Finite Group Databasebeta

← #27151959

Subgroups — #27151959

Order 12416.

370 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#20261212C211yesmaximal, characteristic, complemented
2#20261073C211yesmaximal, characteristic, complemented
3#20261056C211yesmaximal, characteristic
4#20261027C211yesmaximal, characteristic, complemented
5#20261017C211yesmaximal, characteristic, complemented
6#20260954C211yesmaximal, characteristic, complemented
7#20260954C211yesmaximal, characteristic, complemented
8#20260890C211yesmaximal, characteristic, complemented
9#20260873C211yesmaximal, characteristic
10#20260840C211yesmaximal, characteristic, complemented
11#20260835C211yesmaximal, characteristic, complemented
12C4×#13095501C211yesmaximal, characteristic, complemented
13C4×#13095499C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#542×C97C211yesmaximal, characteristic
16C22×Dic194C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095501C2211yescharacteristic
19C2×#13095499C2211yescharacteristic
20C2×#13095499C2211yescharacteristic
21C4×Dic194C2211yescharacteristic
22C4×Dic194C2211yescharacteristic
23C4×Dic194C2211yescharacteristic
24C4×Dic194C2211yescharacteristic
25#15690775—21no—
26#15690775—21no—
27#15690769C2211yescharacteristic
28#15690766—21no—
29#15690766—21no—
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
48C4⋊Q8×C97C2211yescharacteristic
49#126×C97—21no—
50#124×C97—21no—
51#124×C97C2211yescharacteristic
52#123×C97—21no—
53C22⋊Q8×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—22no—
61C2×Dic194—22no—
62C2×Dic194C2311yescharacteristic
63C2×Dic194C2311yescharacteristic
64C22×Dic97—21no—
65#13095501—22no—
66#13095501—22no—
67#13095499—22no—
68#13095499—22no—
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80C4×Dic97—21no—
81C4×Dic97—21no—
82C4×Dic97—21no—
83C4×Dic97—21no—
84C4×Dic97—21no—
85C2×Q8×C97—21no—
86C2×Q8×C97C2311yescharacteristic
87C2×D4×C97—21no—
88C2×D4×C97C2311yescharacteristic
89C22×C4×C97—21no—
90C22×C4×C97C2311yescharacteristic
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97—81no—
113D4×C97—21no—
114D4×C97—21no—
115D4×C97—81no—
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×C97C2×D411yescharacteristic
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C2×Dic97—24no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133Dic194—24no—
134Dic194—21no—
135Dic194—21no—
136Dic194#4112yes—
137Dic194#4112yes—
138#11715895—21no—
139#11715895—21no—
140#11715895—21no—
141#11715895—21no—
142C22×C97—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—41no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97C22×D411yescharacteristic
149C4×C97—21no—
150C4×C97—21no—
151C4×C97—21no—
152C4×C97—21no—
153C4×C97—21no—
154C4×C97—41no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97C2×#4111yescharacteristic
158C4×C97C2×#4111yescharacteristic
159Dic97—22no—
160Dic97—22no—
161Dic97—41no—
162Dic97—41no—
163C2×C97—41no—
164C2×C97—81no—
165C2×C97#52511yescharacteristic
166C2×C97#57711yescharacteristic
167C2×C97#57811yescharacteristic
168#3121—971nomaximal, Sylow
169C97#312111yescharacteristic, complemented, Sylow
170#542D9711yescharacteristic
171#519—971no—
172#485—971no—
173#482—971no—
174#477—971no—
175#475—971no—
176#464—971no—
177#464—971no—
178#451—971no—
179#448—971no—
180#441—971no—
181#440—971no—
182C4×Q16—971no—
183C4×SD16—971no—
184#433—971no—
185C22×Q8—971no—
186C2×Q16—971no—
187C2×Q16—971no—
188C2×SD16—971no—
189C2×SD16—971no—
190C4⋊Q8C2×D9711yescharacteristic
191#126—21no—
192#124—21no—
193#124C2×D9711yescharacteristic
194#123—21no—
195C22⋊Q8—21no—
196C22⋊Q8C2×D9711yescharacteristic
197#121C2×D9711yescharacteristic
198C4×Q8—971no—
199C4×Q8—971no—
200C4×Q8—971no—
201C4×Q8—971no—
202C4×Q8—1941no—
203C4×Q8—1941no—
204C4×Q8C2×D9711yescharacteristic
205C4×D4C2×D9711yescharacteristic
206#117—971no—
207#117—1941no—
208#117—1941no—
209#117C2×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—
228C2×Q8—21no—
229C2×Q8—971no—
230C2×Q8—971no—
231C2×Q8—1942no—
232C2×Q8—1942no—
233C2×Q8—1941no—
234C2×Q8C22×D9711yescharacteristic
235C2×D4—21no—
236C2×D4C22×D9711yescharacteristic
237C22×C4—21no—
238C22×C4—1941no—
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⋊C4C22⋊D9711yescharacteristic
258C4⋊C4C22⋊D9711yescharacteristic
259C4⋊C4C22×D9711yescharacteristic
260C4⋊C4C22×D9711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—1941no—
267C22⋊C4C22⋊D9711yescharacteristic
268C22⋊C4C22⋊D9711yescharacteristic
269C42—21no—
270C42—1941no—
271C42—1941no—
272C42—1941no—
273C42—1941no—
274C42—1941no—
275C42C22×D9711yescharacteristic
276C42C22×D9711yescharacteristic
277C23—21no—
278C23C2×C22⋊D9711yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—81no—
282Q8—972no—
283Q8—972no—
284Q8—1944no—
285Q8—1941no—
286Q8—1941no—
287D4—21no—
288D4—21no—
289D4—81no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1944no—
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—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C22×C22⋊D9711yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—3881no—
327C4—3881no—
328C4—1942no—
329C4—1942no—
330C4#41×D9711yescharacteristic
331C4#41×D9711yescharacteristic
332C2—41no—
333C2—81no—
334C2#2026124211yescharacteristic
335C2#2026154611yescharacteristic
336C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#542×C97C2
socleC22×C97C22×D4
prime radical#121×C97C22
hypercenter#542D97
hypocenterC97#3121
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20261212C2semidirect
2. #20261073C2semidirect
4. #20261027C2semidirect
5. #20261017C2semidirect
6. #20260954C2semidirect
7. #20260954C2semidirect
8. #20260890C2semidirect
10. #20260840C2semidirect
11. #20260835C2semidirect
12. C4×#13095501C2semidirect
13. C4×#13095499C2semidirect
14. #20260796C2semidirect
169. C97#3121semidirect

Maximal subgroups

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

subgroupindex
#202612122
#202610732
#202610562
#202610272
#202610172
#202609542
#202608902
#202608732
#202608402
#202608352
C4×#130955012
C4×#130954992
#202607962
#542×C972
#312197