Finite Group Databasebeta

← #27153542

Subgroups — #27153542

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
1#20261385C211yesmaximal, characteristic, complemented
2D4×C22⋊D97C211yesmaximal, characteristic, complemented
3#20261080C211yesmaximal, characteristic, complemented
4#20261049C211yesmaximal, characteristic, complemented
5#20261047C211yesmaximal, characteristic, complemented
6#20261013C211yesmaximal, characteristic, complemented
7#20260969C211yesmaximal, characteristic, complemented
8#20260968C211yesmaximal, characteristic, complemented
9#20260928C211yesmaximal, characteristic, complemented
10#20260867C211yesmaximal, characteristic, complemented
11D8×Dic97C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic, complemented
13C2×#15690731C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#447×C97C211yesmaximal, characteristic, complemented
16C2×#13095510—21no—
17C22×C22⋊D97—21no—
18C2×D4×D97C2211yescharacteristic
19C2×#13095496C2211yescharacteristic, complemented
20C22×#11715895C2211yescharacteristic, complemented
21C2×C8×D97C2211yescharacteristic, complemented
22#15690809C2211yescharacteristic, complemented
23#15690806C2211yescharacteristic
24#15690792—21no—
25#15690791—21no—
26#15690787—21no—
27#15690784C2211yescharacteristic
28#15690782—21no—
29#15690781C2211yescharacteristic
30#15690781C2211yescharacteristic, complemented
31#15690779—21no—
32#15690778—21no—
33D4×Dic97C2211yescharacteristic
34D4×Dic97C2211yescharacteristic, complemented
35C4×C22⋊D97C2211yescharacteristic
36C2×C4⋊Dic97C2211yescharacteristic, complemented
37C4⋊C4×D97C2211yescharacteristic, complemented
38#15690744C2211yescharacteristic, complemented
39#15690744C2211yescharacteristic, complemented
40#15690743C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690732C2211yescharacteristic, complemented
43#15690732C2211yescharacteristic, complemented
44#15690731C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690731C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic, complemented
49#15690730C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C2×D8×C97C2211yescharacteristic
54C2×D8×C97C2211yescharacteristic, complemented
55#121×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095510—44no—
61C2×Dic194—21no—
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—44no—
70C2×C22⋊D97D411yescharacteristic
71D4×D97—22no—
72D4×D97—41no—
73C22×Dic97—21no—
74C22×Dic97—21no—
75C22×Dic97—21no—
76C2×C4×D97—21no—
77C2×C4×D97C2311yescharacteristic
78#13095496—41no—
79C2×#11715895—41no—
80C2×#11715895C2311yescharacteristic
81C2×#11715895C2311yescharacteristic
82C8×D97—41no—
83C4⋊Dic97—21no—
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2311yescharacteristic
86C4⋊Dic97C2311yescharacteristic
87#13095490—21no—
88#13095490—21no—
89#13095490D411yescharacteristic
90C22⋊Dic97—21no—
91C22⋊Dic97—21no—
92C22⋊Dic97—21no—
93C22⋊Dic97—21no—
94C22⋊Dic97D411yescharacteristic
95#13095488—21no—
96#13095488D411yescharacteristic
97C4×Dic97C2311yescharacteristic
98C24×C97—21no—
99C2×D4×C97—22no—
100C2×D4×C97—21no—
101C2×D4×C97—21no—
102C2×D4×C97—41no—
103C2×D4×C97C2311yescharacteristic
104C2×D4×C97C2311yescharacteristic
105C2×D4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107D8×C97—22no—
108D8×C97—41no—
109C2×C8×C97C2311yescharacteristic
110C2×C8×C97C2311yescharacteristic
111C4⋊C4×C97C2311yescharacteristic
112C22⋊C4×C97—21no—
113C23×C97—21no—
114C23×C97—21no—
115C23×C97—21no—
116C23×C97—42no—
117C23×C97—42no—
118C23×C97—81no—
119C23×C97C2×D411yescharacteristic
120C22×D97—21no—
121C22×D97—42no—
122C22×D97—42no—
123C22×D97—81no—
124C22×D97C2×D411yescharacteristic
125D4×C97—21no—
126D4×C97—21no—
127D4×C97—21no—
128D4×C97—21no—
129D4×C97—21no—
130D4×C97—42no—
131D4×C97—42no—
132D4×C97#4112yescomplemented
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2411yescharacteristic
138C22⋊D97—42no—
139C22⋊D97—42no—
140C22⋊D97—42no—
141C22⋊D97—42no—
142C22⋊D97—42no—
143C22⋊D97—81no—
144C22⋊D97D814yescomplemented
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—42no—
149C2×Dic97—42no—
150C2×Dic97—42no—
151C2×Dic97C2×D411yescharacteristic
152C2×Dic97C2×D411yescharacteristic
153C2×Dic97C2×D411yescharacteristic
154D388—21no—
155D388—21no—
156C4×D97—21no—
157C4×D97—21no—
158C4×D97—42no—
159Dic194—42no—
160C8×C97—21no—
161C8×C97—21no—
162#11715895—21no—
163#11715895—21no—
164C22×C97—21no—
165C22×C97—21no—
166C22×C97—22no—
167C22×C97—41no—
168C22×C97—41no—
169C22×C97—41no—
170C22×C97—41no—
171C22×C97—41no—
172C22×C97—41no—
173C22×C97—41no—
174C22×C97—41no—
175C22×C97—84no—
176C22×C97C2×D812yes—
177C22×C97C22×D411yescharacteristic
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—84no—
184C2×D97C2×D812yes—
185C4×C97—21no—
186C4×C97—41no—
187C4×C97C2×#4111yescharacteristic
188C4×C97H(5,𝔽2)11yescharacteristic
189Dic97—21no—
190Dic97—21no—
191Dic97—41no—
192Dic97—41no—
193Dic97—41no—
194Dic97C2×D812yes—
195C2×C97—22no—
196C2×C97—42no—
197C2×C97—81no—
198C2×C97—81no—
199C2×C97#54611yescharacteristic
200C2×C97C22×D811yescharacteristic
201C2×C97#57711yescharacteristic
202D97—22no—
203D97—81no—
204#3294—971nomaximal, Sylow
205C97#329411yescharacteristic, complemented, Sylow
206#547—971no—
207D42—971no—
208#486—971no—
209#480—971no—
210#480—971no—
211#474—971no—
212#466—971no—
213#466—971no—
214#459—971no—
215#447—971no—
216#447D9711yescharacteristic, complemented
217C4×D8—971no—
218C8×D4—971no—
219C2×#102—971no—
220C2×#102—971no—
221C2×#41—1941no—
222C22×D4—971no—
223C22×D4—1941no—
224C22×D4C2×D9711yescharacteristic
225C2×D8—971no—
226C2×D8C2×D9711yescharacteristic
227C2×D8C2×D9711yescharacteristic, complemented
228C22×C8—971no—
229C22×C8—971no—
230C4⋊Q8—971no—
231#127—971no—
232#123—1941no—
233#123—1941no—
234C22⋊Q8—1941no—
235#121—971no—
236#121—971no—
237#121—971no—
238#121—1941no—
239#121C2×D9711yescharacteristic, complemented
240#120—1941no—
241#120—1941no—
242C4×D4—971no—
243C4×D4—971no—
244C4×D4—971no—
245C2×C4⋊C4—971no—
246C2×C4⋊C4—971no—
247#107—971no—
248#107—971no—
249#107—971no—
250C4⋊C8—971no—
251#102—971no—
252#102—971no—
253#102—971no—
254#102—971no—
255#102—971no—
256#102—971no—
257#102—971no—
258#102—971no—
259#102C2×D9711yescharacteristic
260#102C2×D9711yescharacteristic, complemented
261C22⋊C8—971no—
262C22⋊C8C2×D9711yescharacteristic, complemented
263C4×C8—971no—
264C24—21no—
265C24—1941no—
266#41—3884no—
267C2×Q8—1941no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—971no—
273C2×D4—971no—
274C2×D4—3884no—
275C2×D4—3881no—
276C2×D4—1941no—
277C2×D4—1941no—
278C2×D4—1941no—
279C2×D4—1941no—
280C2×D4—1941no—
281C2×D4—1941no—
282C2×D4—1942no—
283C2×D4C22×D9711yescharacteristic
284C2×D4C22×D9711yescharacteristic
285C2×D4C22×D9711yescharacteristic
286C22×C4—971no—
287C22×C4—1941no—
288C22×C4—1941no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4C22×D9711yescharacteristic
292D8—22no—
293D8—41no—
294D8—3881no—
295C2×C8—971no—
296C2×C8—971no—
297C2×C8—3881no—
298C2×C8—3881no—
299C2×C8C22×D9711yescharacteristic
300C2×C8C22×D9711yescharacteristic
301C4⋊C4—971no—
302C4⋊C4—971no—
303C4⋊C4—971no—
304C4⋊C4—971no—
305C4⋊C4—1941no—
306C4⋊C4—1941no—
307C4⋊C4—1941no—
308C4⋊C4C22×D9711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—971no—
311C22⋊C4—971no—
312C22⋊C4—1941no—
313C22⋊C4—1941no—
314C22⋊C4—1941no—
315C22⋊C4—1941no—
316C22⋊C4—1941no—
317C42—971no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—7761no—
325C23—971no—
326C23—3882no—
327C23—3882no—
328C23—1941no—
329C23D4×D9711yescharacteristic
330Q8—3882no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—7761no—
339D4—974no—
340D4—3882no—
341D4—3882no—
342D4—3882no—
343D4—3882no—
344D4—3882no—
345D4—1941no—
346D4—1941no—
347D4#1309551012yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—971no—
352C2×C4—971no—
353C2×C4—971no—
354C2×C4—3882no—
355C2×C4—3882no—
356C2×C4—3882no—
357C2×C4—3882no—
358C2×C4—1941no—
359C2×C4—1941no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4D4×D9711yescharacteristic
364C2×C4C23×D9711yescharacteristic
365C8—21no—
366C8—21no—
367C8—1941no—
368C8—1941no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—7764no—
382C22—972no—
383C22—3881no—
384C22—3881no—
385C22—3881no—
386C22—1941no—
387C22—1941no—
388C22D8×D9712yes—
389C22C2×D4×D9711yescharacteristic
390C4—21no—
391C4—41no—
392C4—972no—
393C4—3881no—
394C4—3881no—
395C4—3881no—
396C4—1941no—
397C4—1941no—
398C4C2×#1309551011yescharacteristic
399C4#1569085511yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—7761no—
405C2—1942no—
406C2#2026137211yescharacteristic
407C2C2×D8×D9711yescharacteristic
408C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#447×C97C2
socleC22×C97C22×D4
hypercenter#447D97
hypocenterC97#3294
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20261385C2semidirect
2. D4×C22⋊D97C2semidirect
3. #20261080C2semidirect
4. #20261049C2semidirect
5. #20261047C2semidirect
6. #20261013C2semidirect
7. #20260969C2semidirect
8. #20260968C2semidirect
9. #20260928C2semidirect
10. #20260867C2semidirect
11. D8×Dic97C2semidirect
12. #20260798C2semidirect
13. C2×#15690731C2semidirect
14. #102×D97C2semidirect
15. #447×C97C2semidirect
19. C2×#13095496C22semidirect
20. C22×#11715895C22semidirect
21. C2×C8×D97C22semidirect
22. #15690809C22semidirect
30. #15690781C22semidirect
34. D4×Dic97C22semidirect
36. C2×C4⋊Dic97C22semidirect
37. C4⋊C4×D97C22semidirect
38. #15690744C22semidirect
39. #15690744C22semidirect
40. #15690743C22semidirect
41. #15690739C22semidirect
42. #15690732C22semidirect
43. #15690732C22semidirect
46. #15690731C22semidirect
47. #15690731C22semidirect
48. #15690731C22semidirect
49. #15690730C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
54. C2×D8×C97C22semidirect
55. #121×C97C22semidirect
57. #102×C97C22semidirect
58. C22⋊C8×C97C22semidirect
132. D4×C97#41semidirect
144. C22⋊D97D8semidirect
205. C97#3294semidirect
216. #447D97semidirect
227. C2×D8C2×D97semidirect
239. #121C2×D97semidirect
260. #102C2×D97semidirect
262. C22⋊C8C2×D97semidirect
347. D4#13095510semidirect

Maximal subgroups

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

subgroupindex
#202613852
D4×C22⋊D972
#202610802
#202610492
#202610472
#202610132
#202609692
#202609682
#202609282
#202608672
D8×Dic972
#202607982
C2×#156907312
#102×D972
#447×C972
#329497