Finite Group Databasebeta

← #27153582

Subgroups — #27153582

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#20261433C211yesmaximal, characteristic, complemented
2#20261379C211yesmaximal, characteristic, complemented
3#20261088C211yesmaximal, characteristic, complemented
4#20261070C211yesmaximal, characteristic, complemented
5#20261058C211yesmaximal, characteristic, complemented
6#20261037C211yesmaximal, characteristic, complemented
7#20260970C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260949C211yesmaximal, characteristic, complemented
10#20260881C211yesmaximal, characteristic, complemented
11#20260869C211yesmaximal, characteristic, complemented
12#20260815C211yesmaximal, characteristic, complemented
13C8×C22⋊D97C211yesmaximal, characteristic, complemented
14#20260728C211yesmaximal, characteristic, complemented
15C2×#102×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×#13095500C2211yescharacteristic, complemented
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic, complemented
21#15690805—21no—
22#15690801C2211yescharacteristic, complemented
23#15690797C2211yescharacteristic
24#15690795—21no—
25#15690791—21no—
26#15690789C2211yescharacteristic
27#15690785C2211yescharacteristic, complemented
28#15690784—21no—
29#15690784C2211yescharacteristic
30#15690782—21no—
31#15690781C2211yescharacteristic
32#15690779—21no—
33D4×Dic97C2211yescharacteristic
34#15690772C2211yescharacteristic, complemented
35C4×C22⋊D97—21no—
36C4×C22⋊D97C2211yescharacteristic
37#15690768C2211yescharacteristic, complemented
38C2×C22⋊Dic97—21no—
39#15690743C2211yescharacteristic, complemented
40#15690743C2211yescharacteristic, complemented
41#15690742C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690733C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C2×C4⋊C4×C97C2211yescharacteristic
55#102×C97C2211yescharacteristic
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59#13095510—22no—
60#13095510—41no—
61C2×Dic194C2311yescharacteristic
62C2×D388—21no—
63C2×D388C2311yescharacteristic
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67C2×C22⋊D97D411yescharacteristic
68C22×Dic97—21no—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71C2×C4×D97C2311yescharacteristic
72#13095500—41no—
73#13095496—22no—
74#13095496—41no—
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C8×D97—41no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84C22⋊Dic97—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97D411yescharacteristic
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97C24×C97—21no—
98C2×D4×C97—22no—
99C2×D4×C97—21no—
100C2×D4×C97—41no—
101C2×D4×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104C22×C4×C97C2311yescharacteristic
105C2×C8×C97—41no—
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97—21no—
109C4⋊C4×C97C2311yescharacteristic
110C4⋊C4×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—42no—
114C23×C97—42no—
115C23×C97—81no—
116C23×C97C2×D411yescharacteristic
117C22×D97—21no—
118C22×D97C2×D411yescharacteristic
119D4×C97—21no—
120D4×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—81no—
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—22no—
132C22⋊D97—42no—
133C22⋊D97—42no—
134C22⋊D97—81no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—42no—
139C2×Dic97—81no—
140C2×Dic97C2×D411yescharacteristic
141C2×Dic97C2×D411yescharacteristic
142C2×Dic97C2×D411yescharacteristic
143D388—21no—
144D388—21no—
145D388—81no—
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—21no—
156C22×C97—21no—
157C22×C97—22no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—84no—
164C22×C97#13512yes—
165C22×C97C22×D411yescharacteristic
166C2×D97—21no—
167C2×D97—21no—
168C2×D97—21no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—41no—
172C4×C97—21no—
173C4×C97—41no—
174C4×C97—41no—
175C4×C97C2×#4111yescharacteristic
176C4×C97H(5,𝔽2)11yescharacteristic
177Dic97—21no—
178Dic97—21no—
179Dic97—21no—
180Dic97—41no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—22no—
184C2×C97—42no—
185C2×C97—81no—
186C2×C97#54611yescharacteristic
187C2×C97C2×#13511yescharacteristic
188C2×C97#57611yescharacteristic
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×D8—971no—
205C8×D4—971no—
206#416—971no—
207C2×#102D9711yescharacteristic, complemented
208C2×#41—971no—
209C22×D4C2×D9711yescharacteristic
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8—971no—
213C22×C8—971no—
214C22×C8C2×D9711yescharacteristic
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#117—971no—
232C2×C4⋊C4C2×D9711yescharacteristic
233C2×C22⋊C4—1941no—
234#107—971no—
235#107—971no—
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#102C2×D9711yescharacteristic
245#102C2×D9711yescharacteristic
246#102C2×D9711yescharacteristic
247#102C2×D9711yescharacteristic
248C22⋊C8—971no—
249C22⋊C8—971no—
250C4×C8—971no—
251C24—21no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—971no—
259C2×D4—971no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4C22×D9711yescharacteristic
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—21no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—971no—
291C22⋊C4—971no—
292C22⋊C4—1941no—
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—971no—
302C42—1941no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—971no—
309C23—1941no—
310C23D4×D9711yescharacteristic
311Q8—1941no—
312Q8—1941no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—7761no—
318D4—7761no—
319D4—3882no—
320D4—3882no—
321D4—1942no—
322D4—1941no—
323D4—1941no—
324D4#1309551112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—81no—
330C2×C4—7761no—
331C2×C4—7761no—
332C2×C4—971no—
333C2×C4—971no—
334C2×C4—971no—
335C2×C4—3882no—
336C2×C4—1941no—
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—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—84no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—1941no—
360C22—1941no—
361C22—1941no—
362C22#1569083112yes—
363C22C2×D4×D9711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3881no—
368C4—3881no—
369C4—3881no—
370C4—1941no—
371C4—1941no—
372C4—1941no—
373C4C2×#1309551111yescharacteristic
374C4#1569085511yescharacteristic
375C2—22no—
376C2—42no—
377C2—81no—
378C2—7761no—
379C2—3881no—
380C2#2026137511yescharacteristic
381C2C2×#1569083111yescharacteristic
382C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#102×C97C2
socleC22×C97C22×D4
hypercenterC2×#102D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #20261433C2semidirect
2. #20261379C2semidirect
3. #20261088C2semidirect
4. #20261070C2semidirect
5. #20261058C2semidirect
6. #20261037C2semidirect
7. #20260970C2semidirect
8. #20260964C2semidirect
9. #20260949C2semidirect
10. #20260881C2semidirect
11. #20260869C2semidirect
12. #20260815C2semidirect
13. C8×C22⋊D97C2semidirect
14. #20260728C2semidirect
15. C2×#102×C97C2semidirect
17. C2×#13095500C22semidirect
20. C2×C8×D97C22semidirect
22. #15690801C22semidirect
27. #15690785C22semidirect
34. #15690772C22semidirect
37. #15690768C22semidirect
39. #15690743C22semidirect
40. #15690743C22semidirect
41. #15690742C22semidirect
42. #15690739C22semidirect
43. #15690735C22semidirect
44. #15690733C22semidirect
45. #15690733C22semidirect
46. #15690732C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
192. C97#3297semidirect
207. C2×#102D97semidirect

Maximal subgroups

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

subgroupindex
#202614332
#202613792
#202610882
#202610702
#202610582
#202610372
#202609702
#202609642
#202609492
#202608812
#202608692
#202608152
C8×C22⋊D972
#202607282
C2×#102×C972
#329797