirect
63. #480C22semidirect
64. #479C22semidirect
66. #478C22semidirect
67. #477C22semidirect
68. #477C22semidirect
70. #476C22semidirect
71. #476C22semidirect
72. #475C22semidirect
73. #475C22semidirect
78. #474C22semidirect
79. #474C22semidirect
80. #433C22semidirect
81. #432C22semidirect
82. #431C22semidirect

Maximal subgroups

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

subgroupindex
#35032
#34602
#32432
#32412
#32392
#32352
#32332
#32262
#31642
#31362
#30982
#30962
#30792
#30772
#30762
C2×#4742
#29252