Finite Group Databasebeta

← #27153244

Subgroups — #27153244

Order 12416.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690840C211yesmaximal, characteristic, complemented
2C2×SD16×D97C211yesmaximal, characteristic, complemented
3#20261398C211yesmaximal, characteristic, complemented
4#20261373C211yesmaximal, characteristic
5#20261128C211yesmaximal, characteristic, complemented
6#20260983C211yesmaximal, characteristic, complemented
7#20260975C211yesmaximal, characteristic, complemented
8#20260960C211yesmaximal, characteristic, complemented
9#20260938C211yesmaximal, characteristic, complemented
10#20260889C211yesmaximal, characteristic, complemented
11#20260887C211yesmaximal, characteristic, complemented
12#20260876C211yesmaximal, characteristic, complemented
13#20260832C211yesmaximal, characteristic, complemented
14#20260809C211yesmaximal, characteristic, complemented
15#452×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic, complemented
17C2×Q8×D97C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19#15690840—24no—
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095500C2211yescharacteristic
23C2×#13095500C2211yescharacteristic
24C2×#13095499C2211yescharacteristic
25C2×#13095499C2211yescharacteristic
26C2×#13095498C2211yescharacteristic
27SD16×D97—24no—
28C2×#13095495C2211yescharacteristic, complemented
29C2×C8×D97C2211yescharacteristic
30#15690809—21no—
31#15690808C2211yescharacteristic, complemented
32#15690799C2211yescharacteristic
33#15690793—21no—
34#15690790C2211yescharacteristic
35#15690788—21no—
36#15690787—21no—
37#15690786C2211yescharacteristic
38#15690784C2211yescharacteristic
39#15690782—21no—
40#15690778—21no—
41#15690775C2211yescharacteristic, complemented
42D4×Dic97C2211yescharacteristic
43C4×C22⋊D97C2211yescharacteristic
44#15690770—21no—
45C2×#13095488—21no—
46#15690743C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690735C2211yescharacteristic, complemented
50#15690734C2211yescharacteristic, 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—
62C2×Dic194—22no—
63C2×Dic194—21no—
64C2×Dic194—21no—
65C2×Dic194—41no—
66C2×Dic194C2311yescharacteristic
67C2×Dic194C2311yescharacteristic
68Q8×D97—22no—
69Q8×D97—41no—
70C2×D388C2311yescharacteristic
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97D411yescharacteristic
74D4×D97—22no—
75D4×D97—41no—
76C22×Dic97—21no—
77C22×Dic97—21no—
78C2×C4×D97—21no—
79C2×C4×D97C2311yescharacteristic
80#13095501—22no—
81#13095501—22no—
82#13095500—22no—
83#13095500—41no—
84#13095499—22no—
85#13095499D414yescomplemented
86#13095498—22no—
87#13095495—22no—
88C2×#11715895C2311yescharacteristic
89C2×#11715895C2311yescharacteristic
90C8×D97—22no—
91C4⋊Dic97—21no—
92C4⋊Dic97C2311yescharacteristic
93#13095490—21no—
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#13095488D411yescharacteristic
106C4×Dic97—21no—
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—42no—
127C22×D97—42no—
128C22×D97—81no—
129C22×D97C2×D411yescharacteristic
130Q8×C97—21no—
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134D4×C97—21no—
135D4×C97—42no—
136D4×C97C2×D412yes—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—42no—
143C2×C4×C97C2×D411yescharacteristic
144C2×C4×C97C2411yescharacteristic
145C22⋊D97—22no—
146C22⋊D97—42no—
147C22⋊D97—81no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97—42no—
152C2×Dic97—81no—
153C2×Dic97C2×D411yescharacteristic
154C2×Dic97C2×D411yescharacteristic
155C2×Dic97C2×D411yescharacteristic
156D388—21no—
157D388—21no—
158C4×D97—21no—
159C4×D97—21no—
160C4×D97—42no—
161Dic194—21no—
162Dic194—21no—
163Dic194—21no—
164Dic194—42no—
165Dic194—42no—
166Dic194C2×D412yes—
167C8×C97—21no—
168C8×C97—21no—
169#11715895—21no—
170#11715895C2×D412yes—
171C22×C97—21no—
172C22×C97—22no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97C22×D411yescharacteristic
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—84no—
184C2×D97#13712yescomplemented
185C4×C97—21no—
186C4×C97—41no—
187C4×C97—41no—
188C4×C97—41no—
189C4×C97C22×D411yescharacteristic
190C4×C97H(5,𝔽2)11yescharacteristic
191Dic97—22no—
192Dic97—21no—
193Dic97—21no—
194Dic97—21no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—42no—
198C2×C97—41no—
199C2×C97D4211yescharacteristic
200C2×C97C2×#13711yescharacteristic
201C2×C97#57711yescharacteristic
202D97—22no—
203D97—81no—
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137—971no—
207C22×SD16—971no—
208D4×Q8—971no—
209#546—971no—
210#497—971no—
211#468—971no—
212#467—971no—
213#464—971no—
214#461—971no—
215#452D9711yescharacteristic, complemented
216#451—971no—
217#450—971no—
218#448—971no—
219#440—971no—
220#435—971no—
221C2×#41C2×D9711yescharacteristic
222C22×Q8—971no—
223C22×Q8—971no—
224C22×D4—971no—
225#137—1944no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×Q16C2×D9711yescharacteristic, complemented
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×SD16—1944no—
235C2×SD16C2×D9711yescharacteristic
236C2×M16—971no—
237C22×C8—971no—
238C4⋊Q8—971no—
239C4⋊Q8—1941no—
240#124—971no—
241#123—1941no—
242C22⋊Q8—971no—
243C22⋊Q8—971no—
244C22⋊Q8—1941no—
245C22⋊Q8—1941no—
246C22⋊Q8C2×D9711yescharacteristic, complemented
247#121—971no—
248#121—1941no—
249#120—1941no—
250C4×Q8—971no—
251C4×D4—971no—
252C4×D4—971no—
253C4×D4—1941no—
254C2×C22⋊C4—1941no—
255#107—971no—
256C4⋊C8—971no—
257#103—971no—
258#103—971no—
259#103—971no—
260#103C2×D9711yescharacteristic, complemented
261#102—971no—
262#102C2×D9711yescharacteristic
263C22⋊C8—971no—
264C22⋊C8C2×D9711yescharacteristic, complemented
265#97—971no—
266C24—1941no—
267#41—22no—
268#41—41no—
269C2×Q8—971no—
270C2×Q8—971no—
271C2×Q8—3881no—
272C2×Q8—3881no—
273C2×Q8—1941no—
274C2×Q8—1942no—
275C2×Q8—1942no—
276C2×Q8—1941no—
277C2×Q8C22×D9711yescharacteristic
278C2×Q8C22×D9711yescharacteristic
279C2×D4—21no—
280C2×D4—971no—
281C2×D4—971no—
282C2×D4—3881no—
283C2×D4—1941no—
284C2×D4—1941no—
285C2×D4—1942no—
286C2×D4C22×D9711yescharacteristic
287C22×C4—21no—
288C22×C4—971no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4—1941no—
292C22×C4C22×D9711yescharacteristic
293Q16—41no—
294Q16—1942no—
295Q16—1942no—
296SD16—22no—
297SD16—974no—
298SD16—3881no—
299SD16—1942no—
300SD16—1942no—
301SD16—1942no—
302M16—1942no—
303C2×C8—971no—
304C2×C8—971no—
305C2×C8—1942no—
306C2×C8C22×D9711yescharacteristic
307C2×C8C22×D9711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—971no—
310C4⋊C4—971no—
311C4⋊C4—1941no—
312C4⋊C4—1941no—
313C4⋊C4—1941no—
314C4⋊C4—1941no—
315C4⋊C4—1941no—
316C4⋊C4C22×D9711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—971no—
319C22⋊C4—971no—
320C22⋊C4—1941no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C42—971no—
327C42—1941no—
328C23—21no—
329C23—7761no—
330C23—971no—
331C23—3882no—
332C23—3882no—
333C23—1941no—
334C23D4×D9711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—972no—
340Q8—3882no—
341Q8—3882no—
342Q8—1941no—
343Q8—1941no—
344Q8—1941no—
345D4—21no—
346D4—42no—
347D4—7761no—
348D4—3882no—
349D4—1942no—
350D4—1941no—
351D4—1941no—
352D4D4×D9712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—7761no—
360C2×C4—971no—
361C2×C4—971no—
362C2×C4—971no—
363C2×C4—3882no—
364C2×C4—3882no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4—1941no—
368C2×C4—1941no—
369C2×C4—1941no—
370C2×C4D4×D9711yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—21no—
373C8—21no—
374C8—972no—
375C8—1941no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—7764no—
383C22—972no—
384C22—3881no—
385C22—3881no—
386C22—3881no—
387C22—1941no—
388C22—1941no—
389C22C2×D4×D9711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—3881no—
395C4—3881no—
396C4—1941no—
397C4—1941no—
398C4—1941no—
399C4—1942no—
400C4C2×D4×D9711yescharacteristic
401C4#1569085511yescharacteristic
402C2—41no—
403C2—42no—
404C2—7761no—
405C2—1942no—
406C2#2026136711yescharacteristic
407C2#137×D9711yescharacteristic
408C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#452×C97C2
socleC22×C97C22×D4
prime radical#20261373C2
hypercenter#452D97
hypocenterC97#3266
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×#15690840C2semidirect
2. C2×SD16×D97C2semidirect
3. #20261398C2semidirect
5. #20261128C2semidirect
6. #20260983C2semidirect
7. #20260975C2semidirect
8. #20260960C2semidirect
9. #20260938C2semidirect
10. #20260889C2semidirect
11. #20260887C2semidirect
12. #20260876C2semidirect
13. #20260832C2semidirect
14. #20260809C2semidirect
15. #452×C97C2semidirect
16. C22×Dic194C22semidirect
20. C2×#13095501C22semidirect
21. C2×#13095501C22semidirect
28. C2×#13095495C22semidirect
31. #15690808C22semidirect
41. #15690775C22semidirect
46. #15690743C22semidirect
47. #15690739C22semidirect
48. #15690735C22semidirect
49. #15690735C22semidirect
50. #15690734C22semidirect
53. #15690720C22semidirect
55. C2×Q16×C97C22semidirect
57. C22⋊Q8×C97C22semidirect
58. #103×C97C22semidirect
60. C22⋊C8×C97C22semidirect
85. #13095499D4semidirect
184. C2×D97#137semidirect
205. C97#3266semidirect
215. #452D97semidirect
228. C2×Q16C2×D97semidirect
246. C22⋊Q8C2×D97semidirect
260. #103C2×D97semidirect
264. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
C2×SD16×D972
#202613982
#202613732
#202611282
#202609832
#202609752
#202609602
#202609382
#202608892
#202608872
#202608762
#202608322
#202608092
#452×C972
#326697