Finite Group Databasebeta

← #27150608

Subgroups — #27150608

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690831C211yesmaximal, characteristic
2#20261217C211yesmaximal, characteristic, complemented
3#20261214C211yesmaximal, characteristic, complemented
4#20260837C211yesmaximal, characteristic, complemented
5#20260836C211yesmaximal, characteristic, complemented
6#20260834C211yesmaximal, characteristic, complemented
7C4×#13095501C211yesmaximal, characteristic, complemented
8#20260827C211yesmaximal, characteristic, complemented
9C4×#13095496C211yesmaximal, characteristic, complemented
10#20260815C211yesmaximal, characteristic, complemented
11#107×D97C211yesmaximal, characteristic, complemented
12#20260735C211yesmaximal, characteristic, complemented
13#20260733C211yesmaximal, characteristic, complemented
14#20260692C211yesmaximal, characteristic, complemented
15#440×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095510C2211yescharacteristic
18#15690831C418yescomplemented
19C2×#13095501C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21C2×#13095496C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
24Q8×Dic97C2211yescharacteristic
25C4×Dic194C2211yescharacteristic
26#15690775C2211yescharacteristic
27C4×D388—21no—
28C4×D388C2211yescharacteristic
29D4×Dic97C2211yescharacteristic
30#15690772—21no—
31#15690772C2211yescharacteristic
32C4×C22⋊D97—21no—
33#15690770—21no—
34#15690767—21no—
35#15690767C2211yescharacteristic
36#15690766—21no—
37C2×#13095490—21no—
38C4⋊C4×D97—21no—
39C4⋊C4×D97C2211yescharacteristic
40C4⋊C4×D97C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690743C2211yescharacteristic
43#15690743C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690720C2211yescharacteristic
51C4×#11715895C2211yescharacteristic
52C8×Dic97C2211yescharacteristic
53C2×SD16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55C4×D4×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59#97×C97C2211yescharacteristic
60#13095512—22no—
61#13095512C2×C414yescomplemented
62#13095510—22no—
63#13095510C2×C414yescomplemented
64C2×Dic194C2311yescharacteristic
65C2×D388—21no—
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C22×Dic97—21no—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97C2311yescharacteristic
76#13095501C2×C414yescomplemented
77#13095500C2×C414yescomplemented
78#13095496C2×C414yescomplemented
79#13095494—22no—
80C2×#11715895C2311yescharacteristic
81C2×#11715895C2311yescharacteristic
82C8×D97C2×C414yescomplemented
83C4⋊Dic97—21no—
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2311yescharacteristic
86#13095490—21no—
87#13095490—21no—
88#13095490—21no—
89#13095490—21no—
90#13095490D411yescharacteristic
91#13095490D411yescharacteristic
92C22⋊Dic97—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488D411yescharacteristic
96#13095488D411yescharacteristic
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97—21no—
100C4×Dic97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104SD16×C97C2×C414yescomplemented
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97—21no—
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C42×C97—21no—
112C42×C97C2311yescharacteristic
113C23×C97—21no—
114C22×D97—21no—
115C22×D97C2×D411yescharacteristic
116Q8×C97—21no—
117Q8×C97C22×C412yes—
118D4×C97—21no—
119D4×C97C22×C412yes—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—81no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C22⋊D97—24no—
128C2×Dic97—21no—
129C2×Dic97—24no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—81no—
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135C2×Dic97C2×D411yescharacteristic
136D388—24no—
137D388—21no—
138D388C22×C412yes—
139C4×D97—24no—
140C4×D97—81no—
141C4×D97—81no—
142C4×D97—81no—
143C4×D97—81no—
144C4×D97C22×C412yes—
145C4×D97#4112yes—
146Dic194—21no—
147Dic194C22×C412yes—
148C8×C97—21no—
149C8×C97C22×C412yes—
150#11715895—21no—
151#11715895C22×C412yes—
152C22×C97—22no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97C22×D411yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—22no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97C4×D412yes—
162C4×C97—22no—
163C4×C97—21no—
164C4×C97—21no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97C23×C411yescharacteristic
168C4×C97C2×#4111yescharacteristic
169Dic97—22no—
170Dic97—21no—
171Dic97—21no—
172Dic97—41no—
173Dic97—41no—
174Dic97C4×D412yes—
175C2×C97—42no—
176C2×C97C2×C4×D411yescharacteristic
177C2×C97#57611yescharacteristic
178C2×C97#57811yescharacteristic
179D97—22no—
180D97—42no—
181#2951—971nomaximal, Sylow
182C97#295111yescharacteristic, complemented, Sylow
183C2×#135—971no—
184#520—971no—
185#520—971no—
186#440—971no—
187#440—971no—
188#440—971no—
189#440D9711yescharacteristic, complemented
190C4×Q16—971no—
191C4×Q16—971no—
192C4×D8—971no—
193C4×D8—971no—
194C2×#107—971no—
195#417—971no—
196#417—971no—
197#405—971no—
198C2×#41—971no—
199C2×#41—971no—
200#135—978no—
201C2×Q16—971no—
202C2×SD16—971no—
203C2×SD16C2×D9711yescharacteristic
204C2×D8—971no—
205C2×M16—971no—
206C22×C8—971no—
207C4×Q8—971no—
208C4×Q8—971no—
209C4×Q8—971no—
210C4×Q8C2×D9711yescharacteristic
211C4×D4—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—1941no—
215C4×D4—1941no—
216C4×D4—1941no—
217C4×D4—1941no—
218C4×D4C2×D9711yescharacteristic
219#117—971no—
220#117—1941no—
221#117—1941no—
222C2×C4⋊C4—971no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—1941no—
225C2×C4⋊C4—1941no—
226#107—971no—
227#107—971no—
228#107—971no—
229#107C2×D9711yescharacteristic
230#103—971no—
231#103—971no—
232#103—971no—
233#103C2×D9711yescharacteristic
234#102—971no—
235#102—971no—
236#102—971no—
237#102C2×D9711yescharacteristic
238#97—971no—
239#97C2×D9711yescharacteristic
240C4×C8—971no—
241C4×C8—971no—
242#41—974no—
243#41—974no—
244#41—1942no—
245#41—1942no—
246C2×Q8—971no—
247C2×Q8C22×D9711yescharacteristic
248C2×D4—971no—
249C2×D4—1941no—
250C2×D4—1941no—
251C2×D4C22×D9711yescharacteristic
252C22×C4—21no—
253C22×C4—971no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4—1941no—
257C22×C4—1941no—
258C22×C4—1941no—
259C22×C4—1941no—
260C22×C4—1941no—
261Q16—974no—
262SD16—974no—
263SD16C4×D9714yescomplemented
264D8—974no—
265M16—1942no—
266C2×C8—971no—
267C2×C8—974no—
268C2×C8—971no—
269C2×C8C22×D9711yescharacteristic
270C2×C8C22×D9711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—1941no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4C22×D9711yescharacteristic
282C4⋊C4C22×D9711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—971no—
285C22⋊C4—971no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C42—21no—
290C42—971no—
291C42—1941no—
292C42—1941no—
293C42—1941no—
294C42C22×D9711yescharacteristic
295C23—21no—
296C23—971no—
297C23—1941no—
298Q8—21no—
299Q8—972no—
300Q8—1941no—
301Q8C2×C4×D9712yes—
302D4—21no—
303D4—972no—
304D4—1944no—
305D4—1944no—
306D4—1941no—
307D4C2×C4×D9712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—7761no—
313C2×C4—7761no—
314C2×C4—7761no—
315C2×C4—7761no—
316C2×C4—7761no—
317C2×C4—971no—
318C2×C4—972no—
319C2×C4—972no—
320C2×C4—971no—
321C2×C4—971no—
322C2×C4—1944no—
323C2×C4—1944no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4D4×D9711yescharacteristic
328C2×C4D4×D9711yescharacteristic
329C2×C4C23×D9711yescharacteristic
330C8—21no—
331C8—972no—
332C8—1941no—
333C8C2×C4×D9712yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—972no—
338C22—3881no—
339C22—3881no—
340C22—1942no—
341C22—1941no—
342C22—1941no—
343C22C2×D4×D9711yescharacteristic
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—972no—
350C4—3881no—
351C4—3881no—
352C4—1942no—
353C4—1941no—
354C4—1941no—
355C4C22×C4×D9711yescharacteristic
356C4#41×D9711yescharacteristic
357C2—42no—
358C2—3882no—
359C2—1942no—
360C2C4×D4×D9711yescharacteristic
361C2#2026153911yescharacteristic
362C2#2026155111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#440×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690831C2
cosocleC2×C4×C97C24
hypercenter#440D97
hypocenterC97#2951
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20261217C2semidirect
3. #20261214C2semidirect
4. #20260837C2semidirect
5. #20260836C2semidirect
6. #20260834C2semidirect
7. C4×#13095501C2semidirect
8. #20260827C2semidirect
9. C4×#13095496C2semidirect
10. #20260815C2semidirect
11. #107×D97C2semidirect
12. #20260735C2semidirect
13. #20260733C2semidirect
14. #20260692C2semidirect
15. #440×C97C2semidirect
18. #15690831C4semidirect
61. #13095512C2×C4semidirect
63. #13095510C2×C4semidirect
76. #13095501C2×C4semidirect
77. #13095500C2×C4semidirect
78. #13095496C2×C4semidirect
82. C8×D97C2×C4semidirect
104. SD16×C97C2×C4semidirect
182. C97#2951semidirect
189. #440D97semidirect
263. SD16C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908312
#202612172
#202612142
#202608372
#202608362
#202608342
C4×#130955012
#202608272
C4×#130954962
#202608152
#107×D972
#202607352
#202607332
#202606922
#440×C972
#295197