Finite Group Databasebeta

← #27151571

Subgroups — #27151571

Order 12416.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261362C211yesmaximal, characteristic
2#20261211C211yesmaximal, characteristic
3#20261045C211yesmaximal, characteristic, complemented
4#20261043C211yesmaximal, characteristic, complemented
5#20261042C211yesmaximal, characteristic, complemented
6#20261040C211yesmaximal, characteristic, complemented
7#20261028C211yesmaximal, characteristic, complemented
8#20261027C211yesmaximal, characteristic, complemented
9#20261019C211yesmaximal, characteristic, complemented
10#20260774C211yesmaximal, characteristic
11#20260770C211yesmaximal, characteristic
12#20260760C211yesmaximal, characteristic
13#20260742C211yesmaximal, characteristic
14#103×D97C211yesmaximal, characteristic
15#475×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690808—21no—
21#15690802C2211yescharacteristic
22#15690801C2211yescharacteristic
23#15690800—21no—
24#15690788—21no—
25#15690786—21no—
26Q8×Dic97C2211yescharacteristic
27C4×Dic194—21no—
28C4×Dic194C2211yescharacteristic
29#15690775—21no—
30#15690775C2211yescharacteristic
31#15690768—21no—
32#15690768C2211yescharacteristic
33#15690768C2211yescharacteristic
34#15690767—21no—
35#15690767C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690741C2211yescharacteristic
43C4⋊#11715895C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690734C2211yescharacteristic
49#15690734C2211yescharacteristic
50#15690733C2211yescharacteristic
51#15690733C2211yescharacteristic
52C4⋊Q8×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54#106×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62Q8×D97—22no—
63Q8×D97Q814yescomplemented
64C2×C4×D97—21no—
65C2×C4×D97—21no—
66C2×C4×D97C2311yescharacteristic
67#13095494—22no—
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C8×D97—22no—
71C4⋊Dic97—21no—
72C4⋊Dic97—21no—
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84#13095490D411yescharacteristic
85#13095488—21no—
86#13095488—21no—
87#13095488D411yescharacteristic
88#13095488D411yescharacteristic
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97—21no—
96C2×Q8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C4⋊C4×C97—21no—
100C4⋊C4×C97—21no—
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C42×C97—21no—
105C42×C97C2311yescharacteristic
106C22×D97C2×D411yescharacteristic
107Q8×C97—21no—
108Q8×C97—81no—
109Q8×C97C2×Q812yes—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2411yescharacteristic
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97C2×D411yescharacteristic
122C2×Dic97C2×D411yescharacteristic
123C2×Dic97C2×D411yescharacteristic
124C4×D97—24no—
125C4×D97—81no—
126C4×D97C2×Q812yes—
127C4×D97#4112yes—
128Dic194—21no—
129Dic194—24no—
130Dic194—81no—
131Dic194C2×Q812yes—
132C8×C97—21no—
133C8×C97—21no—
134#11715895—21no—
135#11715895—21no—
136C22×C97C22×D411yescharacteristic
137C2×D97—21no—
138C2×D97—21no—
139C2×D97C22⋊Q812yes—
140C4×C97—22no—
141C4×C97—21no—
142C4×C97—21no—
143C4×C97—41no—
144C4×C97—41no—
145C4×C97—41no—
146C4×C97C22×Q811yescharacteristic
147C4×C97C2×#4111yescharacteristic
148Dic97—22no—
149Dic97—21no—
150Dic97—21no—
151Dic97—41no—
152Dic97—41no—
153Dic97—41no—
154Dic97C22⋊Q812yes—
155C2×C97C2×C22⋊Q811yescharacteristic
156C2×C97#57711yescharacteristic
157C2×C97#57811yescharacteristic
158D97—22no—
159#3082—971nomaximal, Sylow
160C97#308211yescharacteristic, complemented, Sylow
161#544—971no—
162#519—971no—
163#479—971no—
164#479—971no—
165#479—971no—
166#479—971no—
167#477—971no—
168#477—971no—
169#475—971no—
170#475D9711yescharacteristic, complemented
171#428—971no—
172#427—971no—
173#424—971no—
174#419—971no—
175C2×#103—971no—
176C22×Q8—971no—
177C2×M16—971no—
178C22×C8—971no—
179C4⋊Q8—971no—
180C4⋊Q8—1941no—
181C4⋊Q8C2×D9711yescharacteristic
182#125—971no—
183#125—971no—
184#125—1941no—
185C22⋊Q8—1941no—
186C22⋊Q8—1941no—
187C4×Q8—971no—
188C4×Q8—971no—
189C4×Q8—971no—
190C4×Q8—1941no—
191C4×Q8—1941no—
192C4×Q8C2×D9711yescharacteristic
193#117—971no—
194#117—971no—
195#117—971no—
196#117—1941no—
197#117—1941no—
198C2×C4⋊C4—971no—
199#107—971no—
200#107—971no—
201#107—971no—
202#107—971no—
203#106—971no—
204#106—971no—
205#106C2×D9711yescharacteristic
206#106C2×D9711yescharacteristic
207C4⋊C8—971no—
208C4⋊C8—971no—
209C4⋊C8—971no—
210C4⋊C8C2×D9711yescharacteristic
211#103—971no—
212#103—971no—
213#103—971no—
214#103—971no—
215#103—971no—
216#103—971no—
217#103C2×D9711yescharacteristic
218#103C2×D9711yescharacteristic
219C2×Q8—21no—
220C2×Q8—974no—
221C2×Q8—971no—
222C2×Q8—1941no—
223C2×Q8—1941no—
224C2×Q8—1942no—
225C2×Q8C22×D9711yescharacteristic
226C22×C4—971no—
227C22×C4—1941no—
228C22×C4—1941no—
229M16—1942no—
230C2×C8—971no—
231C2×C8—971no—
232C2×C8—1942no—
233C2×C8C22×D9711yescharacteristic
234C2×C8C22×D9711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—971no—
238C4⋊C4—971no—
239C4⋊C4—971no—
240C4⋊C4—971no—
241C4⋊C4—971no—
242C4⋊C4—1941no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4C22×D9711yescharacteristic
252C4⋊C4C22×D9711yescharacteristic
253C4⋊C4C22×D9711yescharacteristic
254C22⋊C4—971no—
255C22⋊C4—971no—
256C22⋊C4—1941no—
257C22⋊C4—1941no—
258C42—21no—
259C42—971no—
260C42—1941no—
261C42—1941no—
262C42—1941no—
263C42—1941no—
264C42—1941no—
265C42C22×D9711yescharacteristic
266C23—971no—
267Q8—21no—
268Q8—81no—
269Q8—7761no—
270Q8—972no—
271Q8—1944no—
272Q8—1941no—
273Q8Q8×D9712yes—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—7761no—
279C2×C4—971no—
280C2×C4—972no—
281C2×C4—972no—
282C2×C4—971no—
283C2×C4—971no—
284C2×C4—1944no—
285C2×C4—1941no—
286C2×C4—1941no—
287C2×C4—1941no—
288C2×C4—1941no—
289C2×C4D4×D9711yescharacteristic
290C2×C4D4×D9711yescharacteristic
291C2×C4C23×D9711yescharacteristic
292C8—21no—
293C8—21no—
294C8—1941no—
295C8—1941no—
296C22—972no—
297C22—1941no—
298C22—1941no—
299C22C2×D4×D9711yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—972no—
307C4—3881no—
308C4—3881no—
309C4—3881no—
310C4—1942no—
311C4—1941no—
312C4—1941no—
313C4C2×Q8×D9711yescharacteristic
314C4#41×D9711yescharacteristic
315C2—1942no—
316C2C22⋊Q8×D9711yescharacteristic
317C2#2026154111yescharacteristic
318C2#2026155111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#475×C97C2
socleC22×C97C22×D4
prime radicalC22×D97C2×D4
hypercenter#475D97
hypocenterC97#3082
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #20261045C2semidirect
4. #20261043C2semidirect
5. #20261042C2semidirect
6. #20261040C2semidirect
7. #20261028C2semidirect
8. #20261027C2semidirect
9. #20261019C2semidirect
15. #475×C97C2semidirect
63. Q8×D97Q8semidirect
160. C97#3082semidirect
170. #475D97semidirect

Maximal subgroups

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

subgroupindex
#202613622
#202612112
#202610452
#202610432
#202610422
#202610402
#202610282
#202610272
#202610192
#202607742
#202607702
#202607602
#202607422
#103×D972
#475×C972
#308297