Finite Group Databasebeta

← #27158691

Subgroups — #27158691

Order 12416.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261787C211yesmaximal, characteristic, complemented
2#20261785C211yesmaximal, characteristic, complemented
3#20261769C211yesmaximal, characteristic, complemented
4#20261768C211yesmaximal, characteristic, complemented
5#20261767C211yesmaximal, characteristic, complemented
6#20261765C211yesmaximal, characteristic, complemented
7#20261764C211yesmaximal, characteristic, complemented
8#20261764C211yesmaximal, characteristic, complemented
9#20261760C211yesmaximal, characteristic, complemented
10#20261759C211yesmaximal, characteristic, complemented
11#20261758C211yesmaximal, characteristic, complemented
12#20261742C211yesmaximal, characteristic, complemented
13#20261741C211yesmaximal, characteristic, complemented
14#20261739C211yesmaximal, characteristic, complemented
15#20261268C211yesmaximal, characteristic
16C2×#13095521C2211yescharacteristic
17C2×#13095521C2211yescharacteristic
18C2×#13095521C2211yescharacteristic
19C2×#13095520C2211yescharacteristic
20C2×#13095519C2211yescharacteristic
21C22×#11715897C2211yescharacteristic
22#15690877C2211yescharacteristic
23#15690877C2211yescharacteristic
24#15690877C2211yescharacteristic
25#15690877C2211yescharacteristic
26C4⋊#11715897C2211yescharacteristic
27C4⋊#11715897C2211yescharacteristic
28C4⋊#11715897C2211yescharacteristic
29C4⋊#11715897C2211yescharacteristic
30C4⋊#11715897C2211yescharacteristic
31C4⋊#11715897C2211yescharacteristic
32#15690867C2211yescharacteristic
33#15690867C2211yescharacteristic
34#15690867C2211yescharacteristic
35#15690867C2211yescharacteristic
36C22⋊#11715897C2211yescharacteristic
37C22⋊#11715897C2211yescharacteristic
38#15690865C2211yescharacteristic
39#15690865C2211yescharacteristic
40#15690865C2211yescharacteristic
41#15690865C2211yescharacteristic
42C4×#11715897C2211yescharacteristic
43C4×#11715897C2211yescharacteristic
44#15690804C411yescharacteristic
45#15690800C411yescharacteristic
46#15690796C411yescharacteristic
47#15690795C411yescharacteristic
48#15690792C411yescharacteristic
49#15690788C411yescharacteristic
50#15690783C411yescharacteristic
51#15690775C2211yescharacteristic
52#15690772C2211yescharacteristic
53#15690770C2211yescharacteristic
54#15690770C2211yescharacteristic
55#15690768C2211yescharacteristic
56#15690766C2211yescharacteristic
57C42×D97C2211yescharacteristic
58#126×C97C411yescharacteristic
59#13095521—41no—
60#13095521—41no—
61#13095521—41no—
62#13095520—41no—
63#13095519—41no—
64C2×#11715897—41no—
65C2×#11715897C2311yescharacteristic
66C2×#11715897C2311yescharacteristic
67C2×#11715897C2311yescharacteristic
68C2×#11715897C2311yescharacteristic
69C2×#11715897C2311yescharacteristic
70C2×#11715897C2311yescharacteristic
71C2×#11715897C2311yescharacteristic
72C2×#11715897C2311yescharacteristic
73C2×Dic194C2×C411yescharacteristic
74C2×D388C2×C411yescharacteristic
75C2×C22⋊D97C2×C411yescharacteristic
76C2×C22⋊D97C2×C411yescharacteristic
77C22×Dic97C2311yescharacteristic
78C2×C4×D97C2×C411yescharacteristic
79C2×C4×D97C2×C411yescharacteristic
80C2×C4×D97C2311yescharacteristic
81C2×C4×D97C2311yescharacteristic
82C4⋊Dic97C2×C411yescharacteristic
83#13095490C2×C411yescharacteristic
84#13095490C2×C411yescharacteristic
85#13095490C2×C411yescharacteristic
86#13095490C2×C411yescharacteristic
87#13095490C2×C411yescharacteristic
88#13095490C2×C411yescharacteristic
89C22⋊Dic97C2×C411yescharacteristic
90#13095488C2×C411yescharacteristic
91#13095488C2×C411yescharacteristic
92#13095488C2×C411yescharacteristic
93#13095488C2×C411yescharacteristic
94#13095488C2×C411yescharacteristic
95#13095488C2×C411yescharacteristic
96C4×Dic97C2×C411yescharacteristic
97C4×Dic97C2311yescharacteristic
98C4×Dic97C2311yescharacteristic
99C4×Dic97C2311yescharacteristic
100C4×Dic97C2311yescharacteristic
101C4⋊C4×C97C2×C411yescharacteristic
102C4⋊C4×C97C2×C411yescharacteristic
103C4⋊C4×C97C2×C411yescharacteristic
104C22⋊C4×C97C2×C411yescharacteristic
105C22⋊C4×C97C2×C411yescharacteristic
106C22⋊C4×C97C2×C411yescharacteristic
107C42×C97C2×C411yescharacteristic
108C23×C97C22×C411yescharacteristic
109C22×D97C22×C411yescharacteristic
110C22×D97C22×C411yescharacteristic
111C2×C4×C97C22×C411yescharacteristic
112C2×C4×C97C22×C411yescharacteristic
113C2×C4×C97C22×C411yescharacteristic
114C2×C4×C97C22×C411yescharacteristic
115C2×C4×C97C22×C411yescharacteristic
116C2×C4×C97C22×C411yescharacteristic
117C22⋊D97—41no—
118C22⋊D97—41no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97C22×C411yescharacteristic
122C2×Dic97C22×C411yescharacteristic
123C2×Dic97C22×C411yescharacteristic
124C2×Dic97C22×C411yescharacteristic
125C2×Dic97C22×C411yescharacteristic
126C2×Dic97C2411yescharacteristic
127D388—41no—
128C4×D97—21no—
129C4×D97—21no—
130C4×D97—21no—
131C4×D97—21no—
132C4×D97—41no—
133C4×D97—41no—
134Dic194—41no—
135#11715897—21no—
136#11715897—21no—
137#11715897—21no—
138#11715897—21no—
139#11715897—21no—
140#11715897—21no—
141#11715897—21no—
142#11715897—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—21no—
146C22×C97C23×C411yescharacteristic
147C2×D97—21no—
148C2×D97—21no—
149C2×D97—21no—
150C2×D97—21no—
151C2×D97—21no—
152C2×D97—21no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—21no—
159Dic97—21no—
160Dic97—21no—
161Dic97—21no—
162Dic97—21no—
163Dic97—21no—
164Dic97H(5,𝔽2)11yescharacteristic
165Dic972−1+411yescharacteristic
166C2×C97—41no—
167C2×C97#52011yescharacteristic
168C2×C97#56811yescharacteristic
169C2×C97#56811yescharacteristic
170D97—41no—
171D97—41no—
172#2995—971nomaximal, Sylow
173C97#299511yescharacteristic, complemented, Sylow
174#529—971no—
175#446—971no—
176#446—971no—
177#436—971no—
178#436—971no—
179#435—971no—
180#435—971no—
181#435—971no—
182#435—971no—
183#433—971no—
184#433—971no—
185#432—971no—
186#424—971no—
187#424—971no—
188#423—971no—
189C2×M16—971no—
190C2×M16—971no—
191C2×M16—971no—
192C2×M16—971no—
193C22×C8—971no—
194C22×C8—971no—
195#126—971no—
196#126#8760411yescharacteristic
197#125—971no—
198#124—971no—
199#123—971no—
200#123—971no—
201C22⋊Q8—971no—
202#121—971no—
203C4×Q8—971no—
204C4×D4—971no—
205C4×D4—971no—
206C4×D4—971no—
207#117—971no—
208#117—971no—
209C2×C42—971no—
210C4⋊C8—971no—
211C4⋊C8—971no—
212C4⋊C8—971no—
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217C4⋊C8—971no—
218C4⋊C8—971no—
219C4⋊C8—971no—
220C22⋊C8—971no—
221C22⋊C8—971no—
222C22⋊C8—971no—
223C22⋊C8—971no—
224C22⋊C8—971no—
225C22⋊C8—971no—
226#97—971no—
227#97—971no—
228#97—971no—
229#97—971no—
230C4×C8—971no—
231C4×C8—971no—
232C2×Q8—971no—
233C2×D4—971no—
234C2×D4—971no—
235C2×D4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241M16—3881no—
242M16—3881no—
243M16—3881no—
244M16—3881no—
245C2×C8—971no—
246C2×C8—971no—
247C2×C8—971no—
248C2×C8—971no—
249C2×C8—971no—
250C2×C8—971no—
251C2×C8—971no—
252C2×C8—971no—
253C2×C8—3881no—
254C2×C8—3881no—
255C4⋊C4—971no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4C2×#8760411yescharacteristic
263C4⋊C4C2×#8760411yescharacteristic
264C4⋊C4C2×#8760411yescharacteristic
265C22⋊C4—971no—
266C22⋊C4—971no—
267C22⋊C4—971no—
268C22⋊C4—971no—
269C22⋊C4—971no—
270C22⋊C4—971no—
271C22⋊C4—971no—
272C22⋊C4C2×#8760411yescharacteristic
273C22⋊C4C2×#8760411yescharacteristic
274C22⋊C4C2×#8760411yescharacteristic
275C42—971no—
276C42—971no—
277C42—971no—
278C42—971no—
279C42—971no—
280C42C2×#8760411yescharacteristic
281C23—971no—
282C23—971no—
283C23C22×#8760411yescharacteristic
284Q8—3881no—
285D4—3881no—
286D4—3881no—
287D4—3881no—
288C2×C4—971no—
289C2×C4—971no—
290C2×C4—971no—
291C2×C4—971no—
292C2×C4—971no—
293C2×C4—971no—
294C2×C4—3881no—
295C2×C4—3881no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4C22×#8760411yescharacteristic
303C2×C4C22×#8760411yescharacteristic
304C2×C4C22×#8760411yescharacteristic
305C2×C4C22×#8760411yescharacteristic
306C2×C4C22×#8760411yescharacteristic
307C2×C4C22×#8760411yescharacteristic
308C8—1941no—
309C8—1941no—
310C8—1941no—
311C8—1941no—
312C8—1941no—
313C8—1941no—
314C8—1941no—
315C8—1941no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—1941no—
320C22—1941no—
321C22—1941no—
322C22—1941no—
323C22—1941no—
324C22—1941no—
325C22C23×#8760411yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—971no—
333C4—971no—
334C4—1941no—
335C4—1941no—
336C4—1941no—
337C4—1941no—
338C4—1941no—
339C2—41no—
340C2—3881no—
341C2—3881no—
342C2#2026180411yescharacteristic
343C2#2026181411yescharacteristic
344C2#2026181411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#126×C97C4
prime radical#15690783C4
cosocleC2×Dic97C24
hypercenter#126#87604
hypocenterC97#2995
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #20261787C2semidirect
2. #20261785C2semidirect
3. #20261769C2semidirect
4. #20261768C2semidirect
5. #20261767C2semidirect
6. #20261765C2semidirect
7. #20261764C2semidirect
8. #20261764C2semidirect
9. #20261760C2semidirect
10. #20261759C2semidirect
11. #20261758C2semidirect
12. #20261742C2semidirect
13. #20261741C2semidirect
14. #20261739C2semidirect
173. C97#2995semidirect

Maximal subgroups

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

subgroupindex
#202617872
#202617852
#202617692
#202617682
#202617672
#202617652
#202617642
#202617602
#202617592
#202617582
#202617422
#202617412
#202617392
#202612682
#299597