Finite Group Databasebeta

← #27153591

Subgroups — #27153591

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#20261435C211yesmaximal, characteristic, complemented
2#20261380C211yesmaximal, characteristic, complemented
3#20261090C211yesmaximal, characteristic, complemented
4#20261072C211yesmaximal, characteristic, complemented
5#20261062C211yesmaximal, characteristic, complemented
6#20261035C211yesmaximal, characteristic, complemented
7#20260970C211yesmaximal, characteristic, complemented
8#20260963C211yesmaximal, characteristic, complemented
9#20260950C211yesmaximal, characteristic, complemented
10#20260877C211yesmaximal, characteristic, complemented
11#20260867C211yesmaximal, characteristic, complemented
12C8×D388C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15C4×D8×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095499C2211yescharacteristic, complemented
19C2×#13095496C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic, complemented
22#15690803—21no—
23#15690802C2211yescharacteristic, complemented
24#15690798C2211yescharacteristic
25#15690794—21no—
26#15690793—21no—
27#15690790C2211yescharacteristic
28#15690785C2211yescharacteristic
29#15690784—21no—
30#15690783—21no—
31#15690781C2211yescharacteristic
32#15690781C2211yescharacteristic, complemented
33#15690780—21no—
34C4×D388C2211yescharacteristic
35C4×C22⋊D97—21no—
36#15690768C2211yescharacteristic, complemented
37C4⋊C4×D97C2211yescharacteristic
38C2×#13095488—21no—
39#15690743C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41C4⋊#11715895C2211yescharacteristic, complemented
42#15690735C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690722C2211yescharacteristic, complemented
52C2×D8×C97C2211yescharacteristic
53C4×D4×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic, complemented
55#107×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095510—22no—
61#13095510—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—21no—
64C2×D388D411yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70D4×D97—22no—
71D4×D97—41no—
72C22×Dic97—21no—
73C2×C4×D97—21no—
74C2×C4×D97C2311yescharacteristic
75C2×C4×D97C2311yescharacteristic
76#13095499—41no—
77#13095496—41no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97—41no—
81C8×D97—41no—
82C4⋊Dic97—21no—
83C4⋊Dic97D411yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2311yescharacteristic
86#13095490—21no—
87#13095490—21no—
88#13095490—21no—
89C22⋊Dic97—21no—
90C22⋊Dic97—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488D411yescharacteristic
97#13095488D411yescharacteristic
98C4×Dic97—21no—
99C2×D4×C97C2311yescharacteristic
100C2×D4×C97C2311yescharacteristic
101C22×C4×C97—21no—
102C22×C4×C97—21no—
103D8×C97—22no—
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C4⋊C4×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C22⋊C4×C97—21no—
109C22⋊C4×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C22×D97—21no—
114C22×D97—42no—
115C22×D97—42no—
116C22×D97—81no—
117C22×D97C2×D411yescharacteristic
118C22×D97C2×D411yescharacteristic
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—21no—
122D4×C97#4112yes—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—81no—
126C2×C4×C97—81no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—42no—
131C22⋊D97—42no—
132C22⋊D97—42no—
133C22⋊D97—81no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—42no—
138C2×Dic97C2×D411yescharacteristic
139C2×Dic97C2×D411yescharacteristic
140D388—22no—
141D388—21no—
142D388—21no—
143D388—81no—
144C4×D97—21no—
145C4×D97—21no—
146C4×D97—21no—
147C4×D97—21no—
148C4×D97—81no—
149Dic194—21no—
150Dic194—21no—
151C8×C97—21no—
152C8×C97—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—22no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—21no—
165C2×D97—21no—
166C2×D97—21no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—84no—
171C2×D97#13512yes—
172C4×C97—21no—
173C4×C97—21no—
174C4×C97—41no—
175C4×C97—41no—
176C4×C97C2×#4111yescharacteristic
177C4×C97H(5,𝔽2)11yescharacteristic
178Dic97—21no—
179Dic97—21no—
180Dic97—41no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—42no—
184C2×C97—81no—
185C2×C97#54611yescharacteristic
186C2×C97C2×#13511yescharacteristic
187C2×C97#57611yescharacteristic
188D97—22no—
189D97—41no—
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×D8D9711yescharacteristic, complemented
205C8×D4—971no—
206#416—971no—
207C2×#102—971no—
208C2×#41—971no—
209C22×D4—971no—
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8C2×D9711yescharacteristic
213C22×C8—971no—
214C22×C8—971no—
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—1941no—
229C4×D4C2×D9711yescharacteristic
230C4×D4C2×D9711yescharacteristic, complemented
231#117—971no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234#107—971no—
235#107C2×D9711yescharacteristic, complemented
236#106—971no—
237C4⋊C8—971no—
238#103—971no—
239#103—971no—
240#103—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102C2×D9711yescharacteristic
247#102C2×D9711yescharacteristic, complemented
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8C2×D9711yescharacteristic, complemented
251C24—1941no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—971no—
256C2×D4—971no—
257C2×D4—3881no—
258C2×D4—1941no—
259C2×D4—1941no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1942no—
264C2×D4C22×D9711yescharacteristic
265C2×D4C22×D9711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—1941no—
271C22×C4—1941no—
272SD16—3881no—
273D8—22no—
274D8—3881no—
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—3881no—
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—21no—
292C22⋊C4—971no—
293C22⋊C4—971no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C42—1941no—
302C42C22×D9711yescharacteristic
303C23—21no—
304C23—21no—
305C23—7761no—
306C23—971no—
307C23—971no—
308C23—3882no—
309C23—3882no—
310C23—1941no—
311Q8—1941no—
312Q8—1941no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—7761no—
317D4—7761no—
318D4—3882no—
319D4—3882no—
320D4—3882no—
321D4—1942no—
322D4—1941no—
323D4—1941no—
324D4#1309551112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—7761no—
330C2×C4—971no—
331C2×C4—971no—
332C2×C4—3882no—
333C2×C4—1941no—
334C2×C4—1941no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4D4×D9711yescharacteristic
341C2×C4D4×D9711yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—21no—
344C8—21no—
345C8—1941no—
346C8—1941no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—7764no—
354C22—972no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—1941no—
359C22—1941no—
360C22—1941no—
361C22—1941no—
362C22—1941no—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3881no—
369C4—3881no—
370C4—3881no—
371C4—1941no—
372C4—1941no—
373C4C2×#1309551111yescharacteristic
374C4#1569085611yescharacteristic
375C2—42no—
376C2—81no—
377C2—7761no—
378C2—3881no—
379C2—1942no—
380C2#2026138111yescharacteristic
381C2#135×D9711yescharacteristic
382C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
hypercenterC4×D8D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20261435C2semidirect
2. #20261380C2semidirect
3. #20261090C2semidirect
4. #20261072C2semidirect
5. #20261062C2semidirect
6. #20261035C2semidirect
7. #20260970C2semidirect
8. #20260963C2semidirect
9. #20260950C2semidirect
10. #20260877C2semidirect
11. #20260867C2semidirect
12. C8×D388C2semidirect
13. #20260728C2semidirect
14. #102×D97C2semidirect
15. C4×D8×C97C2semidirect
18. C2×#13095499C22semidirect
21. C2×C8×D97C22semidirect
23. #15690802C22semidirect
32. #15690781C22semidirect
36. #15690768C22semidirect
39. #15690743C22semidirect
40. #15690741C22semidirect
41. C4⋊#11715895C22semidirect
42. #15690735C22semidirect
43. #15690735C22semidirect
44. #15690733C22semidirect
51. #15690722C22semidirect
54. C4×D4×C97C22semidirect
55. #107×C97C22semidirect
57. #102×C97C22semidirect
58. C4×C8×C97C22semidirect
192. C97#3297semidirect
204. C4×D8D97semidirect
230. C4×D4C2×D97semidirect
235. #107C2×D97semidirect
247. #102C2×D97semidirect
250. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614352
#202613802
#202610902
#202610722
#202610622
#202610352
#202609702
#202609632
#202609502
#202608772
#202608672
C8×D3882
#202607282
#102×D972
C4×D8×C972
#329797