Finite Group Databasebeta

← #27154347

Subgroups — #27154347

Order 12416.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261453C211yesmaximal, characteristic
2#20261440C211yesmaximal, characteristic
3#20261132C211yesmaximal, characteristic, complemented
4#20261078C211yesmaximal, characteristic, complemented
5#20261076C211yesmaximal, characteristic, complemented
6#20261067C211yesmaximal, characteristic, complemented
7#20261065C211yesmaximal, characteristic, complemented
8#20261029C211yesmaximal, characteristic, complemented
9#20261019C211yesmaximal, characteristic, complemented
10#20260994C211yesmaximal, characteristic, complemented
11#20260993C211yesmaximal, characteristic, complemented
12#20260829C211yesmaximal, characteristic, complemented
13#20260811C211yesmaximal, characteristic
14#20260776C211yesmaximal, characteristic, complemented
15#428×C97C211yesmaximal, characteristic, complemented
16C2×Dic388C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18#15690809C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690804—21no—
21#15690803—21no—
22#15690800—21no—
23#15690796—21no—
24#15690790C2211yescharacteristic
25#15690790C2211yescharacteristic
26#15690789—21no—
27#15690789C2211yescharacteristic
28#15690789C2211yescharacteristic
29#15690788—21no—
30C4×Dic194—21no—
31#15690775C2211yescharacteristic
32#15690775C2211yescharacteristic
33C4×C22⋊D97—21no—
34C4×C22⋊D97C2211yescharacteristic
35#15690768C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic
37#15690743C2211yescharacteristic
38#15690742C2211yescharacteristic
39#15690741C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690733C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690733C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×M16×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54C2×C4⋊C4×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic
57#106×C97C2211yescharacteristic
58#106×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62C2×Dic194C2311yescharacteristic
63C2×C22⋊D97D411yescharacteristic
64C2×C4×D97—21no—
65C2×C4×D97C2311yescharacteristic
66Dic388—22no—
67#13095494—22no—
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C4⋊Dic97—21no—
71C4⋊Dic97—21no—
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74#13095490—21no—
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82C22⋊Dic97—21no—
83C22⋊Dic97—21no—
84C22⋊Dic97D411yescharacteristic
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488D411yescharacteristic
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97C2311yescharacteristic
93C22×C4×C97—21no—
94C22×C4×C97C2311yescharacteristic
95M16×C97—22no—
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C4⋊C4×C97—21no—
99C4⋊C4×C97C2311yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C42×C97—21no—
105C23×C97C2×D411yescharacteristic
106C22×D97C2×D411yescharacteristic
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—81no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2411yescharacteristic
116C22⋊D97—22no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97C2×D411yescharacteristic
122C2×Dic97C2×D411yescharacteristic
123C2×Dic97C2×D411yescharacteristic
124C4×D97—21no—
125C4×D97—21no—
126C4×D97—81no—
127Dic194—21no—
128Dic194—21no—
129Dic194—21no—
130Dic194—21no—
131Dic194—42no—
132Dic194—81no—
133C8×C97—21no—
134C8×C97#4112yes—
135#11715895—21no—
136#11715895—21no—
137C22×C97—21no—
138C22×C97—21no—
139C22×C97—21no—
140C22×C97C22×D411yescharacteristic
141C2×D97—21no—
142C2×D97—21no—
143C2×D97—21no—
144C4×C97—21no—
145C4×C97—41no—
146C4×C97—41no—
147C4×C97—41no—
148C4×C97—41no—
149C4×C97C2×#4111yescharacteristic
150C4×C972−1+411yescharacteristic
151Dic97—21no—
152Dic97—21no—
153Dic97—41no—
154Dic97—41no—
155Dic97—41no—
156Dic97—41no—
157Dic97#13712yes—
158C2×C97—41no—
159C2×C97#54711yescharacteristic
160C2×C97C2×#13711yescharacteristic
161C2×C97#57711yescharacteristic
162D97—41no—
163#3356—971nomaximal, Sylow
164C97#335611yescharacteristic, complemented, Sylow
165#556—971no—
166#554—971no—
167#498—971no—
168#485—971no—
169#485—971no—
170#483—971no—
171#483—971no—
172#477—971no—
173#475—971no—
174#470—971no—
175#470—971no—
176C4×Q16—971no—
177#436—971no—
178#428—971no—
179#428D9711yescharacteristic, complemented
180C2×Q16—971no—
181C2×M16—971no—
182C2×M16C2×D9711yescharacteristic
183C4⋊Q8—971no—
184C4⋊Q8—971no—
185#126—1941no—
186#126—1941no—
187#125—1941no—
188#124—1941no—
189C22⋊Q8—971no—
190C22⋊Q8—971no—
191C22⋊Q8—971no—
192C22⋊Q8—971no—
193C22⋊Q8—1941no—
194C22⋊Q8—1941no—
195C4×Q8—971no—
196C4×Q8—971no—
197C4×Q8—1941no—
198C4×D4—971no—
199C4×D4—1941no—
200#117—971no—
201#117C2×D9711yescharacteristic
202C2×C4⋊C4—971no—
203C2×C4⋊C4C2×D9711yescharacteristic
204#107—971no—
205#107C2×D9711yescharacteristic
206#107C2×D9711yescharacteristic
207#106—971no—
208#106—971no—
209#106—971no—
210#106—971no—
211#106C2×D9711yescharacteristic
212#106C2×D9711yescharacteristic
213C4⋊C8—971no—
214#103—971no—
215#103—971no—
216#103—971no—
217#103—971no—
218#103—971no—
219#103—971no—
220C22⋊C8—971no—
221C22⋊C8—971no—
222C4×C8—971no—
223C2×Q8—971no—
224C2×Q8—971no—
225C2×Q8—1941no—
226C2×Q8—1941no—
227C2×D4—971no—
228C22×C4—21no—
229C22×C4—971no—
230C22×C4—1941no—
231C22×C4C22×D9711yescharacteristic
232Q16—1942no—
233M16—22no—
234M16—1942no—
235C2×C8—971no—
236C2×C8—971no—
237C2×C8C22×D9711yescharacteristic
238C2×C8C22×D9711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—971no—
241C4⋊C4—971no—
242C4⋊C4—971no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4C22×D9711yescharacteristic
253C4⋊C4C22×D9711yescharacteristic
254C4⋊C4C22×D9711yescharacteristic
255C4⋊C4C22×D9711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—971no—
258C22⋊C4—971no—
259C22⋊C4—1941no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C42—21no—
265C42—971no—
266C42—1941no—
267C42—1941no—
268C42—1941no—
269C23—971no—
270C23D4×D9711yescharacteristic
271Q8—7761no—
272Q8—3882no—
273Q8—1941no—
274Q8—1941no—
275Q8—1941no—
276Q8—1941no—
277D4—1942no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—7761no—
286C2×C4—971no—
287C2×C4—971no—
288C2×C4—971no—
289C2×C4—1941no—
290C2×C4—1941no—
291C2×C4—1941no—
292C2×C4—1941no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4D4×D9711yescharacteristic
296C2×C4C23×D9711yescharacteristic
297C8—21no—
298C8—1941no—
299C8—1941no—
300C8#1309551112yes—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—1941no—
305C22—1941no—
306C22—1941no—
307C22C2×D4×D9711yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—972no—
314C4—3881no—
315C4—3881no—
316C4—3881no—
317C4—3881no—
318C4—1941no—
319C4—1941no—
320C4C2×#1309551111yescharacteristic
321C4#1569085811yescharacteristic
322C2—41no—
323C2—3881no—
324C2#2026138611yescharacteristic
325C2#137×D9711yescharacteristic
326C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radicalC2×C22⋊D97D4
hypercenter#428D97
hypocenterC97#3356
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20261132C2semidirect
4. #20261078C2semidirect
5. #20261076C2semidirect
6. #20261067C2semidirect
7. #20261065C2semidirect
8. #20261029C2semidirect
9. #20261019C2semidirect
10. #20260994C2semidirect
11. #20260993C2semidirect
12. #20260829C2semidirect
14. #20260776C2semidirect
15. #428×C97C2semidirect
164. C97#3356semidirect
179. #428D97semidirect

Maximal subgroups

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

subgroupindex
#202614532
#202614402
#202611322
#202610782
#202610762
#202610672
#202610652
#202610292
#202610192
#202609942
#202609932
#202608292
#202608112
#202607762
#428×C972
#335697