Finite Group Databasebeta

← #27153587

Subgroups — #27153587

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261429C211yesmaximal, characteristic, complemented
2#20261373C211yesmaximal, characteristic, complemented
3#20261091C211yesmaximal, characteristic, complemented
4#20261069C211yesmaximal, characteristic, complemented
5#20261059C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20260971C211yesmaximal, characteristic, complemented
8#20260967C211yesmaximal, characteristic, complemented
9#20260953C211yesmaximal, characteristic, complemented
10#20260880C211yesmaximal, characteristic, complemented
11#20260870C211yesmaximal, characteristic, complemented
12#20260815C211yesmaximal, characteristic, complemented
13C8×C22⋊D97C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#416×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095498C2211yescharacteristic, complemented
18C2×D776C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690804—21no—
22#15690801C2211yescharacteristic, complemented
23#15690799C2211yescharacteristic
24#15690793—21no—
25#15690792—21no—
26#15690786C2211yescharacteristic
27#15690785C2211yescharacteristic
28#15690784C2211yescharacteristic
29#15690784C2211yescharacteristic, complemented
30#15690783—21no—
31#15690782—21no—
32#15690778—21no—
33C4×D388—21no—
34D4×Dic97C2211yescharacteristic
35#15690772C2211yescharacteristic, complemented
36C4×C22⋊D97C2211yescharacteristic
37C4⋊C4×D97C2211yescharacteristic
38C2×#13095488—21no—
39#15690744C2211yescharacteristic, complemented
40#15690743C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690734C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic
45#15690732C2211yescharacteristic, complemented
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic, complemented
52C2×#41×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic, complemented
54#117×C97C2211yescharacteristic, complemented
55#103×C97C2211yescharacteristic, complemented
56#103×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60C2×D388—21no—
61C2×D388C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66C2×C22⋊D97D411yescharacteristic
67D4×D97—22no—
68D4×D97—41no—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97C2311yescharacteristic
73#13095498—41no—
74D776—41no—
75#13095496—22no—
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C8×D97—41no—
79C4⋊Dic97—21no—
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490D411yescharacteristic
86C22⋊Dic97—21no—
87C22⋊Dic97D411yescharacteristic
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488D411yescharacteristic
96C4×Dic97C2311yescharacteristic
97#41×C97—22no—
98#41×C97—41no—
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97—21no—
101C2×D4×C97C2311yescharacteristic
102C22×C4×C97—21no—
103C22×C4×C97C2311yescharacteristic
104C2×C8×C97—41no—
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C22⋊C4×C97—21no—
110C42×C97—21no—
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—81no—
118C22×D97C2×D411yescharacteristic
119Q8×C97—21no—
120Q8×C97—21no—
121D4×C97—21no—
122D4×C97—42no—
123D4×C97#4112yes—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—42no—
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—22no—
133C22⋊D97—42no—
134C22⋊D97—81no—
135C22⋊D97—81no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—81no—
139C2×Dic97C2×D411yescharacteristic
140C2×Dic97C2×D411yescharacteristic
141C2×Dic97C2×D411yescharacteristic
142D388—21no—
143D388—21no—
144D388—21no—
145D388—21no—
146D388—42no—
147C4×D97—21no—
148C4×D97—21no—
149C4×D97—81no—
150C4×D97—81no—
151C8×C97—21no—
152C8×C97—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—22no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—21no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—84no—
171C2×D97#13512yes—
172C4×C97—21no—
173C4×C97—41no—
174C4×C97—41no—
175C4×C97—41no—
176C4×C97C2×#4111yescharacteristic
177C4×C97H(5,𝔽2)11yescharacteristic
178Dic97—21no—
179Dic97—21no—
180Dic97—21no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—42no—
184C2×C97—41no—
185C2×C97#54611yescharacteristic
186C2×C97C2×#13511yescharacteristic
187C2×C97#57611yescharacteristic
188D97—22no—
189D97—81no—
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546—971no—
195#488—971no—
196#484—971no—
197#482—971no—
198#478—971no—
199#466—971no—
200#465—971no—
201#463—971no—
202#449—971no—
203#447—971no—
204C4×D8—971no—
205C8×D4—971no—
206#416D9711yescharacteristic, complemented
207C2×#102—971no—
208C2×#41C2×D9711yescharacteristic
209C22×D4—971no—
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8—971no—
213C22×C8—971no—
214C22×C8C2×D9711yescharacteristic, complemented
215#126—1941no—
216#125—971no—
217#124—971no—
218#123—1941no—
219#123—1941no—
220C22⋊Q8—971no—
221#121—971no—
222#121—971no—
223#121—971no—
224#121—1941no—
225#121—1941no—
226#120—1941no—
227C4×D4—971no—
228C4×D4—971no—
229C4×D4—971no—
230C4×D4—1941no—
231#117C2×D9711yescharacteristic, complemented
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234#107—971no—
235#107—971no—
236#106—971no—
237C4⋊C8—971no—
238#103—971no—
239#103C2×D9711yescharacteristic, complemented
240#103C2×D9711yescharacteristic, complemented
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102C2×D9711yescharacteristic
247#102C2×D9711yescharacteristic
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8—971no—
251C24—1941no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—21no—
256C2×D4—971no—
257C2×D4—971no—
258C2×D4—971no—
259C2×D4—3881no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4—1942no—
265C2×D4C22×D9711yescharacteristic
266C22×C4—21no—
267C22×C4—971no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4C22×D9711yescharacteristic
272SD16—3881no—
273D8—3881no—
274D8—1942no—
275C2×C8—41no—
276C2×C8—971no—
277C2×C8—971no—
278C2×C8—3881no—
279C2×C8C22×D9711yescharacteristic
280C2×C8C22×D9711yescharacteristic
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—971no—
292C22⋊C4—971no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C42—21no—
302C42—971no—
303C23—21no—
304C23—7761no—
305C23—971no—
306C23—3882no—
307C23—3882no—
308C23—1941no—
309C23—1941no—
310C23D4×D9711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—7761no—
316D4—7761no—
317D4—3882no—
318D4—3882no—
319D4—1942no—
320D4—1941no—
321D4—1941no—
322D4—1941no—
323D4—1941no—
324D4#1309551112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—7761no—
332C2×C4—7761no—
333C2×C4—7761no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—971no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1941no—
341C2×C4D4×D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—21no—
344C8—21no—
345C8—1941no—
346C8—1941no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—7764no—
354C22—972no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—3881no—
360C22—3881no—
361C22—1941no—
362C22—1941no—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3881no—
369C4—3881no—
370C4—1941no—
371C4—1941no—
372C4—1941no—
373C4C2×#1309551111yescharacteristic
374C4#1569085511yescharacteristic
375C2—41no—
376C2—42no—
377C2—7761no—
378C2—7761no—
379C2—1942no—
380C2#2026137511yescharacteristic
381C2#135×D9711yescharacteristic
382C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#416×C97C2
socleC22×C97C22×D4
hypercenter#416D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20261429C2semidirect
2. #20261373C2semidirect
3. #20261091C2semidirect
4. #20261069C2semidirect
5. #20261059C2semidirect
6. #20261033C2semidirect
7. #20260971C2semidirect
8. #20260967C2semidirect
9. #20260953C2semidirect
10. #20260880C2semidirect
11. #20260870C2semidirect
12. #20260815C2semidirect
13. C8×C22⋊D97C2semidirect
14. #102×D97C2semidirect
15. #416×C97C2semidirect
17. C2×#13095498C22semidirect
22. #15690801C22semidirect
29. #15690784C22semidirect
35. #15690772C22semidirect
39. #15690744C22semidirect
40. #15690743C22semidirect
41. #15690741C22semidirect
42. #15690739C22semidirect
43. #15690734C22semidirect
45. #15690732C22semidirect
49. C22⋊#11715895C22semidirect
51. C8×Dic97C22semidirect
53. C22×C8×C97C22semidirect
54. #117×C97C22semidirect
55. #103×C97C22semidirect
56. #103×C97C22semidirect
192. C97#3297semidirect
206. #416D97semidirect
214. C22×C8C2×D97semidirect
231. #117C2×D97semidirect
239. #103C2×D97semidirect
240. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614292
#202613732
#202610912
#202610692
#202610592
#202610332
#202609712
#202609672
#202609532
#202608802
#202608702
#202608152
C8×C22⋊D972
#102×D972
#416×C972
#329797