Finite Group Databasebeta

← #27153727

Subgroups — #27153727

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261389C211yesmaximal, characteristic, complemented
2#20261094C211yesmaximal, characteristic, complemented
3#20261068C211yesmaximal, characteristic, complemented
4#20261046C211yesmaximal, characteristic, complemented
5#20261030C211yesmaximal, characteristic, complemented
6#20260983C211yesmaximal, characteristic, complemented
7#20260969C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic, complemented
10#20260874C211yesmaximal, characteristic, complemented
11#20260833C211yesmaximal, characteristic, complemented
12#20260807C211yesmaximal, characteristic, complemented
13#20260736C211yesmaximal, characteristic, complemented
14C2×#15690734C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095510—21no—
17C2×#13095498C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095495C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22#15690810C2211yescharacteristic
23#15690794—21no—
24#15690794—21no—
25#15690790—21no—
26#15690784C2211yescharacteristic
27#15690784C2211yescharacteristic
28C4×D388C2211yescharacteristic
29C2×C4⋊Dic97C2211yescharacteristic
30C2×C4⋊Dic97C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690743C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690734C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690734C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47#15690721C2211yescharacteristic
48C2×#41×C97C2211yescharacteristic
49C22×Q8×C97C2211yescharacteristic
50#124×C97—21no—
51#124×C97C2211yescharacteristic
52C22⋊Q8×C97—21no—
53C22⋊Q8×C97C2211yescharacteristic
54#121×C97—21no—
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97—21no—
58C4×D4×C97C2211yescharacteristic
59#13095510—44no—
60C2×Dic194—21no—
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64C22×Dic97—21no—
65C22×Dic97—21no—
66C2×C4×D97—21no—
67#13095498—22no—
68#13095498—41no—
69#13095496—41no—
70#13095495—41no—
71C2×#11715895—41no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82#13095488—21no—
83#13095488—21no—
84#13095488—21no—
85#41×C97—22no—
86#41×C97—41no—
87C2×Q8×C97—21no—
88C2×Q8×C97—22no—
89C2×Q8×C97—41no—
90C2×Q8×C97C2311yescharacteristic
91C2×Q8×C97C2311yescharacteristic
92C2×D4×C97—21no—
93C2×D4×C97—21no—
94C2×D4×C97D411yescharacteristic
95C2×D4×C97C2311yescharacteristic
96C22×C4×C97—21no—
97C22×C4×C97—21no—
98C22×C4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C4⋊C4×C97—21no—
101C4⋊C4×C97—21no—
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97—21no—
108C22⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C42×C97—21no—
111C42×C97C2311yescharacteristic
112C23×C97—21no—
113C23×C97C2×D411yescharacteristic
114C23×C97C2×D411yescharacteristic
115C22×D97—21no—
116Q8×C97—21no—
117Q8×C97—21no—
118Q8×C97—21no—
119Q8×C97—42no—
120Q8×C97#4112yescomplemented
121D4×C97—22no—
122D4×C97—21no—
123D4×C97—21no—
124D4×C97—42no—
125D4×C97—42no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—42no—
134C2×C4×C97—42no—
135C2×C4×C97C2×D411yescharacteristic
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2411yescharacteristic
140C22⋊D97—42no—
141C22⋊D97—42no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—42no—
146C2×Dic97—42no—
147D388—21no—
148D388—21no—
149C4×D97—42no—
150Dic194—42no—
151#11715895—21no—
152#11715895—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97AGL(1,ℤ/8)12yes—
164C22×C97C22×D411yescharacteristic
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—22no—
171C4×C97—21no—
172C4×C97—21no—
173C4×C97—41no—
174C4×C97—41no—
175C4×C97C2×#4111yescharacteristic
176C4×C97H(5,𝔽2)11yescharacteristic
177Dic97—41no—
178Dic97—41no—
179Dic97—41no—
180C2×C97—22no—
181C2×C97—41no—
182C2×C97—81no—
183C2×C97#54611yescharacteristic
184C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C97#57811yescharacteristic
186D97—81no—
187#3308—971nomaximal, Sylow
188C97#330811yescharacteristic, complemented, Sylow
189#548D9711yescharacteristic, complemented
190#547—971no—
191#489—971no—
192#484—971no—
193#480—971no—
194#477—971no—
195#468—971no—
196#466—971no—
197#463—971no—
198#449—971no—
199#448—971no—
200#440—971no—
201#435—971no—
202#417—971no—
203C2×#103—971no—
204C2×#41—1941no—
205C2×#41C2×D9711yescharacteristic
206C22×Q8C2×D9711yescharacteristic
207C2×SD16—971no—
208C2×SD16—971no—
209C2×D8—971no—
210C2×M16—971no—
211C22×C8—971no—
212C4⋊Q8—971no—
213#124—21no—
214#124C2×D9711yescharacteristic
215#123—1941no—
216#123—1941no—
217C22⋊Q8—21no—
218C22⋊Q8—1941no—
219C22⋊Q8C2×D9711yescharacteristic
220#121—21no—
221#121—971no—
222#121—971no—
223#121C2×D9711yescharacteristic
224C4×Q8C2×D9711yescharacteristic
225C4×D4—21no—
226C4×D4—971no—
227C4×D4C2×D9711yescharacteristic
228C2×C4⋊C4—971no—
229C2×C4⋊C4—971no—
230#107—971no—
231#107—971no—
232#107—971no—
233C4⋊C8—971no—
234#103—971no—
235#103—971no—
236#103—971no—
237#103—971no—
238#103—971no—
239#103—971no—
240#102—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244C22⋊C8—971no—
245C22⋊C8—971no—
246#97—971no—
247#41—22no—
248#41—41no—
249#41—3884no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—1941no—
254C2×Q8C22×D9711yescharacteristic
255C2×Q8C22×D9711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—971no—
259C2×D4—1941no—
260C2×D4—1941no—
261C2×D4C22⋊D9711yescharacteristic
262C2×D4C22×D9711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1941no—
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4C22×D9711yescharacteristic
269C22×C4C22×D9711yescharacteristic
270SD16—3881no—
271SD16—1942no—
272D8—3881no—
273M16—3881no—
274C2×C8—971no—
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—971no—
278C2×C8—3881no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4C22⋊D9711yescharacteristic
288C4⋊C4C22×D9711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4C22⋊D9711yescharacteristic
297C22⋊C4C22⋊D9711yescharacteristic
298C42—21no—
299C42C22×D9711yescharacteristic
300C23—21no—
301C23—1941no—
302C23C2×C22⋊D9711yescharacteristic
303C23C2×C22⋊D9711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—3882no—
309Q8#1309551012yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3882no—
316D4—3882no—
317D4—1941no—
318D4—1941no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3882no—
329C2×C4—3882no—
330C2×C4—3882no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4—1941no—
334C2×C4C2×C22⋊D9711yescharacteristic
335C2×C4C2×C22⋊D9711yescharacteristic
336C2×C4C2×C22⋊D9711yescharacteristic
337C2×C4C2×C22⋊D9711yescharacteristic
338C2×C4C23×D9711yescharacteristic
339C8—1941no—
340C8—1941no—
341C8—1941no—
342C8—1941no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—3881no—
352C22—3881no—
353C22—3881no—
354C22#1569083512yes—
355C22C22×C22⋊D9711yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—3881no—
364C4—3881no—
365C4—3881no—
366C4C2×#1309551011yescharacteristic
367C4#1569085511yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—7761no—
372C2#2026137911yescharacteristic
373C2C2×#1569083511yescharacteristic
374C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#548×C97C2
socleC22×C97C22×D4
hypercenter#548D97
hypocenterC97#3308
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20261389C2semidirect
2. #20261094C2semidirect
3. #20261068C2semidirect
4. #20261046C2semidirect
5. #20261030C2semidirect
6. #20260983C2semidirect
7. #20260969C2semidirect
8. #20260947C2semidirect
9. #20260880C2semidirect
10. #20260874C2semidirect
11. #20260833C2semidirect
12. #20260807C2semidirect
13. #20260736C2semidirect
14. C2×#15690734C2semidirect
15. #548×C97C2semidirect
120. Q8×C97#41semidirect
188. C97#3308semidirect
189. #548D97semidirect
309. Q8#13095510semidirect

Maximal subgroups

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

subgroupindex
#202613892
#202610942
#202610682
#202610462
#202610302
#202609832
#202609692
#202609472
#202608802
#202608742
#202608332
#202608072
#202607362
C2×#156907342
#548×C972
#330897