Finite Group Databasebeta

← #27150552

Subgroups — #27150552

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690842C211yesmaximal, characteristic
2C4×#13095512C211yesmaximal, characteristic
3C4×Q8×D97C211yesmaximal, characteristic, complemented
4#20260840C211yesmaximal, characteristic, complemented
5#20260837C211yesmaximal, characteristic, complemented
6#20260833C211yesmaximal, characteristic, complemented
7Q16×Dic97C211yesmaximal, characteristic, complemented
8#20260827C211yesmaximal, characteristic, complemented
9#20260822C211yesmaximal, characteristic, complemented
10#20260820C211yesmaximal, characteristic, complemented
11#20260774C211yesmaximal, characteristic, complemented
12#20260742C211yesmaximal, characteristic, complemented
13#20260735C211yesmaximal, characteristic
14#20260687C211yesmaximal, characteristic, complemented
15#441×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18#15690842C418yescomplemented
19C2×#13095501C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21C2×#13095498C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×#13095494C2211yescharacteristic
24Q8×Dic97C2211yescharacteristic
25Q8×Dic97C2211yescharacteristic
26C4×Dic194—21no—
27C4×Dic194C2211yescharacteristic
28#15690775—21no—
29#15690775C2211yescharacteristic
30C4×D388—21no—
31C4×D388C2211yescharacteristic
32#15690772—21no—
33#15690772C2211yescharacteristic
34#15690768—21no—
35#15690768C2211yescharacteristic
36C4⋊C4×D97—21no—
37C4⋊C4×D97C2211yescharacteristic
38C42×D97—21no—
39C42×D97—21no—
40C42×D97C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690743C2211yescharacteristic
43#15690741C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690721C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C8×Dic97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55C4×Q8×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59#97×C97C2211yescharacteristic
60#13095512—22no—
61#13095512C2×C414yes—
62C2×Dic194—21no—
63C2×Dic194C2311yescharacteristic
64Q8×D97—22no—
65Q8×D97C2×C414yescomplemented
66C2×D388—21no—
67C2×D388C2311yescharacteristic
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97D411yescharacteristic
74C2×C4×D97D411yescharacteristic
75C2×C4×D97C2311yescharacteristic
76#13095501C2×C414yescomplemented
77#13095500C2×C414yescomplemented
78#13095498C2×C414yescomplemented
79#13095494—22no—
80#13095494C2×C414yescomplemented
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C4⋊Dic97—21no—
84C4⋊Dic97—21no—
85C4⋊Dic97C2311yescharacteristic
86C4⋊Dic97C2311yescharacteristic
87#13095490—21no—
88#13095490—21no—
89#13095490—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C4×Dic97D411yescharacteristic
99C4×Dic97D411yescharacteristic
100C4×Dic97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C2×Q8×C97C2311yescharacteristic
103Q16×C97C2×C414yescomplemented
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C4⋊C4×C97—21no—
107C4⋊C4×C97—21no—
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C42×C97—21no—
111C42×C97—21no—
112C42×C97C2311yescharacteristic
113C22×D97—21no—
114C22×D97C2×D411yescharacteristic
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97C22×C412yes—
118Q8×C97C22×C412yes—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97C2×D411yescharacteristic
130C2×Dic97C2×D411yescharacteristic
131C2×Dic97C2×D411yescharacteristic
132D388—24no—
133D388—21no—
134D388C22×C412yes—
135C4×D97—24no—
136C4×D97—24no—
137C4×D97—22no—
138C4×D97—41no—
139C4×D97—42no—
140C4×D97—42no—
141C4×D97—42no—
142C4×D97C22×C412yes—
143C4×D97#4112yes—
144Dic194—21no—
145Dic194—24no—
146Dic194C22×C412yes—
147C8×C97—21no—
148C8×C97C22×C412yes—
149#11715895—21no—
150#11715895C22×C412yes—
151C22×C97C22×D411yescharacteristic
152C2×D97—21no—
153C2×D97—21no—
154C2×D97—22no—
155C2×D97—41no—
156C2×D97—41no—
157C2×D97C4×D412yes—
158C4×C97—22no—
159C4×C97—22no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97—41no—
163C4×C97—41no—
164C4×C97C23×C411yescharacteristic
165C4×C97C2×#4111yescharacteristic
166Dic97—22no—
167Dic97—21no—
168Dic97—41no—
169Dic97—41no—
170Dic97C4×D412yes—
171Dic97#13712yescomplemented
172C2×C97C2×C4×D411yescharacteristic
173C2×C97C2×#13711yescharacteristic
174C2×C97#57511yescharacteristic
175D97—22no—
176D97—42no—
177C4×#137—971nomaximal, Sylow
178C97C4×#13711yescharacteristic, complemented, Sylow
179C2×#137—971no—
180C4×#41—971no—
181C2×C4×Q8—971no—
182#441—971no—
183#441D9711yescharacteristic, complemented
184#440—971no—
185#440—971no—
186C4×Q16—971no—
187C4×Q16—971no—
188C4×SD16—971no—
189C4×SD16—971no—
190#428—971no—
191#419—971no—
192#417—971no—
193C4×M16—971no—
194C2×#41—971no—
195C22×Q8—971no—
196#137—978no—
197C2×Q16—971no—
198C2×Q16C2×D9711yescharacteristic
199C2×SD16—971no—
200C2×SD16—971no—
201C2×M16—971no—
202C2×M16—971no—
203C4×Q8—971no—
204C4×Q8—971no—
205C4×Q8—971no—
206C4×Q8—971no—
207C4×Q8—1941no—
208C4×Q8—1941no—
209C4×Q8C2×D9711yescharacteristic
210C4×Q8C2×D9711yescharacteristic
211C4×D4—971no—
212C4×D4—971no—
213C4×D4—1941no—
214C4×D4—1941no—
215#117—971no—
216#117—1941no—
217C2×C4⋊C4—971no—
218C2×C4⋊C4—1941no—
219C2×C42—971no—
220C2×C42—1941no—
221C2×C42—1941no—
222#107—971no—
223#107—971no—
224#106—971no—
225#106C2×D9711yescharacteristic
226#103—971no—
227#103—971no—
228#103—971no—
229#103—971no—
230#103C2×D9711yescharacteristic
231#103C2×D9711yescharacteristic
232#102—971no—
233#102—971no—
234#97—971no—
235#97C2×D9711yescharacteristic
236C4×C8—971no—
237C4×C8—971no—
238#41—974no—
239#41—1942no—
240C2×Q8—974no—
241C2×Q8—971no—
242C2×Q8—1941no—
243C2×Q8—1942no—
244C2×Q8C22×D9711yescharacteristic
245C2×Q8C22×D9711yescharacteristic
246C2×D4—971no—
247C2×D4—1941no—
248C22×C4—971no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256Q16—974no—
257Q16C4×D9714yescomplemented
258SD16—974no—
259SD16—974no—
260M16—974no—
261M16—1942no—
262C2×C8—971no—
263C2×C8—971no—
264C2×C8C22×D9711yescharacteristic
265C2×C8C22×D9711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4—1941no—
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C42—21no—
281C42—21no—
282C42—971no—
283C42—971no—
284C42—971no—
285C42—1941no—
286C42—1941no—
287C42—1941no—
288C42—1941no—
289C42—1941no—
290C42C22×D9711yescharacteristic
291C23—971no—
292C23—1941no—
293Q8—21no—
294Q8—21no—
295Q8—972no—
296Q8—1944no—
297Q8—1941no—
298Q8C2×C4×D9712yes—
299Q8C2×C4×D9712yes—
300D4—972no—
301D4—1944no—
302D4—1941no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—971no—
308C2×C4—971no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—971no—
312C2×C4—3882no—
313C2×C4—3881no—
314C2×C4—3882no—
315C2×C4—3882no—
316C2×C4—1942no—
317C2×C4—1944no—
318C2×C4—1944no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4D4×D9711yescharacteristic
323C2×C4D4×D9711yescharacteristic
324C2×C4C23×D9711yescharacteristic
325C8—21no—
326C8—972no—
327C8—1941no—
328C8C2×C4×D9712yes—
329C22—972no—
330C22—3881no—
331C22—3881no—
332C22—1942no—
333C22—1941no—
334C22—1941no—
335C22C2×D4×D9711yescharacteristic
336C4—22no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—972no—
343C4—972no—
344C4—3881no—
345C4—3881no—
346C4—1942no—
347C4—1941no—
348C4C22×C4×D9711yescharacteristic
349C4#41×D9711yescharacteristic
350C2—3882no—
351C2—1942no—
352C2C4×D4×D9711yescharacteristic
353C2#137×D9711yescharacteristic
354C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#441×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095512C22
cosocleC2×C4×C97C24
hypercenter#441D97
hypocenterC97C4×#137
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. C4×Q8×D97C2semidirect
4. #20260840C2semidirect
5. #20260837C2semidirect
6. #20260833C2semidirect
7. Q16×Dic97C2semidirect
8. #20260827C2semidirect
9. #20260822C2semidirect
10. #20260820C2semidirect
11. #20260774C2semidirect
12. #20260742C2semidirect
14. #20260687C2semidirect
15. #441×C97C2semidirect
18. #15690842C4semidirect
65. Q8×D97C2×C4semidirect
76. #13095501C2×C4semidirect
77. #13095500C2×C4semidirect
78. #13095498C2×C4semidirect
80. #13095494C2×C4semidirect
103. Q16×C97C2×C4semidirect
171. Dic97#137semidirect
178. C97C4×#137semidirect
183. #441D97semidirect
257. Q16C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
C4×#130955122
C4×Q8×D972
#202608402
#202608372
#202608332
Q16×Dic972
#202608272
#202608222
#202608202
#202607742
#202607422
#202607352
#202606872
#441×C972
C4×#13797