Finite Group Databasebeta

← #27151950

Subgroups — #27151950

Order 12416.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261347C211yesmaximal, characteristic, complemented
2#20261070C211yesmaximal, characteristic, complemented
3#20261065C211yesmaximal, characteristic, complemented
4#20261042C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20260945C211yesmaximal, characteristic, complemented
7#20260940C211yesmaximal, characteristic, complemented
8#20260897C211yesmaximal, characteristic
9#20260897C211yesmaximal, characteristic
10#20260840C211yesmaximal, characteristic, complemented
11#20260835C211yesmaximal, characteristic, complemented
12C4×#13095501C211yesmaximal, characteristic, complemented
13C4×#13095499C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#520×C97C211yesmaximal, characteristic
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095499C2211yescharacteristic
20#15690810C2211yescharacteristic
21#15690803—21no—
22#15690803—21no—
23#15690802C2211yescharacteristic
24#15690800—21no—
25#15690800—21no—
26#15690789C2211yescharacteristic
27#15690789C2211yescharacteristic
28C4×Dic194C2211yescharacteristic
29C4×Dic194C2211yescharacteristic
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690733C2211yescharacteristic
39#15690733C2211yescharacteristic
40#15690733C2211yescharacteristic
41#15690733C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C2×#41×C97C2211yescharacteristic
49C4×Q8×C97C2211yescharacteristic
50C4×Q8×C97C2211yescharacteristic
51C4×D4×C97—21no—
52C4×D4×C97—21no—
53C4×D4×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55#117×C97—21no—
56#117×C97C2211yescharacteristic
57C2×C4⋊C4×C97—21no—
58C2×C4⋊C4×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60C2×Dic194C2311yescharacteristic
61#13095501—22no—
62#13095501—22no—
63#13095499—22no—
64#13095499—22no—
65C2×#11715895C2311yescharacteristic
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C4⋊Dic97—21no—
70C4⋊Dic97—21no—
71C4⋊Dic97C2311yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73#13095490—21no—
74#13095490—21no—
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80C22⋊Dic97—21no—
81C4×Dic97—21no—
82C4×Dic97—21no—
83#41×C97—22no—
84#41×C97—22no—
85C2×Q8×C97C2311yescharacteristic
86C2×D4×C97—21no—
87C2×D4×C97C2311yescharacteristic
88C22×C4×C97—21no—
89C22×C4×C97—21no—
90C22×C4×C97—21no—
91C22×C4×C97—21no—
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—21no—
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97D411yescharacteristic
97C4⋊C4×C97D411yescharacteristic
98C4⋊C4×C97C2311yescharacteristic
99C4⋊C4×C97C2311yescharacteristic
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97D411yescharacteristic
103C22⋊C4×C97D411yescharacteristic
104C42×C97—21no—
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110Q8×C97—21no—
111Q8×C97#4112yes—
112D4×C97—24no—
113D4×C97—21no—
114D4×C97#4112yes—
115C2×C4×C97—24no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—81no—
122C2×C4×C97—81no—
123C2×C4×C97—81no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134Dic194—21no—
135Dic194—21no—
136Dic194—21no—
137Dic194—21no—
138#11715895—21no—
139#11715895—21no—
140#11715895—21no—
141#11715895—21no—
142C22×C97—21no—
143C22×C97—22no—
144C22×C97—21no—
145C22×C97—21no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97C22×D411yescharacteristic
149C4×C97—21no—
150C4×C97—22no—
151C4×C97—21no—
152C4×C97—21no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97C2×#4111yescharacteristic
158C4×C97C2×#4111yescharacteristic
159Dic97—41no—
160Dic97—41no—
161Dic97—41no—
162Dic97—41no—
163C2×C97—41no—
164C2×C97—42no—
165C2×C97#52511yescharacteristic
166C2×C97#57711yescharacteristic
167C2×C97#57811yescharacteristic
168#3120—971nomaximal, Sylow
169C97#312011yescharacteristic, complemented, Sylow
170#541—971no—
171#520D9711yescharacteristic
172#484—971no—
173#483—971no—
174#479—971no—
175#478—971no—
176#462—971no—
177#461—971no—
178#452—971no—
179#452—971no—
180#441—971no—
181#440—971no—
182C4×Q16—971no—
183C4×SD16—971no—
184#433—971no—
185C2×#41C2×D9711yescharacteristic
186C2×Q16—971no—
187C2×Q16—971no—
188C2×SD16—971no—
189C2×SD16—971no—
190C4⋊Q8—971no—
191#126—1941no—
192#126—1941no—
193#125—971no—
194#125—1941no—
195#125—1941no—
196C22⋊Q8—971no—
197C22⋊Q8—971no—
198C4×Q8—971no—
199C4×Q8—971no—
200C4×Q8C2×D9711yescharacteristic
201C4×Q8C2×D9711yescharacteristic
202C4×D4—21no—
203C4×D4—21no—
204C4×D4C2×D9711yescharacteristic
205C4×D4C2×D9711yescharacteristic
206#117—21no—
207#117C2×D9711yescharacteristic
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×D9711yescharacteristic
210#107—971no—
211#107—971no—
212#106—971no—
213#106—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216#103—971no—
217#103—971no—
218#103—971no—
219#103—971no—
220#103—971no—
221#103—971no—
222#102—971no—
223#102—971no—
224C22⋊C8—971no—
225C22⋊C8—971no—
226#97—971no—
227C4×C8—971no—
228#41—22no—
229#41—22no—
230C2×Q8—971no—
231C2×Q8—971no—
232C2×Q8C22×D9711yescharacteristic
233C2×D4—21no—
234C2×D4C22×D9711yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C22×D9711yescharacteristic
240Q16—1942no—
241Q16—1942no—
242SD16—1942no—
243SD16—1942no—
244C2×C8—971no—
245C2×C8—971no—
246C2×C8—971no—
247C2×C8—971no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—971no—
252C4⋊C4—971no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4C22⋊D9711yescharacteristic
262C4⋊C4C22⋊D9711yescharacteristic
263C4⋊C4C22×D9711yescharacteristic
264C4⋊C4C22×D9711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—1941no—
268C22⋊C4—1941no—
269C22⋊C4C22⋊D9711yescharacteristic
270C22⋊C4C22⋊D9711yescharacteristic
271C42—21no—
272C42—21no—
273C42—1941no—
274C42—1941no—
275C42C22×D9711yescharacteristic
276C42C22×D9711yescharacteristic
277C23—21no—
278C23C2×C22⋊D9711yescharacteristic
279Q8—21no—
280Q8—1941no—
281Q8—1941no—
282Q8—1941no—
283Q8—1941no—
284Q8#1309551112yes—
285D4—24no—
286D4—21no—
287D4#1309551112yes—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4C2×C22⋊D9711yescharacteristic
302C2×C4C2×C22⋊D9711yescharacteristic
303C2×C4C2×C22⋊D9711yescharacteristic
304C2×C4C2×C22⋊D9711yescharacteristic
305C2×C4C2×C22⋊D9711yescharacteristic
306C2×C4C23×D9711yescharacteristic
307C8—1941no—
308C8—1941no—
309C8—1941no—
310C8—1941no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22C22×C22⋊D9711yescharacteristic
318C4—21no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—3881no—
327C4—3881no—
328C4—3881no—
329C4—3881no—
330C4C2×#1309551111yescharacteristic
331C4C2×#1309551111yescharacteristic
332C2—41no—
333C2—42no—
334C2#2026124611yescharacteristic
335C2#2026154611yescharacteristic
336C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#520×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×C97C22
hypercenter#520D97
hypocenterC97#3120
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20261347C2semidirect
2. #20261070C2semidirect
3. #20261065C2semidirect
4. #20261042C2semidirect
5. #20261035C2semidirect
6. #20260945C2semidirect
7. #20260940C2semidirect
10. #20260840C2semidirect
11. #20260835C2semidirect
12. C4×#13095501C2semidirect
13. C4×#13095499C2semidirect
14. #20260796C2semidirect
169. C97#3120semidirect

Maximal subgroups

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

subgroupindex
#202613472
#202610702
#202610652
#202610422
#202610352
#202609452
#202609402
#202608972
#202608402
#202608352
C4×#130955012
C4×#130954992
#202607962
#520×C972
#312097