Finite Group Databasebeta

← #27152263

Subgroups — #27152263

Order 12416.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261365C211yesmaximal, characteristic
2#20261362C211yesmaximal, characteristic
3#20261144C211yesmaximal, characteristic, complemented
4#20261143C211yesmaximal, characteristic, complemented
5#20261142C211yesmaximal, characteristic, complemented
6#20261136C211yesmaximal, characteristic, complemented
7#20261134C211yesmaximal, characteristic, complemented
8#20261132C211yesmaximal, characteristic, complemented
9#20261131C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic
11#20260774C211yesmaximal, characteristic
12#20260772C211yesmaximal, characteristic
13#106×D97C211yesmaximal, characteristic
14#20260692C211yesmaximal, characteristic
15#501×C97C211yesmaximal, characteristic, complemented
16C2×#13095494C2211yescharacteristic
17C2×C8×D97C2211yescharacteristic
18#15690810C2211yescharacteristic
19#15690809—21no—
20#15690809C2211yescharacteristic
21#15690809C2211yescharacteristic
22#15690808—21no—
23#15690802C2211yescharacteristic
24#15690801C2211yescharacteristic
25#15690801C2211yescharacteristic
26#15690800—21no—
27#15690800—21no—
28#15690790—21no—
29#15690788—21no—
30#15690788—21no—
31#15690786—21no—
32#15690768C2211yescharacteristic
33#15690768C2211yescharacteristic
34#15690767C2211yescharacteristic
35C4⋊C4×D97C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690744C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690742C2211yescharacteristic
43#15690741C2211yescharacteristic
44#15690741C2211yescharacteristic
45#15690741C2211yescharacteristic
46#15690741C2211yescharacteristic
47#15690741C2211yescharacteristic
48#15690741C2211yescharacteristic
49#15690720C2211yescharacteristic
50C4×#11715895C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C4⋊Q8×C97C2211yescharacteristic
53#125×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic
57#106×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×C4×D97—21no—
63C2×C4×D97—21no—
64C2×C4×D97C2311yescharacteristic
65#13095494Q814yescomplemented
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C8×D97Q814yescomplemented
69C4⋊Dic97—21no—
70C4⋊Dic97—21no—
71C4⋊Dic97—21no—
72C4⋊Dic97C2311yescharacteristic
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#13095490—21no—
84#13095490D411yescharacteristic
85#13095490D411yescharacteristic
86#13095488—21no—
87#13095488—21no—
88#13095488D411yescharacteristic
89#13095488D411yescharacteristic
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97C2311yescharacteristic
93C2×Q8×C97—21no—
94C2×C8×C97C2311yescharacteristic
95C2×C8×C97C2311yescharacteristic
96C4⋊C4×C97—21no—
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97C2311yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C42×C97C2311yescharacteristic
104C22×D97C2×D411yescharacteristic
105Q8×C97—81no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97C2×D411yescharacteristic
111C2×C4×C97C2×D411yescharacteristic
112C2×C4×C97C2411yescharacteristic
113C2×Dic97—21no—
114C2×Dic97—21no—
115C2×Dic97—21no—
116C2×Dic97—21no—
117C2×Dic97C2×D411yescharacteristic
118C2×Dic97C2×D411yescharacteristic
119C2×Dic97C2×D411yescharacteristic
120C4×D97—81no—
121C4×D97—81no—
122C4×D97C2×Q812yes—
123C4×D97C2×Q812yes—
124Dic194—81no—
125Dic194—81no—
126Dic194—81no—
127C8×C97C2×Q812yes—
128C8×C97C2×Q812yes—
129#11715895C2×Q812yes—
130#11715895C2×Q812yes—
131C22×C97C22×D411yescharacteristic
132C2×D97—21no—
133C2×D97—21no—
134C2×D97C4⋊Q812yes—
135C4×C97—21no—
136C4×C97—21no—
137C4×C97—41no—
138C4×C97—41no—
139C4×C97—41no—
140C4×C97—41no—
141C4×C97C22×Q811yescharacteristic
142C4×C97C22×Q811yescharacteristic
143Dic97—21no—
144Dic97—21no—
145Dic97—41no—
146Dic97—41no—
147Dic97—41no—
148Dic97—41no—
149Dic97C4⋊Q812yes—
150C2×C97C2×C4⋊Q811yescharacteristic
151C2×C97#57711yescharacteristic
152C2×C97#57711yescharacteristic
153D97—22no—
154#3163—971nomaximal, Sylow
155C97#316311yescharacteristic, complemented, Sylow
156#544—971no—
157#544—971no—
158#501—971no—
159#501—971no—
160#501—971no—
161#501D9711yescharacteristic, complemented
162#499—971no—
163#499—971no—
164#498—971no—
165#498—971no—
166#428—971no—
167#428—971no—
168#427—971no—
169C2×#106—971no—
170#405—971no—
171C2×M16—971no—
172C22×C8—971no—
173C4⋊Q8—971no—
174C4⋊Q8—971no—
175C4⋊Q8—971no—
176C4⋊Q8—1941no—
177C4⋊Q8—1941no—
178C4⋊Q8C2×D9711yescharacteristic
179#125—971no—
180#125—971no—
181#125—971no—
182#125—1941no—
183#125—1941no—
184#125C2×D9711yescharacteristic
185C22⋊Q8—1941no—
186C22⋊Q8—1941no—
187C22⋊Q8—1941no—
188C22⋊Q8—1941no—
189#117—971no—
190#117—971no—
191#117—971no—
192C2×C4⋊C4—971no—
193C2×C4⋊C4—971no—
194#107—971no—
195#107—971no—
196#107—971no—
197#107—971no—
198#107C2×D9711yescharacteristic
199#107C2×D9711yescharacteristic
200#106—971no—
201#106—971no—
202#106—971no—
203#106—971no—
204#106—971no—
205#106—971no—
206#106—971no—
207#106—971no—
208#106C2×D9711yescharacteristic
209#106C2×D9711yescharacteristic
210#97—971no—
211#97C2×D9711yescharacteristic
212C4×C8—971no—
213C4×C8—971no—
214C2×Q8—21no—
215C2×Q8—1941no—
216C2×Q8—1941no—
217C2×Q8—1941no—
218C22×C4—971no—
219C22×C4—1941no—
220C22×C4—1941no—
221M16—974no—
222C2×C8—971no—
223C2×C8—971no—
224C2×C8—974no—
225C2×C8C22×D9711yescharacteristic
226C2×C8C22×D9711yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—971no—
231C4⋊C4—971no—
232C4⋊C4—971no—
233C4⋊C4—971no—
234C4⋊C4—971no—
235C4⋊C4—971no—
236C4⋊C4—1941no—
237C4⋊C4—1941no—
238C4⋊C4—1941no—
239C4⋊C4—1941no—
240C4⋊C4—1941no—
241C4⋊C4—1941no—
242C4⋊C4—1941no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4C22×D9711yescharacteristic
248C4⋊C4C22×D9711yescharacteristic
249C4⋊C4C22×D9711yescharacteristic
250C4⋊C4C22×D9711yescharacteristic
251C22⋊C4—971no—
252C22⋊C4—971no—
253C22⋊C4—1941no—
254C22⋊C4—1941no—
255C42—971no—
256C42—1941no—
257C42—1941no—
258C42C22×D9711yescharacteristic
259C23—971no—
260Q8—81no—
261Q8—7761no—
262Q8—7761no—
263Q8—7761no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—7761no—
269C2×C4—7761no—
270C2×C4—971no—
271C2×C4—972no—
272C2×C4—972no—
273C2×C4—971no—
274C2×C4—971no—
275C2×C4—1941no—
276C2×C4—1941no—
277C2×C4—1941no—
278C2×C4—1941no—
279C2×C4D4×D9711yescharacteristic
280C2×C4D4×D9711yescharacteristic
281C2×C4C23×D9711yescharacteristic
282C8—972no—
283C8—972no—
284C8Q8×D9712yes—
285C8Q8×D9712yes—
286C22—972no—
287C22—1941no—
288C22—1941no—
289C22C2×D4×D9711yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—972no—
297C4—3881no—
298C4—3881no—
299C4—3881no—
300C4—3881no—
301C4—1941no—
302C4—1941no—
303C4C2×Q8×D9711yescharacteristic
304C4C2×Q8×D9711yescharacteristic
305C2—1942no—
306C2C4⋊Q8×D9711yescharacteristic
307C2#2026154411yescharacteristic
308C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#501×C97C2
socleC22×C97C22×D4
prime radicalC22×D97C2×D4
hypercenter#501D97
hypocenterC97#3163
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #20261144C2semidirect
4. #20261143C2semidirect
5. #20261142C2semidirect
6. #20261136C2semidirect
7. #20261134C2semidirect
8. #20261132C2semidirect
9. #20261131C2semidirect
15. #501×C97C2semidirect
65. #13095494Q8semidirect
68. C8×D97Q8semidirect
155. C97#3163semidirect
161. #501D97semidirect

Maximal subgroups

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

subgroupindex
#202613652
#202613622
#202611442
#202611432
#202611422
#202611362
#202611342
#202611322
#202611312
#202607762
#202607742
#202607722
#106×D972
#202606922
#501×C972
#316397