Finite Group Databasebeta

← #27153589

Subgroups — #27153589

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#20261382C211yesmaximal, characteristic, complemented
3#20261088C211yesmaximal, characteristic, complemented
4#20261072C211yesmaximal, characteristic, complemented
5#20261034C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260965C211yesmaximal, characteristic, complemented
8#20260949C211yesmaximal, characteristic, complemented
9#20260882C211yesmaximal, characteristic, complemented
10#20260870C211yesmaximal, characteristic, complemented
11#20260817C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic, complemented
13#20260729C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#482×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095500C2211yescharacteristic, complemented
19C2×D776C2211yescharacteristic
20C2×#13095496C2211yescharacteristic
21C22×#11715895C2211yescharacteristic, complemented
22C2×C8×D97C2211yescharacteristic
23#15690804—21no—
24#15690801C2211yescharacteristic, complemented
25#15690797C2211yescharacteristic
26#15690792—21no—
27#15690792—21no—
28#15690790C2211yescharacteristic
29#15690785C2211yescharacteristic
30#15690784C2211yescharacteristic, complemented
31#15690783—21no—
32#15690781—21no—
33#15690778—21no—
34C4×D388—21no—
35#15690772C2211yescharacteristic
36#15690772C2211yescharacteristic, complemented
37C4×C22⋊D97C2211yescharacteristic
38C4⋊C4×D97C2211yescharacteristic
39C22⋊C4×D97—21no—
40#15690743C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690735C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49#15690730C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic, complemented
52#121×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic, complemented
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095511—22no—
61#13095511—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—21no—
64C2×D388C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97D411yescharacteristic
70D4×D97—22no—
71D4×D97—41no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76C2×C4×D97C2311yescharacteristic
77#13095500—41no—
78D776—22no—
79#13095496—41no—
80C2×#11715895—41no—
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C8×D97—41no—
84C4⋊Dic97—21no—
85C4⋊Dic97—21no—
86C4⋊Dic97C2311yescharacteristic
87#13095490—21no—
88#13095490—21no—
89#13095490D411yescharacteristic
90C22⋊Dic97—21no—
91C22⋊Dic97—21no—
92C22⋊Dic97D411yescharacteristic
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97#13095488—21no—
98#13095488D411yescharacteristic
99C4×Dic97C2311yescharacteristic
100C2×D4×C97—21no—
101C2×D4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97C2311yescharacteristic
106C4⋊C4×C97C2311yescharacteristic
107C4⋊C4×C97C2311yescharacteristic
108C22⋊C4×C97—21no—
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
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—81no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—22no—
130C22⋊D97—42no—
131C22⋊D97—42no—
132C22⋊D97—81no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138D388—21no—
139D388—21no—
140D388—21no—
141D388—42no—
142D388#4112yes—
143C4×D97—21no—
144C4×D97—21no—
145C4×D97—42no—
146C4×D97—81no—
147C4×D97—81no—
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—21no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—22no—
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—41no—
184C2×C97—81no—
185C2×C97#54611yescharacteristic
186C2×C97C2×#13511yescharacteristic
187C2×C97#57611yescharacteristic
188D97—22no—
189D97—42no—
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546—971no—
195#488—971no—
196#484—971no—
197#482D9711yescharacteristic, complemented
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×#102—971no—
208C2×#41—971no—
209C22×D4—971no—
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8—971no—
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—1941no—
224#121—1941no—
225#121C2×D9711yescharacteristic
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#107C2×D9711yescharacteristic, complemented
236#106C2×D9711yescharacteristic, complemented
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
248C22⋊C8—971no—
249C22⋊C8C2×D9711yescharacteristic, complemented
250C4×C8—971no—
251C24—1941no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—21no—
256C2×D4—971no—
257C2×D4—971no—
258C2×D4—971no—
259C2×D4—3881no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1942no—
263C2×D4—1941no—
264C2×D4—1941no—
265C2×D4C22×D9711yescharacteristic
266C22×C4—971no—
267C22×C4—1941no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4—1941no—
271C22×C4C22×D9711yescharacteristic
272SD16—3881no—
273D8—3881no—
274D8—1942no—
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—1941no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4C22×D9711yescharacteristic
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—21no—
302C42—971no—
303C23—21no—
304C23—7761no—
305C23—971no—
306C23—3882no—
307C23—3882no—
308C23—1941no—
309C23—1941no—
310C23D4×D9711yescharacteristic
311Q8—1941no—
312Q8—1941no—
313D4—21no—
314D4—21no—
315D4—81no—
316D4—7761no—
317D4—972no—
318D4—3882no—
319D4—3882no—
320D4—3882no—
321D4—1942no—
322D4—1941no—
323D4—1941no—
324D4—1941no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—7761no—
331C2×C4—7761no—
332C2×C4—7761no—
333C2×C4—971no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—3882no—
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—41no—
351C22—41no—
352C22—41no—
353C22—7764no—
354C22—972no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—3881no—
360C22—1942no—
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—
373C4#41×D9711yescharacteristic
374C4#1569085611yescharacteristic
375C2—41no—
376C2—81no—
377C2—7761no—
378C2—3882no—
379C2—1942no—
380C2#2026137711yescharacteristic
381C2#135×D9711yescharacteristic
382C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#482×C97C2
socleC22×C97C22×D4
hypercenter#482D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #20261429C2semidirect
2. #20261382C2semidirect
3. #20261088C2semidirect
4. #20261072C2semidirect
5. #20261034C2semidirect
6. #20260969C2semidirect
7. #20260965C2semidirect
8. #20260949C2semidirect
9. #20260882C2semidirect
10. #20260870C2semidirect
11. #20260817C2semidirect
12. #20260798C2semidirect
13. #20260729C2semidirect
14. #102×D97C2semidirect
15. #482×C97C2semidirect
18. C2×#13095500C22semidirect
21. C22×#11715895C22semidirect
24. #15690801C22semidirect
30. #15690784C22semidirect
36. #15690772C22semidirect
40. #15690743C22semidirect
41. #15690739C22semidirect
42. #15690735C22semidirect
43. #15690733C22semidirect
44. #15690733C22semidirect
49. #15690730C22semidirect
51. C8×Dic97C22semidirect
53. #117×C97C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
58. C22⋊C8×C97C22semidirect
192. C97#3297semidirect
197. #482D97semidirect
231. #117C2×D97semidirect
235. #107C2×D97semidirect
236. #106C2×D97semidirect
249. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614292
#202613822
#202610882
#202610722
#202610342
#202609692
#202609652
#202609492
#202608822
#202608702
#202608172
#202607982
#202607292
#102×D972
#482×C972
#329797