Finite Group Databasebeta

← #27153276

Subgroups — #27153276

Order 12416.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2#20261404C211yesmaximal, characteristic, complemented
3#20261373C211yesmaximal, characteristic, complemented
4#20261121C211yesmaximal, characteristic, complemented
5#20260978C211yesmaximal, characteristic, complemented
6#20260971C211yesmaximal, characteristic, complemented
7#20260959C211yesmaximal, characteristic, complemented
8#20260928C211yesmaximal, characteristic, complemented
9#20260896C211yesmaximal, characteristic, complemented
10#20260880C211yesmaximal, characteristic, complemented
11#20260880C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260836C211yesmaximal, characteristic, complemented
14#20260806C211yesmaximal, characteristic, complemented
15C2×#135×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18#15690837—24no—
19C2×#13095498C2211yescharacteristic
20C2×#13095498C2211yescharacteristic, complemented
21C2×D776C2211yescharacteristic
22C2×#13095496C2211yescharacteristic
23C2×#13095496C2211yescharacteristic
24C2×#13095494C2211yescharacteristic
25#15690806—21no—
26#15690806C2211yescharacteristic, complemented
27#15690799C2211yescharacteristic
28#15690793—21no—
29#15690786C2211yescharacteristic
30#15690784—21no—
31#15690784C2211yescharacteristic
32#15690784C2211yescharacteristic, complemented
33#15690782—21no—
34#15690781—21no—
35#15690781C2211yescharacteristic
36#15690778—21no—
37Q8×Dic97C2211yescharacteristic, complemented
38D4×Dic97C2211yescharacteristic
39C4×C22⋊D97—21no—
40C4×C22⋊D97C2211yescharacteristic
41C2×#13095488—21no—
42#15690744C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic
47#15690732C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic
49#15690731C2211yescharacteristic, complemented
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic
52#15690720C2211yescharacteristic, complemented
53C2×#41×C97C2211yescharacteristic
54C2×#41×C97C2211yescharacteristic, complemented
55#135×C97—24no—
56C2×Q16×C97C2211yescharacteristic, complemented
57C2×SD16×C97C2211yescharacteristic
58C2×SD16×C97C2211yescharacteristic, complemented
59C2×D8×C97C2211yescharacteristic
60C22×C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095512—22no—
63#13095512—41no—
64C2×D388—21no—
65C2×D388C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97D411yescharacteristic
74D4×D97—22no—
75D4×D97—41no—
76C22×Dic97—21no—
77C2×C4×D97—21no—
78C2×C4×D97C2311yescharacteristic
79#13095498—22no—
80#13095498—41no—
81D776—22no—
82#13095496—22no—
83#13095496—41no—
84#13095494—22no—
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C4⋊Dic97—21no—
88C4⋊Dic97C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91#13095490D411yescharacteristic
92C22⋊Dic97—21no—
93C22⋊Dic97—21no—
94C22⋊Dic97D411yescharacteristic
95#13095488—21no—
96#13095488—21no—
97#13095488—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488D411yescharacteristic
101C4×Dic97—21no—
102C4×Dic97C2311yescharacteristic
103#41×C97—22no—
104#41×C97—22no—
105#41×C97—41no—
106#41×C97—41no—
107C2×Q8×C97C2311yescharacteristic
108C2×Q8×C97C2311yescharacteristic
109C2×D4×C97—21no—
110C2×D4×C97—21no—
111C2×D4×C97C2311yescharacteristic
112C2×D4×C97C2311yescharacteristic
113C22×C4×C97—21no—
114C22×C4×C97—21no—
115C22×C4×C97C2311yescharacteristic
116Q16×C97—22no—
117SD16×C97—22no—
118SD16×C97D414yescomplemented
119D8×C97—22no—
120C2×C8×C97—22no—
121C2×C8×C97C2311yescharacteristic
122C2×C8×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97—21no—
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97—81no—
131C22×D97C2×D411yescharacteristic
132Q8×C97—21no—
133Q8×C97—21no—
134Q8×C97—21no—
135Q8×C97C2×D412yes—
136D4×C97—21no—
137D4×C97—21no—
138D4×C97—21no—
139D4×C97—42no—
140D4×C97—42no—
141D4×C97C2×D412yes—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—42no—
147C2×C4×C97—42no—
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2411yescharacteristic
150C22⋊D97—41no—
151C22⋊D97—42no—
152C22⋊D97—42no—
153C22⋊D97—81no—
154C22⋊D97—81no—
155C22⋊D97—81no—
156C22⋊D97—81no—
157C2×Dic97—21no—
158C2×Dic97—21no—
159C2×Dic97—81no—
160C2×Dic97C2×D411yescharacteristic
161C2×Dic97C2×D411yescharacteristic
162C2×Dic97C2×D411yescharacteristic
163D388—21no—
164D388—21no—
165D388—21no—
166D388—21no—
167D388—42no—
168C4×D97—21no—
169C4×D97—21no—
170C4×D97—42no—
171C8×C97—21no—
172C8×C97C2×D412yes—
173#11715895—21no—
174#11715895—21no—
175C22×C97—22no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97—21no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97C22×D411yescharacteristic
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—21no—
188C2×D97—41no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—84no—
195C4×C97—22no—
196C4×C97—21no—
197C4×C97—41no—
198C4×C97C22×D411yescharacteristic
199C4×C97H(5,𝔽2)11yescharacteristic
200Dic97—21no—
201Dic97—21no—
202Dic97—21no—
203Dic97—41no—
204Dic97—41no—
205C2×C97—42no—
206C2×C97—41no—
207C2×C97—81no—
208C2×C97D4211yescharacteristic
209C2×C97#57511yescharacteristic
210C2×C97#57611yescharacteristic
211D97—41no—
212D97—81no—
213D97—81no—
214#3268—971nomaximal, Sylow
215C97#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—971no—
217C2×#135D9711yescharacteristic, complemented
218#550—971no—
219#546—971no—
220#496—971no—
221#468—971no—
222#466—971no—
223#464—971no—
224#459—971no—
225#452—971no—
226#449—971no—
227#449—971no—
228#447—971no—
229#440—971no—
230#435—971no—
231C2×#41—971no—
232C2×#41C2×D9711yescharacteristic
233C2×#41C2×D9711yescharacteristic, complemented
234C22×D4—971no—
235AGL(1,ℤ/8)—1944no—
236#135—24no—
237C2×Q16C2×D9711yescharacteristic, complemented
238C2×SD16—971no—
239C2×SD16—971no—
240C2×SD16C2×D9711yescharacteristic
241C2×SD16C2×D9711yescharacteristic, complemented
242C2×D8—971no—
243C2×D8—971no—
244C2×D8—971no—
245C2×D8C2×D9711yescharacteristic
246C2×M16—971no—
247C22×C8C2×D9711yescharacteristic, complemented
248#127—971no—
249#127—1941no—
250#124—971no—
251#123—1941no—
252C22⋊Q8—971no—
253#121—971no—
254#121—971no—
255#121—971no—
256#121—1941no—
257#121—1941no—
258#121—1941no—
259#120—1941no—
260C4×Q8—971no—
261C4×D4—971no—
262C4×D4—971no—
263C4×D4—1941no—
264C2×C22⋊C4—1941no—
265#107—971no—
266C4⋊C8—971no—
267#103—971no—
268#103—971no—
269#102—971no—
270#102—971no—
271#102—971no—
272#102—971no—
273C22⋊C8—971no—
274C22⋊C8—971no—
275#97—971no—
276C24—1941no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—3881no—
282#41—1942no—
283C2×Q8C22×D9711yescharacteristic
284C2×Q8C22×D9711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—971no—
288C2×D4—971no—
289C2×D4—971no—
290C2×D4—3881no—
291C2×D4—1941no—
292C2×D4—1941no—
293C2×D4—1941no—
294C2×D4—1941no—
295C2×D4—1941no—
296C2×D4—1941no—
297C2×D4—1941no—
298C2×D4—1942no—
299C2×D4C22×D9711yescharacteristic
300C2×D4C22×D9711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—971no—
304C22×C4—1941no—
305C22×C4—1941no—
306C22×C4C22×D9711yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—3881no—
310SD16—1942no—
311SD16C22⋊D9714yescomplemented
312D8—22no—
313D8—3881no—
314D8—1942no—
315D8—1942no—
316M16—1942no—
317C2×C8—22no—
318C2×C8—971no—
319C2×C8—971no—
320C2×C8C22×D9711yescharacteristic
321C2×C8C22×D9711yescharacteristic
322C4⋊C4—971no—
323C4⋊C4—971no—
324C4⋊C4—971no—
325C4⋊C4—1941no—
326C4⋊C4—1941no—
327C22⋊C4—971no—
328C22⋊C4—971no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4—1941no—
335C22⋊C4—1941no—
336C42—971no—
337C42—1941no—
338C23—21no—
339C23—21no—
340C23—7761no—
341C23—971no—
342C23—3882no—
343C23—3882no—
344C23—1941no—
345C23—1941no—
346C23D4×D9711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D9712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—7761no—
357D4—7761no—
358D4—7761no—
359D4—7761no—
360D4—3882no—
361D4—3882no—
362D4—3881no—
363D4—3882no—
364D4—1941no—
365D4—1941no—
366D4—1941no—
367D4—1941no—
368D4C2×C22⋊D9712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—7761no—
376C2×C4—971no—
377C2×C4—971no—
378C2×C4—971no—
379C2×C4—3882no—
380C2×C4—1941no—
381C2×C4—1941no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4D4×D9711yescharacteristic
385C2×C4C23×D9711yescharacteristic
386C8—21no—
387C8—1941no—
388C8—1941no—
389C8C2×C22⋊D9712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—7764no—
400C22—3881no—
401C22—3881no—
402C22—3881no—
403C22—3881no—
404C22—3881no—
405C22—3881no—
406C22—1941no—
407C22—1941no—
408C22—1941no—
409C22C2×D4×D9711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—3881no—
414C4—3881no—
415C4—1941no—
416C4—1941no—
417C4—1941no—
418C4C22×C22⋊D9711yescharacteristic
419C4#1569085511yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—7761no—
424C2—7761no—
425C2—3881no—
426C2D4×C22⋊D9711yescharacteristic
427C2#2026153211yescharacteristic
428C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#135×C97C2
socleC22×C97C22×D4
hypercenterC2×#135D97
hypocenterC97#3268
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. #20261404C2semidirect
3. #20261373C2semidirect
4. #20261121C2semidirect
5. #20260978C2semidirect
6. #20260971C2semidirect
7. #20260959C2semidirect
8. #20260928C2semidirect
9. #20260896C2semidirect
10. #20260880C2semidirect
11. #20260880C2semidirect
12. #20260867C2semidirect
13. #20260836C2semidirect
14. #20260806C2semidirect
15. C2×#135×C97C2semidirect
20. C2×#13095498C22semidirect
26. #15690806C22semidirect
32. #15690784C22semidirect
37. Q8×Dic97C22semidirect
42. #15690744C22semidirect
43. #15690739C22semidirect
44. #15690734C22semidirect
45. #15690734C22semidirect
47. #15690732C22semidirect
49. #15690731C22semidirect
50. C22⋊#11715895C22semidirect
52. #15690720C22semidirect
54. C2×#41×C97C22semidirect
56. C2×Q16×C97C22semidirect
58. C2×SD16×C97C22semidirect
60. C22×C8×C97C22semidirect
118. SD16×C97D4semidirect
215. C97#3268semidirect
217. C2×#135D97semidirect
233. C2×#41C2×D97semidirect
237. C2×Q16C2×D97semidirect
241. C2×SD16C2×D97semidirect
247. C22×C8C2×D97semidirect
311. SD16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
#202614042
#202613732
#202611212
#202609782
#202609712
#202609592
#202609282
#202608962
#202608802
#202608672
#202608362
#202608062
C2×#135×C972
#326897