Finite Group Databasebeta

← #60873

Subgroups — #60873

Order 256.

871 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3494C211yesmaximal, characteristic
2C2×#544C211yesmaximal, characteristic, complemented
3#3244C214yesmaximal, complemented
4#3241C214yesmaximal, complemented
5#3239C212yesmaximal, complemented
6#3234C212yesmaximal, complemented
7#3232C212yesmaximal, complemented
8#3227C212yesmaximal, complemented
9#3163C212yesmaximal, complemented
10#3136C212yesmaximal, complemented
11#3094C211yesmaximal, characteristic, complemented
12#3092C211yesmaximal, characteristic, complemented
13#3091C211yesmaximal, characteristic, complemented
14C2×#481C211yesmaximal, characteristic, complemented
15#3084C212yesmaximal, complemented
16#3081C212yesmaximal, complemented
17#2924C211yesmaximal, characteristic, complemented
18#555—21no—
19#553—21no—
20#553—21no—
21#551—22no—
22#547—21no—
23#546—22no—
24#544—22no—
25#544—21no—
26#544—21no—
27#544C2212yes—
28#544C2212yes—
29#544C2211yescharacteristic
30#544C2214yescomplemented
31#542C2212yes—
32#540C2212yes—
33#535C2211yescharacteristic
34#532C2212yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8C2212yescomplemented
37#529C2212yes—
38C2×#125—21no—
39C2×#125C2212yescomplemented
40#525C2211yescharacteristic
41C2×C22⋊Q8—22no—
42C2×#121C2211yescharacteristic
43#520C2211yescharacteristic
44#518C2211yescharacteristic
45C2×#117C2211yescharacteristic
46C2×#117C2211yescharacteristic
47C22×C4⋊C4C2211yescharacteristic
48#501C2214yescomplemented
49#501C2214yescomplemented
50#499C2214yescomplemented
51#498C2214yescomplemented
52#490C2214yescomplemented
53#489C2214yescomplemented
54#488C2214yescomplemented
55#486C2214yescomplemented
56#485C2212yescomplemented
57#484C2211yescharacteristic, complemented
58#484C2212yescomplemented
59#483C2211yescharacteristic, complemented
60#483C2212yescomplemented
61#482C2214yes—
62#482C2212yescomplemented
63#481C2212yes—
64#481C2212yes—
65#481C2214yes—
66#481C2212yes—
67#481C2211yescharacteristic, complemented
68#481C2212yescomplemented
69#480C2212yes—
70#480C2211yescharacteristic, complemented
71#480C2212yescomplemented
72#479C2214yescomplemented
73#478C2214yescomplemented
74#478C2214yescomplemented
75#478C2214yescomplemented
76#476C2214yescomplemented
77#476C2214yescomplemented
78#475C2214yescomplemented
79#474C2214yescomplemented
80#433C2214yescomplemented
81#432C2212yescomplemented
82#431C2212yescomplemented
83#428C2212yes—
84#428C2212yes—
85#427C2212yes—
86C2×#106C2212yes—
87#424C2212yes—
88#418C2212yes—
89#417C2212yes—
90#416C2212yes—
91C2×#102C2212yes—
92#407C2211yescharacteristic
93C2×C22⋊C8C2211yescharacteristic
94#405C2212yes—
95C2×#41—22no—
96C2×#41C2311yescharacteristic
97C22×Q8—22no—
98C22×D4—21no—
99C22×D4C2311yescharacteristic
100C23×C4—21no—
101C23×C4C2311yescharacteristic
102C2×M16C2312yes—
103C22×C8C2312yes—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—21no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8C2314yes—
111C4⋊Q8C2314yes—
112C4⋊Q8C2312yes—
113#127C2312yes—
114#126—22no—
115#126—22no—
116#125—22no—
117#125—22no—
118#125—24no—
119#125—21no—
120#125—22no—
121#125—22no—
122#125C2314yes—
123#125C2314yes—
124#125C2312yes—
125#124—22no—
126#124—22no—
127#124C2312yes—
128#123—21no—
129#123—22no—
130#123—22no—
131#123—22no—
132#123—22no—
133C22⋊Q8—22no—
134C22⋊Q8—24no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8—24no—
138C22⋊Q8—22no—
139C22⋊Q8C2311yescharacteristic
140C22⋊Q8C2311yescharacteristic
141#121—21no—
142#121—21no—
143#121—22no—
144#121—21no—
145#121—21no—
146#121—22no—
147#121—22no—
148#121C2312yes—
149#121C2312yes—
150#121C2311yescharacteristic
151#121C2311yescharacteristic
152#120—21no—
153#120—22no—
154C4×Q8C2312yes—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—22no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4C2312yes—
163C4×D4C2312yes—
164C4×D4C2312yes—
165#117—21no—
166#117—21no—
167#117—21no—
168#117—22no—
169#117D412yes—
170#117D411yescharacteristic
171#117D411yescharacteristic
172#117C2312yes—
173#117C2312yes—
174#117C2311yescharacteristic
175#117C2311yescharacteristic
176#117C2311yescharacteristic
177#117C2312yes—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4D411yescharacteristic
186C2×C4⋊C4D411yescharacteristic
187C2×C4⋊C4C2312yes—
188C2×C4⋊C4C2312yes—
189C2×C4⋊C4C2312yes—
190C2×C4⋊C4C2311yescharacteristic
191C2×C4⋊C4C2311yescharacteristic
192C2×C4⋊C4C2311yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4D412yes—
197C2×C42—21no—
198C2×C42C2312yes—
199#107C2312yes—
200#107C2312yes—
201#107C2314yes—
202#107C2314yes—
203#106C2314yes—
204#106C2314yes—
205#106C2312yes—
206#106C2312yes—
207#106C2314yes—
208#106C2314yes—
209C4⋊C8C2314yes—
210C4⋊C8C2314yes—
211#103C2312yes—
212#103C2312yes—
213#103C2314yes—
214#102C2314yes—
215#102C2314yes—
216#102C2312yes—
217#102C2312yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2311yescharacteristic
223C22⋊C8C2312yes—
224C22⋊C8C2312yes—
225C22⋊C8C2312yes—
226#97C2314yes—
227C4×C8C2314yes—
228C24—21no—
229C24C2×D411yescharacteristic
230#41—24no—
231#41—84no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—84no—
236C2×Q8C2411yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—82no—
247C2×D4C2412yes—
248C2×D4C2411yescharacteristic
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—82no—
261C22×C4C2×D411yescharacteristic
262C22×C4C2×D412yes—
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D412yes—
266C22×C4C2411yescharacteristic
267C22×C4C2411yescharacteristic
268C22×C4C2411yescharacteristic
269M16—24no—
270C2×C8—24no—
271C2×C8C2412yes—
272C2×C8C2414yes—
273C2×C8C2412yes—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—21no—
292C4⋊C4C2×D411yescharacteristic
293C4⋊C4C2×D411yescharacteristic
294C4⋊C4C2×D411yescharacteristic
295C4⋊C4C2×D411yescharacteristic
296C4⋊C4C2×D412yes—
297C4⋊C4C2×D412yes—
298C4⋊C4C2414yes—
299C4⋊C4C2412yes—
300C4⋊C4C2412yes—
301C4⋊C4C2412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2411yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—21no—
316C22⋊C4—22no—
317C22⋊C4—21no—
318C22⋊C4C2×D412yes—
319C22⋊C4C2×D412yes—
320C22⋊C4C2×D412yes—
321C22⋊C4C2×D412yes—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—21no—
326C42C2×D412yes—
327C42C2×D412yes—
328C42C2412yes—
329C42C2412yes—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23C22×D411yescharacteristic
339C23C22×D411yescharacteristic
340C23C22×D411yescharacteristic
341Q8—22no—
342Q8—84no—
343Q8—84no—
344Q8—82no—
345D4—24no—
346D4—22no—
347D4—42no—
348D4—44no—
349D4—42no—
350D4—42no—
351D4—82no—
352D4—82no—
353D4—81no—
354D4—82no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4C22×D411yescharacteristic
380C2×C4C22×D412yes—
381C2×C4C22×D411yescharacteristic
382C2×C4C22×D412yes—
383C2×C4C22×D412yes—
384C2×C4C22×D412yes—
385C2×C4C22×D411yescharacteristic
386C2×C42−1+414yes—
387C2×C4C2511yescharacteristic
388C8—22no—
389C8—22no—
390C8—24no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—44no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—42no—
403C22—41no—
404C22—42no—
405C22—162no—
406C22#53612yes—
407C22#57712yes—
408C22C23×D411yescharacteristic
409C4—22no—
410C4—22no—
411C4—21no—
412C4—21no—
413C4—22no—
414C4—22no—
415C4—44no—
416C4—42no—
417C4—41no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4#53612yes—
423C4C2×2−1+412yes—
424C2—22no—
425C2—22no—
426C2—41no—
427C2—81no—
428C2—82no—
429C2C2×#53611yescharacteristic
430C2C2×#57711yescharacteristic
431C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3494C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#544C2semidirect
3. #3244C2semidirect
4. #3241C2semidirect
5. #3239C2semidirect
6. #3234C2semidirect
7. #3232C2semidirect
8. #3227C2semidirect
9. #3163C2semidirect
10. #3136C2semidirect
11. #3094C2semidirect
12. #3092C2semidirect
13. #3091C2semidirect
14. C2×#481C2semidirect
15. #3084C2semidirect
16. #3081C2semidirect
17. #2924C2semidirect
30. #544C22semidirect
36. C2×C4⋊Q8C22semidirect
39. C2×#125C22semidirect
48. #501C22semidirect
49. #501C22semidirect
50. #499C22semidirect
51. #498C22semidirect
52. #490C22semidirect
53. #489C22semidirect
54. #488C22semidirect
55. #486C22semidirect
56. #485C22semidirect
57. #484C22semidirect
58. #484C22semidirect
59. #483C22semidirect
60. #483C22semidirect
62. #482C22semidirect
67. #481C22semidirect
68. #481C22semidirect
70. #480C22semidirect
71. #480C22semidirect
72. #479C22semidirect
73. #478C22semidirect
74. #478C22semidirect
75. #478C22semidirect
76. #476C22semidirect
77. #476C22semidirect
78. #475C22semidirect
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
#34942
C2×#5442
#32442
#32412
#32392
#32342
#32322
#32272
#31632
#31362
#30942
#30922
#30912
C2×#4812
#30842
#30812
#29242