Finite Group Databasebeta

← #27154318

Subgroups — #27154318

Order 12416.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261453C211yesmaximal, characteristic, complemented
2#20261385C211yesmaximal, characteristic
3#20261139C211yesmaximal, characteristic, complemented
4#20261072C211yesmaximal, characteristic, complemented
5#20261071C211yesmaximal, characteristic, complemented
6#20261057C211yesmaximal, characteristic, complemented
7#20261019C211yesmaximal, characteristic, complemented
8#20261013C211yesmaximal, characteristic, complemented
9#20260980C211yesmaximal, characteristic, complemented
10#20260979C211yesmaximal, characteristic, complemented
11#20260826C211yesmaximal, characteristic, complemented
12#20260810C211yesmaximal, characteristic, complemented
13#20260778C211yesmaximal, characteristic, complemented
14#20260774C211yesmaximal, characteristic, complemented
15#482×C97C211yesmaximal, characteristic, complemented
16C2×#13095510—21no—
17C2×#13095499C2211yescharacteristic
18C2×#13095495C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20#15690809C2211yescharacteristic
21#15690809C2211yescharacteristic
22#15690804—21no—
23#15690803—21no—
24#15690796—21no—
25#15690792—21no—
26#15690791—21no—
27#15690790C2211yescharacteristic
28#15690789C2211yescharacteristic
29#15690787—21no—
30#15690781C2211yescharacteristic
31C4×Dic194—21no—
32#15690775C2211yescharacteristic
33D4×Dic97C2211yescharacteristic
34C4×C22⋊D97C2211yescharacteristic
35#15690768C2211yescharacteristic
36C2×C4⋊Dic97C2211yescharacteristic
37C4⋊C4×D97C2211yescharacteristic
38#15690744C2211yescharacteristic
39#15690744C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690743C2211yescharacteristic
42#15690743C2211yescharacteristic
43#15690741C2211yescharacteristic
44#15690741C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690733C2211yescharacteristic
49#15690731C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52#121×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59#13095510—44no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97D411yescharacteristic
65C22×Dic97—21no—
66C22×Dic97—21no—
67C2×C4×D97—21no—
68C2×C4×D97C2311yescharacteristic
69#13095499—22no—
70#13095495—22no—
71#13095494—22no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C4⋊Dic97—21no—
75C4⋊Dic97—21no—
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490D411yescharacteristic
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88C22⋊Dic97D411yescharacteristic
89#13095488—21no—
90#13095488—21no—
91#13095488D411yescharacteristic
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×D4×C97—21no—
96C2×D4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C42×C97—21no—
106C23×C97—21no—
107C23×C97C2×D411yescharacteristic
108C22×D97C2×D411yescharacteristic
109D4×C97—21no—
110D4×C97—21no—
111D4×C97—42no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97C2×D411yescharacteristic
118C2×C4×C97C2411yescharacteristic
119C22⋊D97—22no—
120C22⋊D97—42no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—42no—
126C2×Dic97—42no—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129C2×Dic97C2×D411yescharacteristic
130C4×D97—21no—
131C4×D97—21no—
132C4×D97—42no—
133Dic194—21no—
134Dic194—21no—
135Dic194—42no—
136Dic194—42no—
137C8×C97—21no—
138C8×C97—21no—
139#11715895—21no—
140#11715895#4112yescomplemented
141C22×C97—21no—
142C22×C97—21no—
143C22×C97—21no—
144C22×C97—41no—
145C22×C97—41no—
146C22×C97—41no—
147C22×C97C22×D411yescharacteristic
148C2×D97—21no—
149C2×D97—21no—
150C2×D97—21no—
151C4×C97—21no—
152C4×C97—41no—
153C4×C97—41no—
154C4×C97—41no—
155C4×C97C2×#4111yescharacteristic
156C4×C972−1+411yescharacteristic
157Dic97—21no—
158Dic97—21no—
159Dic97—41no—
160Dic97—41no—
161Dic97—41no—
162Dic97—41no—
163Dic97AGL(1,ℤ/8)12yes—
164C2×C97—41no—
165C2×C97—81no—
166C2×C97#54711yescharacteristic
167C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C97#57811yescharacteristic
169D97—41no—
170#3353—971nomaximal, Sylow
171C97#335311yescharacteristic, complemented, Sylow
172#556—971no—
173#547—971no—
174#500—971no—
175#484—971no—
176#484—971no—
177#482—971no—
178#482D9711yescharacteristic, complemented
179#475—971no—
180#474—971no—
181#468—971no—
182#468—971no—
183C4×SD16—971no—
184#436—971no—
185#428—971no—
186#428—971no—
187C2×#41—1941no—
188C2×SD16—971no—
189C2×M16—971no—
190C2×M16—971no—
191C4⋊Q8—971no—
192C4⋊Q8—971no—
193#126—1941no—
194#126—1941no—
195#124—1941no—
196#123—1941no—
197#123—1941no—
198C22⋊Q8—971no—
199C22⋊Q8—971no—
200C22⋊Q8—1941no—
201#121—971no—
202#121C2×D9711yescharacteristic
203C4×Q8—971no—
204C4×Q8—1941no—
205C4×D4—971no—
206C4×D4—971no—
207#117—971no—
208#117C2×D9711yescharacteristic
209C2×C4⋊C4—971no—
210C2×C4⋊C4—971no—
211#107—971no—
212#107—971no—
213#107—971no—
214#107—971no—
215#107—971no—
216#107C2×D9711yescharacteristic
217#106—971no—
218#106—971no—
219#106C2×D9711yescharacteristic
220C4⋊C8—971no—
221#103—971no—
222#103—971no—
223#103—971no—
224#102—971no—
225#102C2×D9711yescharacteristic
226#102C2×D9711yescharacteristic
227C22⋊C8—971no—
228C22⋊C8C2×D9711yescharacteristic
229C4×C8—971no—
230#41—3884no—
231C2×Q8—971no—
232C2×Q8—1941no—
233C2×Q8—1941no—
234C2×D4—21no—
235C2×D4—971no—
236C2×D4—1941no—
237C2×D4C22×D9711yescharacteristic
238C22×C4—971no—
239C22×C4—1941no—
240C22×C4—1941no—
241C22×C4—1941no—
242C22×C4C22×D9711yescharacteristic
243SD16—1942no—
244M16—1942no—
245M16—1942no—
246C2×C8—971no—
247C2×C8—971no—
248C2×C8C22×D9711yescharacteristic
249C2×C8C22×D9711yescharacteristic
250C4⋊C4—971no—
251C4⋊C4—971no—
252C4⋊C4—971no—
253C4⋊C4—971no—
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
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—971no—
267C22⋊C4—971no—
268C22⋊C4—1941no—
269C22⋊C4—1941no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C42—21no—
274C42—971no—
275C42—1941no—
276C42—1941no—
277C23—21no—
278C23—971no—
279C23D4×D9711yescharacteristic
280Q8—3882no—
281Q8—3882no—
282Q8—1941no—
283Q8—1941no—
284D4—21no—
285D4—21no—
286D4—42no—
287D4—3882no—
288D4—1942no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—971no—
297C2×C4—3882no—
298C2×C4—3882no—
299C2×C4—3882no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4D4×D9711yescharacteristic
307C2×C4C23×D9711yescharacteristic
308C8—21no—
309C8—21no—
310C8—972no—
311C8—1941no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—1941no—
319C22—1941no—
320C22—1941no—
321C22C2×D4×D9711yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—972no—
327C4—3881no—
328C4—3881no—
329C4—3881no—
330C4—3881no—
331C4—1941no—
332C4—1941no—
333C4#41×D9711yescharacteristic
334C4#1569085711yescharacteristic
335C2—41no—
336C2—81no—
337C2—3881no—
338C2#2026138311yescharacteristic
339C2AGL(1,ℤ/8)×D9711yescharacteristic
340C2#2026154911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#482×C97C2
socleC22×C97C22×D4
prime radical#20261385C2
hypercenter#482D97
hypocenterC97#3353
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #20261453C2semidirect
3. #20261139C2semidirect
4. #20261072C2semidirect
5. #20261071C2semidirect
6. #20261057C2semidirect
7. #20261019C2semidirect
8. #20261013C2semidirect
9. #20260980C2semidirect
10. #20260979C2semidirect
11. #20260826C2semidirect
12. #20260810C2semidirect
13. #20260778C2semidirect
14. #20260774C2semidirect
15. #482×C97C2semidirect
140. #11715895#41semidirect
171. C97#3353semidirect
178. #482D97semidirect

Maximal subgroups

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

subgroupindex
#202614532
#202613852
#202611392
#202610722
#202610712
#202610572
#202610192
#202610132
#202609802
#202609792
#202608262
#202608102
#202607782
#202607742
#482×C972
#335397