Finite Group Databasebeta

← #27151845

Subgroups — #27151845

Order 12416.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261269C211yesmaximal, characteristic, complemented
2#20261076C211yesmaximal, characteristic, complemented
3#20261059C211yesmaximal, characteristic, complemented
4#20261042C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20260957C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260897C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic
10#20260845C211yesmaximal, characteristic, complemented
11#20260840C211yesmaximal, characteristic, complemented
12#20260835C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260792C211yesmaximal, characteristic, complemented
15C4×#41×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20#15690804—21no—
21#15690802C2211yescharacteristic
22#15690798C2211yescharacteristic
23#15690793—21no—
24#15690789C2211yescharacteristic
25#15690784C2211yescharacteristic
26C4×Dic194C2211yescharacteristic
27C4×D388C2211yescharacteristic
28C4×C22⋊D97—21no—
29#15690767—21no—
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690733C2211yescharacteristic
39#15690733C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic
47#15690721C2211yescharacteristic
48C2×#41×C97C2211yescharacteristic
49C4×Q8×C97C2211yescharacteristic
50C4×Q8×C97C2211yescharacteristic
51C4×D4×C97—21no—
52C4×D4×C97—21no—
53C4×D4×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55#117×C97—21no—
56#117×C97C2211yescharacteristic
57C2×C42×C97—21no—
58C2×C42×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C2×C4×D97—21no—
63#13095501—22no—
64#13095499—22no—
65#13095498—22no—
66#13095496—22no—
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C4⋊Dic97C2311yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73#13095490—21no—
74#13095490—21no—
75#13095490—21no—
76#13095490—21no—
77C22⋊Dic97—21no—
78#13095488—21no—
79#13095488—21no—
80#13095488—21no—
81#13095488—21no—
82C4×Dic97—21no—
83#41×C97—22no—
84#41×C97—22no—
85C2×Q8×C97C2311yescharacteristic
86C2×D4×C97—21no—
87C2×D4×C97C2311yescharacteristic
88C22×C4×C97—21no—
89C22×C4×C97—21no—
90C22×C4×C97—21no—
91C22×C4×C97D411yescharacteristic
92C22×C4×C97D411yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C42×C97—21no—
102C42×C97—21no—
103C42×C97—21no—
104C42×C97D411yescharacteristic
105C42×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111Q8×C97—21no—
112Q8×C97#4112yes—
113D4×C97—24no—
114D4×C97—21no—
115D4×C97#4112yes—
116C2×C4×C97—24no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—41no—
123C2×C4×C97—41no—
124C2×C4×C97—42no—
125C2×C4×C97—42no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—42no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136D388—21no—
137D388—21no—
138C4×D97—42no—
139Dic194—21no—
140Dic194—21no—
141#11715895—21no—
142#11715895—21no—
143#11715895—21no—
144#11715895—21no—
145C22×C97—21no—
146C22×C97—22no—
147C22×C97—21no—
148C22×C97—21no—
149C22×C97—41no—
150C22×C97—41no—
151C22×C97C22×D411yescharacteristic
152C2×D97—41no—
153C2×D97—41no—
154C2×D97—41no—
155C4×C97—21no—
156C4×C97—22no—
157C4×C97—21no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—41no—
162C4×C97—41no—
163C4×C97C2×#4111yescharacteristic
164C4×C97C2×#4111yescharacteristic
165Dic97—41no—
166Dic97—41no—
167Dic97—41no—
168C2×C97—41no—
169C2×C97—42no—
170C2×C97#52511yescharacteristic
171C2×C97#57511yescharacteristic
172C2×C97#57511yescharacteristic
173D97—81no—
174#3111—971nomaximal, Sylow
175C97#311111yescharacteristic, complemented, Sylow
176#529—971no—
177C4×#41D9711yescharacteristic, complemented
178#485—971no—
179#482—971no—
180#479—971no—
181#478—971no—
182#464—971no—
183#463—971no—
184#452—971no—
185#449—971no—
186#442—971no—
187#441—971no—
188#440—971no—
189#440—971no—
190#432—971no—
191C2×#41C2×D9711yescharacteristic
192C2×Q16—971no—
193C2×SD16—971no—
194C2×SD16—971no—
195C2×D8—971no—
196#126—1941no—
197#125—971no—
198#124—971no—
199#123—1941no—
200C22⋊Q8—971no—
201#121—971no—
202C4×Q8—971no—
203C4×Q8C2×D9711yescharacteristic
204C4×Q8C2×D9711yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—971no—
208C4×D4—1941no—
209C4×D4C2×D9711yescharacteristic
210C4×D4C2×D9711yescharacteristic
211#117—21no—
212#117—1941no—
213#117C2×D9711yescharacteristic
214C2×C42—21no—
215C2×C42C2×D9711yescharacteristic
216#107—971no—
217#107—971no—
218#106—971no—
219#106—971no—
220C4⋊C8—971no—
221C4⋊C8—971no—
222#103—971no—
223#103—971no—
224#103—971no—
225#103—971no—
226#102—971no—
227#102—971no—
228#102—971no—
229#102—971no—
230C22⋊C8—971no—
231C22⋊C8—971no—
232#97—971no—
233#97—971no—
234#41—22no—
235#41—22no—
236C2×Q8—971no—
237C2×Q8C22×D9711yescharacteristic
238C2×D4—21no—
239C2×D4—971no—
240C2×D4—1941no—
241C2×D4C22×D9711yescharacteristic
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—1941no—
246C22×C4C22⋊D9711yescharacteristic
247C22×C4C22⋊D9711yescharacteristic
248C22×C4C22×D9711yescharacteristic
249Q16—1942no—
250SD16—1942no—
251SD16—1942no—
252D8—1942no—
253C2×C8—971no—
254C2×C8—971no—
255C2×C8—971no—
256C2×C8—971no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—1941no—
262C4⋊C4—1941no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4C22×D9711yescharacteristic
266C4⋊C4C22×D9711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—1941no—
279C42C22⋊D9711yescharacteristic
280C42C22⋊D9711yescharacteristic
281C42C22×D9711yescharacteristic
282C42C22×D9711yescharacteristic
283C23—21no—
284C23—1941no—
285C23C2×C22⋊D9711yescharacteristic
286Q8—21no—
287Q8—1941no—
288Q8—1941no—
289Q8#1309551112yes—
290D4—24no—
291D4—21no—
292D4—3882no—
293D4—1941no—
294D4—1941no—
295D4#1309551112yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—3882no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4C2×C22⋊D9711yescharacteristic
311C2×C4C2×C22⋊D9711yescharacteristic
312C2×C4C2×C22⋊D9711yescharacteristic
313C2×C4C2×C22⋊D9711yescharacteristic
314C2×C4C2×C22⋊D9711yescharacteristic
315C2×C4C23×D9711yescharacteristic
316C8—1941no—
317C8—1941no—
318C8—1941no—
319C8—1941no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—3881no—
327C22—3881no—
328C22—3881no—
329C22C22×C22⋊D9711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—3881no—
339C4—3881no—
340C4—3881no—
341C4C2×#1309551111yescharacteristic
342C4C2×#1309551111yescharacteristic
343C2—41no—
344C2—42no—
345C2—7761no—
346C2#2026124611yescharacteristic
347C2#2026153311yescharacteristic
348C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×#41×C97C2
socleC22×C97C22×D4
prime radical#20260880C2
hypercenterC4×#41D97
hypocenterC97#3111
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #20261269C2semidirect
2. #20261076C2semidirect
3. #20261059C2semidirect
4. #20261042C2semidirect
5. #20261035C2semidirect
6. #20260957C2semidirect
7. #20260950C2semidirect
8. #20260897C2semidirect
10. #20260845C2semidirect
11. #20260840C2semidirect
12. #20260835C2semidirect
13. #20260833C2semidirect
14. #20260792C2semidirect
15. C4×#41×C97C2semidirect
175. C97#3111semidirect
177. C4×#41D97semidirect

Maximal subgroups

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

subgroupindex
#202612692
#202610762
#202610592
#202610422
#202610352
#202609572
#202609502
#202608972
#202608802
#202608452
#202608402
#202608352
#202608332
#202607922
C4×#41×C972
#311197