Finite Group Databasebeta

← #27150574

Subgroups — #27150574

Order 12416.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690842C211yesmaximal, characteristic
2#20261217C211yesmaximal, characteristic
3#20261211C211yesmaximal, characteristic, complemented
4#20260842C211yesmaximal, characteristic, complemented
5#20260840C211yesmaximal, characteristic, complemented
6#20260834C211yesmaximal, characteristic, complemented
7#20260833C211yesmaximal, characteristic, complemented
8#20260827C211yesmaximal, characteristic, complemented
9C4×#13095500C211yesmaximal, characteristic, complemented
10#20260820C211yesmaximal, characteristic, complemented
11#20260776C211yesmaximal, characteristic, complemented
12#20260730C211yesmaximal, characteristic
13#103×D97C211yesmaximal, characteristic, complemented
14#20260689C211yesmaximal, characteristic, complemented
15C4×Q16×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18#15690842C418yescomplemented
19C2×#13095501C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21C2×#13095498C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
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
36#15690767—21no—
37#15690767—21no—
38#15690767C2211yescharacteristic
39C4⋊C4×D97—21no—
40C4⋊C4×D97C2211yescharacteristic
41#15690743C2211yescharacteristic
42#15690742C2211yescharacteristic
43#15690741C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690721C2211yescharacteristic
51#15690720C2211yescharacteristic
52C8×Dic97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55C4×Q8×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59C4×C8×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×D97C2311yescharacteristic
74#13095501C2×C414yescomplemented
75#13095500C2×C414yescomplemented
76#13095498C2×C414yescomplemented
77#13095494C2×C414yescomplemented
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97—22no—
81C4⋊Dic97—21no—
82C4⋊Dic97—21no—
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490D411yescharacteristic
89#13095490D411yescharacteristic
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488D411yescharacteristic
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C4×Dic97—21no—
99C4×Dic97—21no—
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—81no—
138C4×D97—81no—
139C4×D97—81no—
140C4×D97C22×C412yes—
141C4×D97#4112yes—
142Dic194—21no—
143Dic194—24no—
144Dic194C22×C412yes—
145C8×C97—21no—
146C8×C97C22×C412yes—
147#11715895—21no—
148#11715895C22×C412yes—
149C22×C97C22×D411yescharacteristic
150C2×D97—21no—
151C2×D97—21no—
152C2×D97—22no—
153C2×D97—41no—
154C2×D97—41no—
155C2×D97C4×D412yes—
156C4×C97—22no—
157C4×C97—22no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97C23×C411yescharacteristic
163C4×C97C2×#4111yescharacteristic
164Dic97—22no—
165Dic97—21no—
166Dic97—21no—
167Dic97—41no—
168Dic97—41no—
169Dic97C4×D412yes—
170C2×C97C2×C4×D411yescharacteristic
171C2×C97#57711yescharacteristic
172C2×C97#57811yescharacteristic
173D97—22no—
174D97—42no—
175#2947—971nomaximal, Sylow
176C97#294711yescharacteristic, complemented, Sylow
177C2×#137—971no—
178#520—971no—
179#519—971no—
180#441—971no—
181#441—971no—
182#440—971no—
183#440—971no—
184C4×Q16—971no—
185C4×Q16D9711yescharacteristic, complemented
186C4×SD16—971no—
187C4×SD16—971no—
188#428—971no—
189#416—971no—
190C2×#103—971no—
191#405—971no—
192C2×#41—971no—
193C22×Q8—971no—
194#137—978no—
195C2×Q16—971no—
196C2×Q16C2×D9711yescharacteristic
197C2×SD16—971no—
198C2×SD16—971no—
199C2×M16—971no—
200C22×C8—971no—
201C4×Q8—971no—
202C4×Q8—971no—
203C4×Q8—971no—
204C4×Q8—971no—
205C4×Q8—1941no—
206C4×Q8—1941no—
207C4×Q8C2×D9711yescharacteristic
208C4×Q8C2×D9711yescharacteristic
209C4×D4—971no—
210C4×D4—971no—
211C4×D4—1941no—
212C4×D4—1941no—
213#117—971no—
214#117—971no—
215#117—1941no—
216#117—1941no—
217#117—1941no—
218C2×C4⋊C4—971no—
219C2×C4⋊C4—1941no—
220#107—971no—
221#107C2×D9711yescharacteristic
222#106—971no—
223#106—971no—
224#103—971no—
225#103—971no—
226#103—971no—
227#103—971no—
228#103C2×D9711yescharacteristic
229#103C2×D9711yescharacteristic
230#102—971no—
231#102—971no—
232#97—971no—
233#97—971no—
234C4×C8—971no—
235C4×C8C2×D9711yescharacteristic
236#41—974no—
237#41—1942no—
238C2×Q8—974no—
239C2×Q8—971no—
240C2×Q8—1941no—
241C2×Q8—1942no—
242C2×Q8C22×D9711yescharacteristic
243C2×Q8C22×D9711yescharacteristic
244C2×D4—971no—
245C2×D4—1941no—
246C22×C4—971no—
247C22×C4—1941no—
248C22×C4—1941no—
249C22×C4—1941no—
250C22×C4—1941no—
251C22×C4—1941no—
252Q16—974no—
253Q16C4×D9714yescomplemented
254SD16—974no—
255SD16—974no—
256M16—974no—
257C2×C8—971no—
258C2×C8—971no—
259C2×C8—1942no—
260C2×C8C22×D9711yescharacteristic
261C2×C8C22×D9711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22×D9711yescharacteristic
274C4⋊C4C22×D9711yescharacteristic
275C22⋊C4—971no—
276C22⋊C4—971no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C42—21no—
281C42—21no—
282C42—971no—
283C42—1941no—
284C42—1941no—
285C42—1941no—
286C42—1941no—
287C42—1941no—
288C42C22×D9711yescharacteristic
289C23—971no—
290C23—1941no—
291Q8—21no—
292Q8—21no—
293Q8—972no—
294Q8—1944no—
295Q8—1941no—
296Q8C2×C4×D9712yes—
297Q8C2×C4×D9712yes—
298D4—972no—
299D4—1944no—
300D4—1941no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—7761no—
306C2×C4—7761no—
307C2×C4—7761no—
308C2×C4—971no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—971no—
312C2×C4—971no—
313C2×C4—1944no—
314C2×C4—1944no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4D4×D9711yescharacteristic
319C2×C4D4×D9711yescharacteristic
320C2×C4C23×D9711yescharacteristic
321C8—21no—
322C8—972no—
323C8—1941no—
324C8C2×C4×D9712yes—
325C22—972no—
326C22—3881no—
327C22—3881no—
328C22—1942no—
329C22—1941no—
330C22—1941no—
331C22C2×D4×D9711yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—972no—
339C4—3881no—
340C4—3881no—
341C4—1942no—
342C4—1941no—
343C4—1941no—
344C4C22×C4×D9711yescharacteristic
345C4#41×D9711yescharacteristic
346C2—3882no—
347C2—1942no—
348C2C4×D4×D9711yescharacteristic
349C2#2026154211yescharacteristic
350C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC4×Q16×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095512C22
cosocleC2×C4×C97C24
hypercenterC4×Q16D97
hypocenterC97#2947
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #20261211C2semidirect
4. #20260842C2semidirect
5. #20260840C2semidirect
6. #20260834C2semidirect
7. #20260833C2semidirect
8. #20260827C2semidirect
9. C4×#13095500C2semidirect
10. #20260820C2semidirect
11. #20260776C2semidirect
13. #103×D97C2semidirect
14. #20260689C2semidirect
15. C4×Q16×C97C2semidirect
18. #15690842C4semidirect
65. Q8×D97C2×C4semidirect
74. #13095501C2×C4semidirect
75. #13095500C2×C4semidirect
76. #13095498C2×C4semidirect
77. #13095494C2×C4semidirect
103. Q16×C97C2×C4semidirect
176. C97#2947semidirect
185. C4×Q16D97semidirect
253. Q16C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
#202612172
#202612112
#202608422
#202608402
#202608342
#202608332
#202608272
C4×#130955002
#202608202
#202607762
#202607302
#103×D972
#202606892
C4×Q16×C972
#294797