Finite Group Databasebeta

← #27153156

Subgroups — #27153156

Order 12416.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261356C211yesmaximal, characteristic
2#20261039C211yesmaximal, characteristic, complemented
3#20261032C211yesmaximal, characteristic, complemented
4#20261030C211yesmaximal, characteristic, complemented
5#20261020C211yesmaximal, characteristic, complemented
6#20260991C211yesmaximal, characteristic, complemented
7#20260983C211yesmaximal, characteristic
8#20260974C211yesmaximal, characteristic, complemented
9#20260966C211yesmaximal, characteristic
10#20260957C211yesmaximal, characteristic, complemented
11#20260957C211yesmaximal, characteristic, complemented
12#20260942C211yesmaximal, characteristic, complemented
13#20260937C211yesmaximal, characteristic, complemented
14#20260762C211yesmaximal, characteristic, complemented
15#541×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C2×#13095495C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22#15690810C2211yescharacteristic
23#15690805—21no—
24#15690798C2211yescharacteristic
25#15690796—21no—
26#15690792—21no—
27#15690789C2211yescharacteristic
28#15690788—21no—
29#15690784C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31C4×D388C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690743C2211yescharacteristic
34#15690741C2211yescharacteristic
35#15690741C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39C4⋊#11715895C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690734C2211yescharacteristic
42#15690734C2211yescharacteristic
43#15690734C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690733C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic
48C4⋊Q8×C97C2211yescharacteristic
49#126×C97—21no—
50#126×C97—21no—
51#125×C97—21no—
52#125×C97—21no—
53#125×C97C2211yescharacteristic
54C22⋊Q8×C97C2211yescharacteristic
55C22⋊Q8×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×Q8×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194C2311yescharacteristic
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C4×D97—21no—
64#13095501—41no—
65#13095501—41no—
66#13095498—41no—
67#13095498—41no—
68#13095495—41no—
69C2×#11715895—41no—
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C4⋊Dic97—21no—
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84C4×Dic97—21no—
85C2×Q8×C97C2311yescharacteristic
86C2×Q8×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C4⋊C4×C97—21no—
89C4⋊C4×C97—21no—
90C4⋊C4×C97—21no—
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
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×C97C2×D411yescharacteristic
109C22×D97—21no—
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97C2×D411yescharacteristic
121C2×C4×C97C2×D411yescharacteristic
122C2×C4×C97C2×D411yescharacteristic
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C22⋊D97—81no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130D388—21no—
131D388—21no—
132C4×D97—81no—
133Dic194—21no—
134Dic194—21no—
135Dic194—81no—
136#11715895—21no—
137#11715895—21no—
138#11715895—21no—
139#11715895—21no—
140C22×C97—21no—
141C22×C97—21no—
142C22×C97—21no—
143C22×C97C22×D411yescharacteristic
144C2×D97—41no—
145C2×D97—41no—
146C2×D97—41no—
147C4×C97—21no—
148C4×C97—21no—
149C4×C97—21no—
150C4×C97—21no—
151C4×C97—21no—
152C4×C97—41no—
153C4×C97—41no—
154C4×C97—41no—
155C4×C97—41no—
156C4×C97H(5,𝔽2)11yescharacteristic
157C4×C972−1+411yescharacteristic
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161C2×C97—41no—
162C2×C97#53711yescharacteristic
163C2×C97#57711yescharacteristic
164C2×C97#57811yescharacteristic
165D97—81no—
166#3258—971nomaximal, Sylow
167C97#325811yescharacteristic, complemented, Sylow
168#542—971no—
169#541D9711yescharacteristic, complemented
170#479—971no—
171#478—971no—
172#477—971no—
173#475—971no—
174#470—971no—
175#468—971no—
176#467—971no—
177#465—971no—
178#464—971no—
179#464—971no—
180#462—971no—
181#461—971no—
182#424—971no—
183C2×Q16—971no—
184C2×Q16—971no—
185C2×SD16—971no—
186C2×SD16—971no—
187C2×M16—971no—
188C22×C8—971no—
189C4⋊Q8—971no—
190C4⋊Q8C2×D9711yescharacteristic
191#126—21no—
192#126—21no—
193#126—1941no—
194#125—21no—
195#125—21no—
196#125C2×D9711yescharacteristic
197#124—971no—
198#124—1941no—
199#123—1941no—
200C22⋊Q8—971no—
201C22⋊Q8—1941no—
202C22⋊Q8C2×D9711yescharacteristic
203C22⋊Q8C2×D9711yescharacteristic
204#121—971no—
205C4×Q8—971no—
206C4×Q8C2×D9711yescharacteristic
207C4×Q8C2×D9711yescharacteristic
208C4×D4—971no—
209#117C2×D9711yescharacteristic
210#107—971no—
211#107—971no—
212#106—971no—
213#106—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217C4⋊C8—971no—
218#103—971no—
219#103—971no—
220#103—971no—
221#103—971no—
222#103—971no—
223#103—971no—
224#102—971no—
225#102—971no—
226C2×Q8—971no—
227C2×Q8—1941no—
228C2×Q8C22×D9711yescharacteristic
229C2×Q8C22×D9711yescharacteristic
230C2×D4—971no—
231C2×D4—1941no—
232C22×C4—1941no—
233C22×C4C22×D9711yescharacteristic
234Q16—3881no—
235Q16—3881no—
236SD16—3881no—
237SD16—3881no—
238M16—3881no—
239C2×C8—971no—
240C2×C8—971no—
241C2×C8—971no—
242C2×C8—971no—
243C2×C8—3881no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—971no—
253C4⋊C4—971no—
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—1941no—
264C22⋊C4—1941no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C22⋊C4—1941no—
268C22⋊C4C22⋊D9711yescharacteristic
269C22⋊C4C22⋊D9711yescharacteristic
270C42—21no—
271C42—21no—
272C42—1941no—
273C42C22×D9711yescharacteristic
274C42C22×D9711yescharacteristic
275C23—1941no—
276C23C2×C22⋊D9711yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—7761no—
282Q8—1941no—
283Q8—1941no—
284D4—7761no—
285D4—1941no—
286D4—1941no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—7761no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4C2×C22⋊D9711yescharacteristic
298C2×C4C2×C22⋊D9711yescharacteristic
299C2×C4C2×C22⋊D9711yescharacteristic
300C2×C4C2×C22⋊D9711yescharacteristic
301C2×C4C2×C22⋊D9711yescharacteristic
302C2×C4C23×D9711yescharacteristic
303C8—1941no—
304C8—1941no—
305C8—1941no—
306C8—1941no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—3881no—
311C22—3881no—
312C22—3881no—
313C22C22×C22⋊D9711yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3881no—
324C4—3881no—
325C4—3881no—
326C4#1569085611yescharacteristic
327C4#1569085711yescharacteristic
328C2—41no—
329C2—7761no—
330C2#2026131311yescharacteristic
331C2#2026154311yescharacteristic
332C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#541×C97C2
socleC22×C97C22×D4
prime radical#15690784C22
hypercenter#541D97
hypocenterC97#3258
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20261039C2semidirect
3. #20261032C2semidirect
4. #20261030C2semidirect
5. #20261020C2semidirect
6. #20260991C2semidirect
8. #20260974C2semidirect
10. #20260957C2semidirect
11. #20260957C2semidirect
12. #20260942C2semidirect
13. #20260937C2semidirect
14. #20260762C2semidirect
15. #541×C97C2semidirect
167. C97#3258semidirect
169. #541D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202610392
#202610322
#202610302
#202610202
#202609912
#202609832
#202609742
#202609662
#202609572
#202609422
#202609372
#202607622
#541×C972
#325897