Finite Group Databasebeta

← #27153265

Subgroups — #27153265

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C2×#15690830C211yesmaximal, characteristic, complemented
3#20261387C211yesmaximal, characteristic, complemented
4#20261373C211yesmaximal, characteristic, complemented
5#20261128C211yesmaximal, characteristic, complemented
6#20260992C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260953C211yesmaximal, characteristic, complemented
9#20260938C211yesmaximal, characteristic, complemented
10#20260896C211yesmaximal, characteristic, complemented
11#20260887C211yesmaximal, characteristic, complemented
12#20260882C211yesmaximal, characteristic, complemented
13#20260844C211yesmaximal, characteristic, complemented
14#20260803C211yesmaximal, characteristic, complemented
15#452×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095511C2211yescharacteristic, complemented
18C2×D4×D97C2211yescharacteristic
19#15690837—24no—
20#15690830—24no—
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095500C2211yescharacteristic
23C2×#13095499C2211yescharacteristic
24C2×#13095498C2211yescharacteristic
25C2×#13095498C2211yescharacteristic, complemented
26C2×D776C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095494C2211yescharacteristic
29C22×#11715895C2211yescharacteristic, complemented
30#15690808C2211yescharacteristic, complemented
31#15690799C2211yescharacteristic
32#15690795—21no—
33#15690793—21no—
34#15690792—21no—
35#15690790C2211yescharacteristic
36#15690786C2211yescharacteristic
37#15690784C2211yescharacteristic
38#15690783—21no—
39#15690782—21no—
40#15690778—21no—
41D4×Dic97C2211yescharacteristic
42#15690772C2211yescharacteristic, complemented
43C4×C22⋊D97C2211yescharacteristic
44C4⋊C4×D97—21no—
45C2×#13095488—21no—
46#15690741C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690734C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic
52#15690722C2211yescharacteristic
53#15690720C2211yescharacteristic, complemented
54C2×#41×C97C2211yescharacteristic
55C2×Q16×C97C2211yescharacteristic, complemented
56C2×SD16×C97C2211yescharacteristic
57C22⋊Q8×C97C2211yescharacteristic, complemented
58#103×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095512—22no—
63#13095512—41no—
64#13095511—22no—
65#13095511—41no—
66C2×Dic194C2311yescharacteristic
67C2×D388—21no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
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×D97—21no—
79C2×C4×D97—21no—
80C2×C4×D97C2311yescharacteristic
81#13095501—22no—
82#13095500—41no—
83#13095499—22no—
84#13095498—22no—
85#13095498—22no—
86D776—22no—
87#13095496D414yescomplemented
88#13095494—22no—
89C2×#11715895—22no—
90C2×#11715895C2311yescharacteristic
91C2×#11715895C2311yescharacteristic
92C4⋊Dic97—21no—
93C4⋊Dic97C2311yescharacteristic
94#13095490—21no—
95#13095490—21no—
96#13095490—21no—
97#13095490D411yescharacteristic
98C22⋊Dic97—21no—
99C22⋊Dic97D411yescharacteristic
100#13095488—21no—
101#13095488—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488D411yescharacteristic
107C4×Dic97C2311yescharacteristic
108#41×C97—22no—
109#41×C97—41no—
110C2×Q8×C97C2311yescharacteristic
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97—21no—
113C2×D4×C97C2311yescharacteristic
114C22×C4×C97—21no—
115C22×C4×C97C2311yescharacteristic
116Q16×C97—41no—
117SD16×C97—22no—
118C2×C8×C97C2311yescharacteristic
119C2×C8×C97C2311yescharacteristic
120C4⋊C4×C97—21no—
121C4⋊C4×C97C2311yescharacteristic
122C22⋊C4×C97—21no—
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—42no—
128C22×D97—42no—
129C22×D97—81no—
130C22×D97C2×D411yescharacteristic
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134Q8×C97—21no—
135D4×C97—21no—
136D4×C97—42no—
137D4×C97C2×D412yes—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—42no—
144C2×C4×C97C2×D411yescharacteristic
145C2×C4×C97C2411yescharacteristic
146C22⋊D97—41no—
147C22⋊D97—42no—
148C22⋊D97—42no—
149C22⋊D97—81no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—81no—
153C2×Dic97C2×D411yescharacteristic
154C2×Dic97C2×D411yescharacteristic
155C2×Dic97C2×D411yescharacteristic
156D388—21no—
157D388—21no—
158D388—21no—
159D388—42no—
160D388C2×D412yes—
161C4×D97—21no—
162C4×D97—21no—
163C4×D97—42no—
164C4×D97—42no—
165C4×D97—42no—
166Dic194—21no—
167Dic194—21no—
168C8×C97—21no—
169C8×C97—21no—
170#11715895—21no—
171#11715895C2×D412yes—
172C22×C97—21no—
173C22×C97—22no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97C22×D411yescharacteristic
179C2×D97—21no—
180C2×D97—22no—
181C2×D97—21no—
182C2×D97—21no—
183C2×D97—41no—
184C2×D97—41no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—84no—
189C4×C97—21no—
190C4×C97—41no—
191C4×C97—41no—
192C4×C97—41no—
193C4×C97C22×D411yescharacteristic
194C4×C97H(5,𝔽2)11yescharacteristic
195Dic97—21no—
196Dic97—21no—
197Dic97—21no—
198Dic97—41no—
199Dic97—41no—
200C2×C97—42no—
201C2×C97—41no—
202C2×C97D4211yescharacteristic
203C2×C97#57511yescharacteristic
204C2×C97#57711yescharacteristic
205D97—41no—
206D97—42no—
207D97—81no—
208#3267—971nomaximal, Sylow
209C97#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—971no—
211C2×#135—971no—
212#547—971no—
213#546—971no—
214#497—971no—
215#470—971no—
216#465—971no—
217#463—971no—
218#461—971no—
219#452—971no—
220#452D9711yescharacteristic, complemented
221#450—971no—
222#449—971no—
223#442—971no—
224#435—971no—
225C2×#41—971no—
226C2×#41—971no—
227C2×#41C2×D9711yescharacteristic
228C22×D4—971no—
229AGL(1,ℤ/8)—1944no—
230#135—1944no—
231C2×Q16—971no—
232C2×Q16C2×D9711yescharacteristic, complemented
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—971no—
236C2×SD16—971no—
237C2×SD16C2×D9711yescharacteristic
238C2×D8—971no—
239C2×D8—971no—
240C2×M16—971no—
241C22×C8—971no—
242C4⋊Q8—971no—
243#124—971no—
244#123—1941no—
245#123—1941no—
246#123—1941no—
247C22⋊Q8—971no—
248C22⋊Q8—971no—
249C22⋊Q8C2×D9711yescharacteristic, complemented
250#121—971no—
251#121—1941no—
252#121—1941no—
253#120—1941no—
254C4×D4—971no—
255C4×D4—971no—
256C4×D4—971no—
257C2×C4⋊C4—1941no—
258C2×C22⋊C4—1941no—
259#106—971no—
260C4⋊C8—971no—
261#103—971no—
262#103—971no—
263#103C2×D9711yescharacteristic, complemented
264#102—971no—
265#102—971no—
266#102C2×D9711yescharacteristic
267C22⋊C8—971no—
268C22⋊C8C2×D9711yescharacteristic, complemented
269#97—971no—
270C24—1941no—
271#41—22no—
272#41—41no—
273#41—3881no—
274#41—3881no—
275#41—1942no—
276#41—1942no—
277C2×Q8—971no—
278C2×Q8C22×D9711yescharacteristic
279C2×Q8C22×D9711yescharacteristic
280C2×D4—21no—
281C2×D4—971no—
282C2×D4—971no—
283C2×D4—971no—
284C2×D4—3881no—
285C2×D4—1941no—
286C2×D4—1941no—
287C2×D4—1941no—
288C2×D4—1941no—
289C2×D4—1942no—
290C2×D4C22×D9711yescharacteristic
291C22×C4—21no—
292C22×C4—971no—
293C22×C4—1941no—
294C22×C4—1941no—
295C22×C4—1941no—
296C22×C4—1941no—
297C22×C4C22×D9711yescharacteristic
298Q16—41no—
299Q16—1942no—
300SD16—22no—
301SD16—3881no—
302SD16—1942no—
303SD16—1942no—
304SD16—1942no—
305D8—974no—
306D8—1942no—
307M16—1942no—
308C2×C8—971no—
309C2×C8—971no—
310C2×C8—1942no—
311C2×C8C22×D9711yescharacteristic
312C2×C8C22×D9711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—971no—
315C4⋊C4—971no—
316C4⋊C4—1941no—
317C4⋊C4—1941no—
318C4⋊C4—1941no—
319C4⋊C4—1941no—
320C4⋊C4C22×D9711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—971no—
323C22⋊C4—971no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C42—971no—
332C23—21no—
333C23—7761no—
334C23—971no—
335C23—3882no—
336C23—3882no—
337C23—1941no—
338C23—1941no—
339C23D4×D9711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1941no—
345Q8—1941no—
346D4—21no—
347D4—42no—
348D4—7761no—
349D4—972no—
350D4—3882no—
351D4—3882no—
352D4—3882no—
353D4—3881no—
354D4—1941no—
355D4—1941no—
356D4—1941no—
357D4D4×D9712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—7761no—
365C2×C4—971no—
366C2×C4—971no—
367C2×C4—971no—
368C2×C4—3882no—
369C2×C4—3882no—
370C2×C4—3882no—
371C2×C4—1941no—
372C2×C4—1941no—
373C2×C4—1941no—
374C2×C4—1941no—
375C2×C4D4×D9711yescharacteristic
376C2×C4C23×D9711yescharacteristic
377C8—21no—
378C8—21no—
379C8—972no—
380C8—1941no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—7764no—
388C22—3881no—
389C22—3881no—
390C22—3881no—
391C22—3881no—
392C22—3881no—
393C22—1941no—
394C22—1941no—
395C22—1942no—
396C22—1941no—
397C22C2×D4×D9711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—3881no—
403C4—3881no—
404C4—1941no—
405C4—1941no—
406C4—1941no—
407C4C2×D4×D9711yescharacteristic
408C4#1569085511yescharacteristic
409C2—41no—
410C2—42no—
411C2—7761no—
412C2—3881no—
413C2—3882no—
414C2#2026136711yescharacteristic
415C2#2026153211yescharacteristic
416C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#452×C97C2
socleC22×C97C22×D4
hypercenter#452D97
hypocenterC97#3267
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690830C2semidirect
3. #20261387C2semidirect
4. #20261373C2semidirect
5. #20261128C2semidirect
6. #20260992C2semidirect
7. #20260966C2semidirect
8. #20260953C2semidirect
9. #20260938C2semidirect
10. #20260896C2semidirect
11. #20260887C2semidirect
12. #20260882C2semidirect
13. #20260844C2semidirect
14. #20260803C2semidirect
15. #452×C97C2semidirect
17. C2×#13095511C22semidirect
21. C2×#13095501C22semidirect
25. C2×#13095498C22semidirect
29. C22×#11715895C22semidirect
30. #15690808C22semidirect
42. #15690772C22semidirect
46. #15690741C22semidirect
47. #15690739C22semidirect
48. #15690735C22semidirect
49. #15690734C22semidirect
50. #15690732C22semidirect
53. #15690720C22semidirect
55. C2×Q16×C97C22semidirect
57. C22⋊Q8×C97C22semidirect
58. #103×C97C22semidirect
60. C22⋊C8×C97C22semidirect
87. #13095496D4semidirect
209. C97#3267semidirect
220. #452D97semidirect
232. C2×Q16C2×D97semidirect
249. C22⋊Q8C2×D97semidirect
263. #103C2×D97semidirect
268. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908302
#202613872
#202613732
#202611282
#202609922
#202609662
#202609532
#202609382
#202608962
#202608872
#202608822
#202608442
#202608032
#452×C972
#326797