Finite Group Databasebeta

← #27150561

Subgroups — #27150561

Order 12416.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690834C211yesmaximal, characteristic
2#20261214C211yesmaximal, characteristic, complemented
3#20261207C211yesmaximal, characteristic, complemented
4#20260847C211yesmaximal, characteristic, complemented
5#20260845C211yesmaximal, characteristic, complemented
6#20260835C211yesmaximal, characteristic, complemented
7#20260834C211yesmaximal, characteristic, complemented
8#20260826C211yesmaximal, characteristic, complemented
9C4×#13095500C211yesmaximal, characteristic, complemented
10#20260815C211yesmaximal, characteristic, complemented
11#20260776C211yesmaximal, characteristic, complemented
12#20260728C211yesmaximal, characteristic, complemented
13#102×D97C211yesmaximal, characteristic, complemented
14#20260689C211yesmaximal, characteristic, complemented
15C4×D8×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18#15690834C418yescomplemented
19C2×#13095500C2211yescharacteristic
20C2×#13095499C2211yescharacteristic
21C2×#13095496C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
24C4×Dic194C2211yescharacteristic, complemented
25#15690775C2211yescharacteristic, complemented
26C4×D388C2211yescharacteristic
27D4×Dic97C2211yescharacteristic
28D4×Dic97C2211yescharacteristic, complemented
29#15690772C2211yescharacteristic
30C4×C22⋊D97—21no—
31C4×C22⋊D97—21no—
32#15690770—21no—
33#15690770—21no—
34#15690768C2211yescharacteristic
35#15690767C2211yescharacteristic
36#15690766—21no—
37C2×#13095490—21no—
38C4⋊C4×D97C2211yescharacteristic
39C2×#13095488—21no—
40C22⋊C4×D97—21no—
41#15690743C2211yescharacteristic, complemented
42#15690742C2211yescharacteristic, complemented
43#15690741C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690733C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic, complemented
49#15690730C2211yescharacteristic
50#15690721C2211yescharacteristic, complemented
51#15690720C2211yescharacteristic, complemented
52C8×Dic97C2211yescharacteristic, complemented
53C2×D8×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55C4×D4×C97C2211yescharacteristic, complemented
56#107×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic, complemented
59C4×C8×C97C2211yescharacteristic, complemented
60C23×D97—21no—
61#13095510—22no—
62#13095510C2×C414yescomplemented
63C2×Dic194C2311yescharacteristic
64C2×D388C2311yescharacteristic
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67D4×D97—22no—
68D4×D97C2×C414yescomplemented
69C22×Dic97—21no—
70C22×Dic97—21no—
71C22×Dic97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97C2311yescharacteristic
76#13095500C2×C414yescomplemented
77#13095499C2×C414yescomplemented
78#13095496C2×C414yescomplemented
79#13095494C2×C414yescomplemented
80C2×#11715895C2311yescharacteristic
81C2×#11715895C2311yescharacteristic
82C8×D97—22no—
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490D411yescharacteristic
89#13095490D411yescharacteristic
90C22⋊Dic97—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488D411yescharacteristic
96#13095488D411yescharacteristic
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97—21no—
100C4×Dic97C2311yescharacteristic
101C2×D4×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104C22×C4×C97—21no—
105D8×C97C2×C414yescomplemented
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C22⋊C4×C97—21no—
112C42×C97C2311yescharacteristic
113C23×C97—21no—
114C23×C97—21no—
115C22×D97—21no—
116C22×D97—24no—
117C22×D97—81no—
118C22×D97—81no—
119C22×D97C2×D411yescharacteristic
120D4×C97—21no—
121D4×C97—21no—
122D4×C97C22×C412yes—
123D4×C97C22×C412yes—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—81no—
127C2×C4×C97—81no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—24no—
132C22⋊D97—24no—
133C2×Dic97—21no—
134C2×Dic97—24no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—81no—
138C2×Dic97—81no—
139C2×Dic97C2×D411yescharacteristic
140C2×Dic97C2×D411yescharacteristic
141C2×Dic97C2×D411yescharacteristic
142D388—21no—
143D388C22×C412yes—
144C4×D97—81no—
145C4×D97—81no—
146C4×D97—81no—
147C4×D97C22×C412yes—
148C4×D97#4112yes—
149Dic194—21no—
150Dic194C22×C412yes—
151C8×C97—21no—
152C8×C97C22×C412yes—
153#11715895—21no—
154#11715895C22×C412yes—
155C22×C97—22no—
156C22×C97—22no—
157C22×C97—41no—
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—84no—
168C2×D97C4×D412yes—
169C4×C97—21no—
170C4×C97—21no—
171C4×C97—41no—
172C4×C97—41no—
173C4×C97C23×C411yescharacteristic
174C4×C97C2×#4111yescharacteristic
175Dic97—22no—
176Dic97—21no—
177Dic97—21no—
178Dic97—41no—
179Dic97—41no—
180Dic97C4×D412yes—
181C2×C97—42no—
182C2×C97—42no—
183C2×C97C2×C4×D411yescharacteristic
184C2×C97#57611yescharacteristic
185C2×C97#57711yescharacteristic
186D97—22no—
187D97—42no—
188#2946—971nomaximal, Sylow
189C97#294611yescharacteristic, complemented, Sylow
190C2×AGL(1,ℤ/8)—971no—
191#520—971no—
192#518—971no—
193#442—971no—
194#442—971no—
195#440—971no—
196#440—971no—
197C4×SD16—971no—
198C4×SD16—971no—
199C4×D8—971no—
200C4×D8D9711yescharacteristic, complemented
201#428—971no—
202#416—971no—
203C2×#102—971no—
204#405—971no—
205C2×#41—971no—
206C22×D4—971no—
207AGL(1,ℤ/8)—978no—
208C2×SD16—971no—
209C2×SD16—971no—
210C2×D8—971no—
211C2×D8C2×D9711yescharacteristic
212C2×M16—971no—
213C22×C8—971no—
214C4×Q8—971no—
215C4×Q8—971no—
216C4×D4—971no—
217C4×D4—971no—
218C4×D4—971no—
219C4×D4—971no—
220C4×D4—1941no—
221C4×D4—1941no—
222C4×D4—1941no—
223C4×D4—1941no—
224C4×D4C2×D9711yescharacteristic
225C4×D4C2×D9711yescharacteristic, complemented
226#117—971no—
227#117—971no—
228#117—1941no—
229C2×C4⋊C4—971no—
230C2×C4⋊C4—1941no—
231C2×C22⋊C4—1941no—
232C2×C22⋊C4—1941no—
233#107—971no—
234#107C2×D9711yescharacteristic, complemented
235#106—971no—
236#106—971no—
237#103—971no—
238#103—971no—
239#102—971no—
240#102—971no—
241#102—971no—
242#102—971no—
243#102C2×D9711yescharacteristic
244#102C2×D9711yescharacteristic, complemented
245#97—971no—
246#97—971no—
247C4×C8—971no—
248C4×C8C2×D9711yescharacteristic, complemented
249C24—1941no—
250#41—974no—
251#41—1942no—
252C2×Q8—971no—
253C2×D4—974no—
254C2×D4—971no—
255C2×D4—1942no—
256C2×D4—1941no—
257C2×D4—1941no—
258C2×D4C22×D9711yescharacteristic
259C2×D4C22×D9711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—971no—
263C22×C4—1941no—
264C22×C4—1941no—
265C22×C4—1941no—
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4—1941no—
269SD16—974no—
270SD16—974no—
271D8—974no—
272D8C4×D9714yescomplemented
273M16—974no—
274C2×C8—971no—
275C2×C8—971no—
276C2×C8—1942no—
277C2×C8C22×D9711yescharacteristic
278C2×C8C22×D9711yescharacteristic
279C4⋊C4—971no—
280C4⋊C4—971no—
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—1941no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4C22×D9711yescharacteristic
287C4⋊C4C22×D9711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—971no—
291C22⋊C4—971no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C42—971no—
298C42—1941no—
299C42—1941no—
300C42—1941no—
301C42C22×D9711yescharacteristic
302C23—21no—
303C23—21no—
304C23—7761no—
305C23—7761no—
306C23—971no—
307C23—1944no—
308C23—1941no—
309Q8—972no—
310Q8—1941no—
311D4—21no—
312D4—21no—
313D4—972no—
314D4—1944no—
315D4—1944no—
316D4—1941no—
317D4C2×C4×D9712yes—
318D4C2×C4×D9712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—7761no—
324C2×C4—7761no—
325C2×C4—7761no—
326C2×C4—7761no—
327C2×C4—7761no—
328C2×C4—971no—
329C2×C4—972no—
330C2×C4—972no—
331C2×C4—971no—
332C2×C4—971no—
333C2×C4—1944no—
334C2×C4—1941no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4D4×D9711yescharacteristic
338C2×C4D4×D9711yescharacteristic
339C2×C4C23×D9711yescharacteristic
340C8—21no—
341C8—972no—
342C8—1941no—
343C8C2×C4×D9712yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—7764no—
351C22—972no—
352C22—3881no—
353C22—3881no—
354C22—1942no—
355C22—1941no—
356C22—1941no—
357C22C2×D4×D9711yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—972no—
363C4—3881no—
364C4—3881no—
365C4—1942no—
366C4—1941no—
367C4—1941no—
368C4C22×C4×D9711yescharacteristic
369C4#41×D9711yescharacteristic
370C2—42no—
371C2—42no—
372C2—3882no—
373C2—1942no—
374C2C4×D4×D9711yescharacteristic
375C2#2026153611yescharacteristic
376C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690834C2
cosocleC2×C4×C97C24
hypercenterC4×D8D97
hypocenterC97#2946
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
2. #20261214C2semidirect
3. #20261207C2semidirect
4. #20260847C2semidirect
5. #20260845C2semidirect
6. #20260835C2semidirect
7. #20260834C2semidirect
8. #20260826C2semidirect
9. C4×#13095500C2semidirect
10. #20260815C2semidirect
11. #20260776C2semidirect
12. #20260728C2semidirect
13. #102×D97C2semidirect
14. #20260689C2semidirect
15. C4×D8×C97C2semidirect
18. #15690834C4semidirect
24. C4×Dic194C22semidirect
25. #15690775C22semidirect
28. D4×Dic97C22semidirect
41. #15690743C22semidirect
42. #15690742C22semidirect
43. #15690741C22semidirect
44. #15690735C22semidirect
45. #15690733C22semidirect
48. #15690731C22semidirect
50. #15690721C22semidirect
51. #15690720C22semidirect
52. C8×Dic97C22semidirect
55. C4×D4×C97C22semidirect
56. #107×C97C22semidirect
58. #102×C97C22semidirect
59. C4×C8×C97C22semidirect
62. #13095510C2×C4semidirect
68. D4×D97C2×C4semidirect
76. #13095500C2×C4semidirect
77. #13095499C2×C4semidirect
78. #13095496C2×C4semidirect
79. #13095494C2×C4semidirect
105. D8×C97C2×C4semidirect
189. C97#2946semidirect
200. C4×D8D97semidirect
225. C4×D4C2×D97semidirect
234. #107C2×D97semidirect
244. #102C2×D97semidirect
248. C4×C8C2×D97semidirect
272. D8C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202612142
#202612072
#202608472
#202608452
#202608352
#202608342
#202608262
C4×#130955002
#202608152
#202607762
#202607282
#102×D972
#202606892
C4×D8×C972
#294697