Finite Group Databasebeta

← #27153599

Subgroups — #27153599

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261398C211yesmaximal, characteristic, complemented
2#20261386C211yesmaximal, characteristic
3#20261084C211yesmaximal, characteristic, complemented
4#20261067C211yesmaximal, characteristic, complemented
5#20261019C211yesmaximal, characteristic, complemented
6#20260966C211yesmaximal, characteristic, complemented
7#20260965C211yesmaximal, characteristic, complemented
8#20260939C211yesmaximal, characteristic, complemented
9#20260876C211yesmaximal, characteristic, complemented
10#20260875C211yesmaximal, characteristic, complemented
11#20260822C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic, complemented
13C2×#15690733C211yesmaximal, characteristic, complemented
14#103×D97C211yesmaximal, characteristic, complemented
15#483×C97C211yesmaximal, characteristic, complemented
16C2×#13095511—21no—
17C22×Dic194C2211yescharacteristic
18C2×Q8×D97C2211yescharacteristic
19C2×#13095500C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21C2×#13095498C2211yescharacteristic
22C22×#11715895C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
24#15690809—21no—
25#15690809C2211yescharacteristic
26#15690808C2211yescharacteristic
27#15690795—21no—
28#15690793—21no—
29#15690790C2211yescharacteristic
30#15690788—21no—
31#15690788—21no—
32#15690787—21no—
33#15690785C2211yescharacteristic
34#15690784C2211yescharacteristic
35#15690775C2211yescharacteristic
36#15690772C2211yescharacteristic
37C4×C22⋊D97C2211yescharacteristic
38#15690770—21no—
39C4⋊C4×D97C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690734C2211yescharacteristic
45#15690733C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690733C2211yescharacteristic
49#15690730C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C22⋊Q8×C97C2211yescharacteristic
53C2×C4⋊C4×C97C2211yescharacteristic
54#106×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59#13095511—44no—
60C2×Dic194—22no—
61C2×Dic194—21no—
62C2×Dic194—21no—
63C2×Dic194—21no—
64C2×Dic194—41no—
65C2×Dic194C2311yescharacteristic
66C2×Dic194C2311yescharacteristic
67Q8×D97—22no—
68Q8×D97—41no—
69C2×D388—21no—
70C2×D388C2311yescharacteristic
71C2×C22⋊D97—21no—
72C2×C22⋊D97D411yescharacteristic
73C22×Dic97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76C2×C4×D97—21no—
77C2×C4×D97C2311yescharacteristic
78#13095500—22no—
79#13095500—41no—
80#13095498—41no—
81C2×#11715895—41no—
82C2×#11715895C2311yescharacteristic
83C2×#11715895C2311yescharacteristic
84C8×D97—41no—
85C4⋊Dic97—21no—
86C4⋊Dic97C2311yescharacteristic
87#13095490—21no—
88#13095490—21no—
89#13095490—21no—
90#13095490—21no—
91#13095490—21no—
92#13095490D411yescharacteristic
93C22⋊Dic97D411yescharacteristic
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97#13095488—21no—
98#13095488D411yescharacteristic
99C4×Dic97—21no—
100C4×Dic97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C22×C4×C97—21no—
103C22×C4×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C4⋊C4×C97—21no—
107C4⋊C4×C97—21no—
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C4⋊C4×C97C2311yescharacteristic
111C22⋊C4×C97—21no—
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97C2×D411yescharacteristic
115Q8×C97—21no—
116Q8×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—42no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C22⋊D97—42no—
127C22⋊D97SD1614yescomplemented
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—42no—
132C2×Dic97C2×D411yescharacteristic
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135D388—21no—
136D388—21no—
137D388—42no—
138C4×D97—21no—
139C4×D97—21no—
140C4×D97—42no—
141C4×D97—42no—
142C4×D97—42no—
143Dic194—21no—
144Dic194—21no—
145Dic194—21no—
146Dic194—42no—
147Dic194—42no—
148Dic194—42no—
149Dic194#4112yescomplemented
150C8×C97—21no—
151C8×C97—21no—
152#11715895—21no—
153#11715895—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97C2×SD1612yes—
157C22×C97C22×D411yescharacteristic
158C2×D97—21no—
159C2×D97—21no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97C2×SD1612yes—
164C4×C97—21no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C2×#4111yescharacteristic
170C4×C97H(5,𝔽2)11yescharacteristic
171Dic97—22no—
172Dic97—21no—
173Dic97—21no—
174Dic97—41no—
175Dic97—41no—
176Dic97C2×SD1612yes—
177C2×C97—22no—
178C2×C97#54611yescharacteristic
179C2×C97C22×SD1611yescharacteristic
180C2×C97#57811yescharacteristic
181D97—22no—
182D97—81no—
183#3298—971nomaximal, Sylow
184C97#329811yescharacteristic, complemented, Sylow
185D4×Q8—971no—
186#547—971no—
187#487—971no—
188#483—971no—
189#483D9711yescharacteristic, complemented
190#475—971no—
191#465—971no—
192#465—971no—
193#461—971no—
194#448—971no—
195#448—971no—
196C4×SD16—971no—
197C8×D4—971no—
198C2×#103—971no—
199C2×#103—971no—
200C2×#41—1941no—
201C22×Q8—971no—
202C22×Q8—971no—
203C2×SD16—971no—
204C2×SD16—971no—
205C2×SD16—971no—
206C22×C8—971no—
207C22×C8—971no—
208C4⋊Q8—971no—
209C4⋊Q8—971no—
210C4⋊Q8—1941no—
211#123—1941no—
212#123—1941no—
213C22⋊Q8—971no—
214C22⋊Q8—1941no—
215C22⋊Q8—1941no—
216C22⋊Q8—1941no—
217C22⋊Q8C2×D9711yescharacteristic
218#121—971no—
219#121—971no—
220C4×Q8—971no—
221C4×D4—971no—
222C4×D4—971no—
223C4×D4—1941no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4C2×D9711yescharacteristic
226#106—971no—
227#106C2×D9711yescharacteristic
228#106C2×D9711yescharacteristic
229C4⋊C8—971no—
230#103—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103—971no—
236#103—971no—
237#103C2×D9711yescharacteristic
238#103C2×D9711yescharacteristic
239#102—971no—
240C22⋊C8—971no—
241C22⋊C8C2×D9711yescharacteristic
242C4×C8—971no—
243#41—3884no—
244C2×Q8—971no—
245C2×Q8—971no—
246C2×Q8—3881no—
247C2×Q8—3881no—
248C2×Q8—1941no—
249C2×Q8—1941no—
250C2×Q8—1942no—
251C2×Q8—1942no—
252C2×Q8—1941no—
253C2×Q8C22×D9711yescharacteristic
254C2×D4—971no—
255C2×D4—971no—
256C2×D4—1941no—
257C2×D4—1941no—
258C22×C4—21no—
259C22×C4—971no—
260C22×C4—1941no—
261C22×C4—1941no—
262C22×C4—1941no—
263C22×C4—1941no—
264C22×C4C22×D9711yescharacteristic
265SD16—3881no—
266SD16—3881no—
267SD16—1942no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—3881no—
271C2×C8—3881no—
272C2×C8C22×D9711yescharacteristic
273C2×C8C22×D9711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—971no—
277C4⋊C4—971no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4—1941no—
283C4⋊C4—1941no—
284C4⋊C4C22×D9711yescharacteristic
285C4⋊C4C22×D9711yescharacteristic
286C4⋊C4C22×D9711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—971no—
289C22⋊C4—971no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C42—971no—
295C42—1941no—
296C23—971no—
297C23—1941no—
298C23D4×D9711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—972no—
302Q8—3882no—
303Q8—3882no—
304Q8—3882no—
305Q8—1941no—
306Q8—1941no—
307Q8—1941no—
308D4—974no—
309D4—3882no—
310D4—3882no—
311D4—1941no—
312D4—1941no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—971no—
321C2×C4—971no—
322C2×C4—971no—
323C2×C4—3882no—
324C2×C4—3882no—
325C2×C4—3882no—
326C2×C4—3882no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4D4×D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—21no—
335C8—21no—
336C8—1941no—
337C8—1941no—
338C22—21no—
339C22—21no—
340C22—972no—
341C22—3881no—
342C22—3881no—
343C22—3881no—
344C22—1941no—
345C22—1941no—
346C22SD16×D9712yes—
347C22C2×D4×D9711yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—972no—
354C4—3881no—
355C4—3881no—
356C4—1941no—
357C4—1942no—
358C4—1941no—
359C4#41×D9711yescharacteristic
360C4#1569085611yescharacteristic
361C2—22no—
362C2—7761no—
363C2—1942no—
364C2#2026137711yescharacteristic
365C2C2×SD16×D9711yescharacteristic
366C2#2026155111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#483×C97C2
socleC22×C97C22×D4
prime radical#20261386C2
hypercenter#483D97
hypocenterC97#3298
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #20261398C2semidirect
3. #20261084C2semidirect
4. #20261067C2semidirect
5. #20261019C2semidirect
6. #20260966C2semidirect
7. #20260965C2semidirect
8. #20260939C2semidirect
9. #20260876C2semidirect
10. #20260875C2semidirect
11. #20260822C2semidirect
12. #20260798C2semidirect
13. C2×#15690733C2semidirect
14. #103×D97C2semidirect
15. #483×C97C2semidirect
127. C22⋊D97SD16semidirect
149. Dic194#41semidirect
184. C97#3298semidirect
189. #483D97semidirect

Maximal subgroups

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

subgroupindex
#202613982
#202613862
#202610842
#202610672
#202610192
#202609662
#202609652
#202609392
#202608762
#202608752
#202608222
#202607982
C2×#156907332
#103×D972
#483×C972
#329897