Finite Group Databasebeta

← #27153691

Subgroups — #27153691

Order 12416.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261381C211yesmaximal, characteristic, complemented
2#20261097C211yesmaximal, characteristic, complemented
3#20261065C211yesmaximal, characteristic, complemented
4#20261053C211yesmaximal, characteristic, complemented
5#20261022C211yesmaximal, characteristic, complemented
6#20260986C211yesmaximal, characteristic, complemented
7#20260962C211yesmaximal, characteristic, complemented
8#20260950C211yesmaximal, characteristic, complemented
9#20260878C211yesmaximal, characteristic, complemented
10#20260868C211yesmaximal, characteristic, complemented
11#20260845C211yesmaximal, characteristic, complemented
12#20260807C211yesmaximal, characteristic, complemented
13#20260734C211yesmaximal, characteristic, complemented
14C2×#15690730C211yesmaximal, characteristic, complemented
15#547×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×#13095499C2211yescharacteristic, complemented
19C2×#13095496C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C2×#13095495C2211yescharacteristic, complemented
22C22×#11715895C2211yescharacteristic
23#15690798C2211yescharacteristic
24#15690795—21no—
25#15690789C2211yescharacteristic
26#15690784C2211yescharacteristic
27#15690783—21no—
28#15690778—21no—
29C4×D388C2211yescharacteristic
30C22⋊C4×D97—21no—
31#15690741C2211yescharacteristic, complemented
32#15690741C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690733C2211yescharacteristic, complemented
36#15690733C2211yescharacteristic, complemented
37#15690733C2211yescharacteristic, complemented
38#15690731C2211yescharacteristic, complemented
39#15690730C2211yescharacteristic
40#15690730C2211yescharacteristic
41#15690730C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47#15690721C2211yescharacteristic, complemented
48C2×#41×C97—21no—
49C4⋊Q8×C97C2211yescharacteristic, complemented
50#123×C97—21no—
51#123×C97—21no—
52C22⋊Q8×C97—21no—
53#121×C97C2211yescharacteristic
54#121×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic, complemented
57C2×C4⋊C4×C97C2211yescharacteristic
58C2×C4⋊C4×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095511—22no—
61#13095511—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—22no—
64C2×D388—21no—
65C2×D388—41no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72#13095499—41no—
73#13095496—22no—
74#13095496—41no—
75#13095495—41no—
76C2×#11715895—41no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C2×#11715895C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#41×C97—44no—
89C2×Q8×C97—21no—
90C2×D4×C97—21no—
91C2×D4×C97—21no—
92C2×D4×C97D411yescharacteristic
93C2×D4×C97C2311yescharacteristic
94C22×C4×C97—21no—
95C22×C4×C97—21no—
96C22×C4×C97—21no—
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97—21no—
100C4⋊C4×C97—21no—
101C4⋊C4×C97—21no—
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C4⋊C4×C97C2311yescharacteristic
106C22⋊C4×C97—21no—
107C22⋊C4×C97—21no—
108C22⋊C4×C97—21no—
109C22⋊C4×C97D411yescharacteristic
110C22⋊C4×C97D411yescharacteristic
111C42×C97C2311yescharacteristic
112C23×C97—21no—
113C23×C97C2×D411yescharacteristic
114C23×C97C2×D411yescharacteristic
115C22×D97—21no—
116C22×D97—21no—
117C22×D97—42no—
118C22×D97—42no—
119C22×D97—81no—
120Q8×C97—42no—
121D4×C97—22no—
122D4×C97—21no—
123D4×C97—21no—
124D4×C97—42no—
125D4×C97—42no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—42no—
134C2×C4×C97—42no—
135C2×C4×C97—42no—
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2411yescharacteristic
141C22⋊D97—42no—
142C22⋊D97—81no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145D388—21no—
146D388—21no—
147D388—21no—
148D388—42no—
149D388#4112yescomplemented
150C4×D97—42no—
151C4×D97—81no—
152Dic194—21no—
153Dic194—21no—
154#11715895—21no—
155#11715895—21no—
156#11715895—21no—
157#11715895—21no—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97—21no—
161C22×C97—21no—
162C22×C97—21no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97AGL(1,ℤ/8)12yes—
167C22×C97C22×D411yescharacteristic
168C2×D97—22no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—41no—
172C2×D97—41no—
173C2×D97—41no—
174C2×D97—84no—
175C4×C97—21no—
176C4×C97—21no—
177C4×C97—21no—
178C4×C97—21no—
179C4×C97—41no—
180C4×C97—41no—
181C4×C97—41no—
182C4×C97C2×#4111yescharacteristic
183C4×C97H(5,𝔽2)11yescharacteristic
184Dic97—41no—
185Dic97—41no—
186C2×C97—22no—
187C2×C97—41no—
188C2×C97—81no—
189C2×C97#54611yescharacteristic
190C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C97#57711yescharacteristic
192D97—42no—
193D97—81no—
194#3306—971nomaximal, Sylow
195C97#330611yescharacteristic, complemented, Sylow
196#547D9711yescharacteristic, complemented
197#546—971no—
198#489—971no—
199#483—971no—
200#481—971no—
201#476—971no—
202#469—971no—
203#465—971no—
204#463—971no—
205#449—971no—
206#447—971no—
207#442—971no—
208#435—971no—
209#417—971no—
210C2×#102—971no—
211C2×#41—21no—
212C2×#41—971no—
213C22×D4—971no—
214C2×SD16—971no—
215C2×D8—971no—
216C2×D8—971no—
217C2×M16—971no—
218C22×C8—971no—
219C4⋊Q8C2×D9711yescharacteristic, complemented
220#124—971no—
221#123—21no—
222#123—21no—
223#123—1941no—
224C22⋊Q8—21no—
225C22⋊Q8—971no—
226#121—971no—
227#121—1941no—
228#121C2×D9711yescharacteristic
229#121C2×D9711yescharacteristic, complemented
230#120—1941no—
231C4×D4—971no—
232C4×D4C2×D9711yescharacteristic
233C4×D4C2×D9711yescharacteristic, complemented
234C2×C4⋊C4C2×D9711yescharacteristic
235C2×C4⋊C4C2×D9711yescharacteristic, complemented
236C2×C22⋊C4—1941no—
237#106—971no—
238#106—971no—
239#106—971no—
240C4⋊C8—971no—
241#103—971no—
242#103—971no—
243#103—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102—971no—
248#102—971no—
249#102—971no—
250#102—971no—
251C22⋊C8—971no—
252C22⋊C8—971no—
253#97—971no—
254C24—1941no—
255#41—44no—
256#41—3881no—
257#41—1942no—
258C2×Q8—21no—
259C2×Q8—971no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—971no—
263C2×D4—971no—
264C2×D4—3881no—
265C2×D4—1942no—
266C2×D4—1941no—
267C2×D4—1941no—
268C2×D4—1941no—
269C2×D4C22⋊D9711yescharacteristic
270C2×D4C22×D9711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—1941no—
275C22×C4—1941no—
276C22×C4C22×D9711yescharacteristic
277C22×C4C22×D9711yescharacteristic
278SD16—3881no—
279D8—3881no—
280D8—1942no—
281M16—3881no—
282C2×C8—971no—
283C2×C8—971no—
284C2×C8—971no—
285C2×C8—971no—
286C2×C8—3881no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—971no—
291C4⋊C4—1941no—
292C4⋊C4C22⋊D9711yescharacteristic
293C4⋊C4C22×D9711yescharacteristic
294C4⋊C4C22×D9711yescharacteristic
295C4⋊C4C22×D9711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C22⋊C4—1941no—
302C22⋊C4—1941no—
303C22⋊C4—1941no—
304C22⋊C4C22⋊D9711yescharacteristic
305C22⋊C4C22⋊D9711yescharacteristic
306C42C22×D9711yescharacteristic
307C23—21no—
308C23—7761no—
309C23—3882no—
310C23—3882no—
311C23—1941no—
312C23—1941no—
313C23C2×C22⋊D9711yescharacteristic
314C23C2×C22⋊D9711yescharacteristic
315Q8—42no—
316Q8—1941no—
317Q8—1941no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—7761no—
324D4—972no—
325D4—3882no—
326D4—3882no—
327D4—1941no—
328D4—1941no—
329D4—1941no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—7761no—
341C2×C4—3882no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4C2×C22⋊D9711yescharacteristic
345C2×C4C2×C22⋊D9711yescharacteristic
346C2×C4C2×C22⋊D9711yescharacteristic
347C2×C4C2×C22⋊D9711yescharacteristic
348C2×C4C23×D9711yescharacteristic
349C8—1941no—
350C8—1941no—
351C8—1941no—
352C8—1941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—7764no—
362C22—3881no—
363C22—3881no—
364C22—3881no—
365C22—3881no—
366C22—3881no—
367C22—1942no—
368C22#1569083612yes—
369C22C22×C22⋊D9711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3881no—
378C4—3881no—
379C4#41×D9711yescharacteristic
380C4#1569085611yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—7761no—
385C2—3882no—
386C2#2026137311yescharacteristic
387C2C2×#1569083611yescharacteristic
388C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#547×C97C2
socleC22×C97C22×D4
hypercenter#547D97
hypocenterC97#3306
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20261381C2semidirect
2. #20261097C2semidirect
3. #20261065C2semidirect
4. #20261053C2semidirect
5. #20261022C2semidirect
6. #20260986C2semidirect
7. #20260962C2semidirect
8. #20260950C2semidirect
9. #20260878C2semidirect
10. #20260868C2semidirect
11. #20260845C2semidirect
12. #20260807C2semidirect
13. #20260734C2semidirect
14. C2×#15690730C2semidirect
15. #547×C97C2semidirect
18. C2×#13095499C22semidirect
21. C2×#13095495C22semidirect
31. #15690741C22semidirect
32. #15690741C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690733C22semidirect
36. #15690733C22semidirect
37. #15690733C22semidirect
38. #15690731C22semidirect
46. C22⋊#11715895C22semidirect
47. #15690721C22semidirect
49. C4⋊Q8×C97C22semidirect
54. #121×C97C22semidirect
56. C4×D4×C97C22semidirect
58. C2×C4⋊C4×C97C22semidirect
149. D388#41semidirect
195. C97#3306semidirect
196. #547D97semidirect
219. C4⋊Q8C2×D97semidirect
229. #121C2×D97semidirect
233. C4×D4C2×D97semidirect
235. C2×C4⋊C4C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613812
#202610972
#202610652
#202610532
#202610222
#202609862
#202609622
#202609502
#202608782
#202608682
#202608452
#202608072
#202607342
C2×#156907302
#547×C972
#330697