Finite Group Databasebeta

← #27150436

Subgroups — #27150436

Order 12416.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690818C211yesmaximal, characteristic, complemented
2C2×M16×D97C211yesmaximal, characteristic, complemented
3#20261244C211yesmaximal, characteristic
4#20260814C211yesmaximal, characteristic, complemented
5#20260811C211yesmaximal, characteristic, complemented
6#20260811C211yesmaximal, characteristic, complemented
7#20260809C211yesmaximal, characteristic, complemented
8#20260807C211yesmaximal, characteristic, complemented
9#20260805C211yesmaximal, characteristic, complemented
10#20260803C211yesmaximal, characteristic, complemented
11#20260760C211yesmaximal, characteristic, complemented
12#20260700C211yesmaximal, characteristic, complemented
13#20260697C211yesmaximal, characteristic, complemented
14#20260689C211yesmaximal, characteristic, complemented
15#436×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18#15690818—24no—
19C2×#13095495C2211yescharacteristic
20C2×#13095494C2211yescharacteristic
21C2×#13095494C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×#13095494C2211yescharacteristic
24C2×#13095494C2211yescharacteristic, complemented
25M16×D97—24no—
26C22×#11715895C2211yescharacteristic, complemented
27C2×C8×D97C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic
29C2×C8×D97C2211yescharacteristic, complemented
30#15690795C411yescharacteristic
31#15690792C411yescharacteristic
32#15690788C411yescharacteristic
33#15690787C411yescharacteristic
34#15690783C411yescharacteristic
35#15690782C411yescharacteristic
36#15690780C411yescharacteristic
37#15690778C411yescharacteristic
38C4×D388C2211yescharacteristic
39C4×C22⋊D97C2211yescharacteristic
40C4×C22⋊D97C2211yescharacteristic
41#15690767C2211yescharacteristic
42C4⋊#11715895C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690739C2211yescharacteristic, complemented
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic, complemented
47#15690722C2211yescharacteristic
48#15690722C2211yescharacteristic
49#15690722C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51#15690721C2211yescharacteristic, complemented
52#15690720C2211yescharacteristic, complemented
53C8×Dic97C2211yescharacteristic, complemented
54C2×M16×C97C2211yescharacteristic
55C2×M16×C97C2211yescharacteristic, complemented
56C4×D4×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic
59C22⋊C8×C97C2211yescharacteristic, complemented
60C4×C8×C97C2211yescharacteristic, complemented
61C23×D97C2×C411yescharacteristic
62#13095511—22no—
63#13095511—22no—
64C2×Dic194C2×C411yescharacteristic
65C2×D388C2×C411yescharacteristic
66C2×D388C2×C411yescharacteristic
67C2×C22⋊D97C2×C411yescharacteristic
68C2×C22⋊D97C2×C411yescharacteristic
69C2×C22⋊D97C2×C411yescharacteristic
70C2×C22⋊D97C2×C411yescharacteristic
71C22×Dic97C2×C411yescharacteristic
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97C2×C411yescharacteristic
75C2×C4×D97C2×C411yescharacteristic
76C2×C4×D97C2311yescharacteristic
77C2×C4×D97C2311yescharacteristic
78C2×C4×D97C2311yescharacteristic
79#13095495—22no—
80#13095494—22no—
81#13095494—22no—
82#13095494—22no—
83#13095494—22no—
84#13095494—22no—
85C2×#11715895—22no—
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C2×#11715895C2311yescharacteristic
90C8×D97—22no—
91C8×D97—22no—
92C8×D97D414yescomplemented
93C4⋊Dic97C2×C411yescharacteristic
94#13095490C2×C411yescharacteristic
95#13095490C2×C411yescharacteristic
96#13095490C2×C411yescharacteristic
97#13095490C2×C411yescharacteristic
98C22⋊Dic97C2×C411yescharacteristic
99C22⋊Dic97C2×C411yescharacteristic
100#13095488C2×C411yescharacteristic
101#13095488C2×C411yescharacteristic
102#13095488C2×C411yescharacteristic
103#13095488C2×C411yescharacteristic
104#13095488C2×C411yescharacteristic
105#13095488C2×C411yescharacteristic
106C4×Dic97C2311yescharacteristic
107C2×D4×C97C2×C411yescharacteristic
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110M16×C97—22no—
111M16×C97—22no—
112C2×C8×C97C2311yescharacteristic
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C2×C8×C97C2311yescharacteristic
116C4⋊C4×C97C2×C411yescharacteristic
117C22⋊C4×C97C2×C411yescharacteristic
118C22⋊C4×C97C2×C411yescharacteristic
119C42×C97C2311yescharacteristic
120C23×C97C22×C411yescharacteristic
121C23×C97C22×C411yescharacteristic
122C22×D97—22no—
123C22×D97—41no—
124C22×D97—41no—
125C22×D97C22×C411yescharacteristic
126C22×D97C22×C411yescharacteristic
127C22×D97C22×C411yescharacteristic
128D4×C97—41no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97C22×C411yescharacteristic
134C2×C4×C97C22×C411yescharacteristic
135C2×C4×C97C22×C411yescharacteristic
136C2×C4×C97C22×C411yescharacteristic
137C2×C4×C97C2411yescharacteristic
138C22⋊D97—22no—
139C22⋊D97—22no—
140C22⋊D97—41no—
141C22⋊D97—41no—
142C2×Dic97—22no—
143C2×Dic97C22×C411yescharacteristic
144C2×Dic97C22×C411yescharacteristic
145C2×Dic97C22×C411yescharacteristic
146C2×Dic97C22×C411yescharacteristic
147C2×Dic97C22×C411yescharacteristic
148D388—22no—
149D388—41no—
150C4×D97—22no—
151C4×D97—21no—
152C4×D97—21no—
153C4×D97—21no—
154C4×D97—21no—
155C4×D97—22no—
156C4×D97C2×D412yes—
157C4×D97#4112yes—
158Dic194—22no—
159C8×C97—21no—
160C8×C97—21no—
161C8×C97—21no—
162C8×C97C2×D412yes—
163#11715895—21no—
164#11715895—21no—
165#11715895—21no—
166#11715895C2×D412yes—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97C23×C411yescharacteristic
174C2×D97—21no—
175C2×D97—21no—
176C2×D97—21no—
177C2×D97—21no—
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—21no—
181C2×D97—21no—
182C2×D97—44no—
183C2×D97C4×D412yes—
184C4×C97—21no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97—21no—
188C4×C97C22×D411yescharacteristic
189C4×C97C2×#4111yescharacteristic
190Dic97—21no—
191Dic97—21no—
192Dic97—21no—
193Dic97—21no—
194Dic97C4×D412yes—
195C2×C97—41no—
196C2×C97—41no—
197C2×C97C2×C4×D411yescharacteristic
198C2×C97#56811yescharacteristic
199C2×C97#56811yescharacteristic
200D97—22no—
201D97—41no—
202D97—41no—
203#2930—971nomaximal, Sylow
204C97#293011yescharacteristic, complemented, Sylow
205C2×#131—971no—
206C22×M16—971no—
207#525—971no—
208#436—971no—
209#436—971no—
210#436—971no—
211#436D9711yescharacteristic, complemented
212#435—971no—
213#435—971no—
214#435—971no—
215#435—971no—
216#424—971no—
217#408—971no—
218#407—971no—
219#405—971no—
220C2×#41—971no—
221C23×C4—971no—
222#131—1944no—
223C2×M16—971no—
224C2×M16—971no—
225C2×M16—971no—
226C2×M16—971no—
227C2×M16—971no—
228C2×M16—971no—
229C2×M16—1944no—
230C2×M16C2×D9711yescharacteristic
231C2×M16C2×D9711yescharacteristic, complemented
232C22×C8—971no—
233C22×C8—971no—
234C22×C8—971no—
235C22×C8—971no—
236#123—971no—
237#123—971no—
238C22⋊Q8—971no—
239C22⋊Q8—971no—
240#121—971no—
241#121—971no—
242#120—971no—
243#120—971no—
244C4×D4—971no—
245C4×D4—971no—
246C4×D4—971no—
247C4×D4C2×D9711yescharacteristic
248#117—971no—
249C4⋊C8—971no—
250C4⋊C8—971no—
251C4⋊C8—971no—
252C4⋊C8C2×D9711yescharacteristic, complemented
253C22⋊C8—971no—
254C22⋊C8—971no—
255C22⋊C8—971no—
256C22⋊C8—971no—
257C22⋊C8—971no—
258C22⋊C8—971no—
259C22⋊C8C2×D9711yescharacteristic
260C22⋊C8C2×D9711yescharacteristic, complemented
261#97—971no—
262#97—971no—
263C4×C8—971no—
264C4×C8C2×D9711yescharacteristic, complemented
265C24—971no—
266#41—1942no—
267#41—1942no—
268C2×Q8—971no—
269C2×D4—971no—
270C2×D4—971no—
271C2×D4—971no—
272C2×D4—971no—
273C2×D4—971no—
274C2×D4—971no—
275C2×D4C4×D9711yescharacteristic
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—1942no—
283C22×C4—1942no—
284C22×C4C22×D9711yescharacteristic
285C22×C4C22×D9711yescharacteristic
286M16—22no—
287M16—22no—
288M16—1942no—
289M16—1942no—
290M16—1942no—
291M16—1942no—
292M16—1942no—
293M16—1942no—
294C2×C8—971no—
295C2×C8—974no—
296C2×C8—971no—
297C2×C8—971no—
298C2×C8—971no—
299C2×C8—1942no—
300C2×C8—1942no—
301C2×C8—1942no—
302C2×C8C22×D9711yescharacteristic
303C2×C8C22×D9711yescharacteristic
304C2×C8C22×D9711yescharacteristic
305C2×C8C22×D9711yescharacteristic
306C4⋊C4—971no—
307C4⋊C4—971no—
308C4⋊C4—971no—
309C4⋊C4—971no—
310C4⋊C4—971no—
311C4⋊C4C4×D9711yescharacteristic
312C22⋊C4—971no—
313C22⋊C4—971no—
314C22⋊C4—971no—
315C22⋊C4—971no—
316C22⋊C4—971no—
317C22⋊C4—971no—
318C22⋊C4—971no—
319C22⋊C4—971no—
320C22⋊C4C4×D9711yescharacteristic
321C22⋊C4C4×D9711yescharacteristic
322C42—971no—
323C42C22×D9711yescharacteristic
324C23—971no—
325C23—971no—
326C23—971no—
327C23—3881no—
328C23—3881no—
329C23—1942no—
330C23C2×C4×D9711yescharacteristic
331C23C2×C4×D9711yescharacteristic
332Q8—1942no—
333D4—41no—
334D4—3881no—
335D4—3881no—
336D4—3881no—
337D4—1942no—
338D4—1942no—
339D4—1942no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—971no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—971no—
348C2×C4—971no—
349C2×C4—971no—
350C2×C4—971no—
351C2×C4—1942no—
352C2×C4—1942no—
353C2×C4—1942no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4—1941no—
358C2×C4C2×C4×D9711yescharacteristic
359C2×C4C2×C4×D9711yescharacteristic
360C2×C4C2×C4×D9711yescharacteristic
361C2×C4C2×C4×D9711yescharacteristic
362C2×C4C23×D9711yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—972no—
367C8—1941no—
368C8—1941no—
369C8—1941no—
370C8D4×D9712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—972no—
378C22—3884no—
379C22—1941no—
380C22—1941no—
381C22—1941no—
382C22—1941no—
383C22—1941no—
384C22—1941no—
385C22—1941no—
386C22—1941no—
387C22C22×C4×D9711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—972no—
393C4—1941no—
394C4—1941no—
395C4—1941no—
396C4—1941no—
397C4C2×D4×D9711yescharacteristic
398C4#41×D9711yescharacteristic
399C2—41no—
400C2—41no—
401C2—3881no—
402C2—3881no—
403C2—1942no—
404C2C4×D4×D9711yescharacteristic
405C2#2026150011yescharacteristic
406C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#436×C97C2
prime radical#20261244C2
cosocleC2×C4×C97C24
hypercenter#436D97
hypocenterC97#2930
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#15690818C2semidirect
2. C2×M16×D97C2semidirect
4. #20260814C2semidirect
5. #20260811C2semidirect
6. #20260811C2semidirect
7. #20260809C2semidirect
8. #20260807C2semidirect
9. #20260805C2semidirect
10. #20260803C2semidirect
11. #20260760C2semidirect
12. #20260700C2semidirect
13. #20260697C2semidirect
14. #20260689C2semidirect
15. #436×C97C2semidirect
24. C2×#13095494C22semidirect
26. C22×#11715895C22semidirect
29. C2×C8×D97C22semidirect
42. C4⋊#11715895C22semidirect
43. #15690739C22semidirect
44. #15690739C22semidirect
46. C22⋊#11715895C22semidirect
49. #15690722C22semidirect
50. #15690722C22semidirect
51. #15690721C22semidirect
52. #15690720C22semidirect
53. C8×Dic97C22semidirect
55. C2×M16×C97C22semidirect
57. C4⋊C8×C97C22semidirect
59. C22⋊C8×C97C22semidirect
60. C4×C8×C97C22semidirect
92. C8×D97D4semidirect
204. C97#2930semidirect
211. #436D97semidirect
231. C2×M16C2×D97semidirect
252. C4⋊C8C2×D97semidirect
260. C22⋊C8C2×D97semidirect
264. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908182
C2×M16×D972
#202612442
#202608142
#202608112
#202608092
#202608072
#202608052
#202608032
#202607602
#202607002
#202606972
#202606892
#436×C972
#293097