Finite Group Databasebeta

← #27153307

Subgroups — #27153307

Order 12416.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690841C211yesmaximal, characteristic, complemented
2C22×#13095501C211yesmaximal, characteristic, complemented
3#20261394C211yesmaximal, characteristic
4#20261127C211yesmaximal, characteristic, complemented
5#20260991C211yesmaximal, characteristic, complemented
6#20260966C211yesmaximal, characteristic, complemented
7#20260957C211yesmaximal, characteristic, complemented
8#20260942C211yesmaximal, characteristic, complemented
9#20260895C211yesmaximal, characteristic, complemented
10#20260891C211yesmaximal, characteristic, complemented
11#20260890C211yesmaximal, characteristic, complemented
12#20260874C211yesmaximal, characteristic, complemented
13#20260840C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15D4×Q8×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18#15690841—24no—
19C2×#13095501—24no—
20C2×#13095501C2211yescharacteristic
21C2×#13095501C2211yescharacteristic
22C2×#13095501C2211yescharacteristic
23C2×#13095501C2211yescharacteristic
24C2×#13095501C2211yescharacteristic
25C2×#13095501C2211yescharacteristic
26C2×#13095498C2211yescharacteristic
27C2×#13095498C2211yescharacteristic
28C2×#13095498C2211yescharacteristic
29C2×#13095495C2211yescharacteristic
30C22×#11715895C2211yescharacteristic
31#15690798C2211yescharacteristic
32#15690796—21no—
33#15690789C2211yescharacteristic
34#15690788—21no—
35#15690784C2211yescharacteristic
36#15690782—21no—
37C4×Dic194C2211yescharacteristic
38#15690770—21no—
39#15690741C2211yescharacteristic
40C4⋊#11715895C2211yescharacteristic
41#15690734C2211yescharacteristic
42#15690734C2211yescharacteristic
43#15690734C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690733C2211yescharacteristic
46#15690730C2211yescharacteristic
47C22⋊#11715895C2211yescharacteristic
48C22⋊#11715895C2211yescharacteristic
49#15690721C2211yescharacteristic
50C22×Q8×C97C2211yescharacteristic
51C22×Q8×C97C2211yescharacteristic
52C4⋊Q8×C97—21no—
53C4⋊Q8×C97C2211yescharacteristic
54C22⋊Q8×C97—21no—
55C22⋊Q8×C97—21no—
56C22⋊Q8×C97C2211yescharacteristic
57C22⋊Q8×C97C2211yescharacteristic
58C4×Q8×C97C2211yescharacteristic
59C4×D4×C97—21no—
60C4×D4×C97C2211yescharacteristic
61#13095511—22no—
62#13095511—41no—
63C2×Dic194—22no—
64C2×Dic194—21no—
65C2×Dic194—41no—
66C2×Dic194C2311yescharacteristic
67C2×Dic194C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C22×Dic97—21no—
72C2×C4×D97—21no—
73#13095501—22no—
74#13095501—22no—
75#13095501—22no—
76#13095501—22no—
77#13095501—41no—
78#13095501D414yescomplemented
79#13095498—22no—
80#13095498—22no—
81#13095498—41no—
82#13095495—22no—
83C2×#11715895—22no—
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C4⋊Dic97C2311yescharacteristic
89#13095490—21no—
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95C4×Dic97—21no—
96C2×Q8×C97—21no—
97C2×Q8×C97—21no—
98C2×Q8×C97—22no—
99C2×Q8×C97—22no—
100C2×Q8×C97—41no—
101C2×Q8×C97—41no—
102C2×Q8×C97C2311yescharacteristic
103C2×Q8×C97C2311yescharacteristic
104C2×Q8×C97C2311yescharacteristic
105C2×D4×C97D411yescharacteristic
106C22×C4×C97—21no—
107C22×C4×C97—21no—
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C4⋊C4×C97—21no—
111C4⋊C4×C97—21no—
112C4⋊C4×C97—21no—
113C4⋊C4×C97—21no—
114C4⋊C4×C97—21no—
115C4⋊C4×C97D411yescharacteristic
116C4⋊C4×C97C2311yescharacteristic
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97D411yescharacteristic
120C22⋊C4×C97D411yescharacteristic
121C42×C97—21no—
122C42×C97C2311yescharacteristic
123C23×C97C2×D411yescharacteristic
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126Q8×C97—21no—
127Q8×C97—21no—
128Q8×C97—21no—
129Q8×C97—21no—
130Q8×C97—21no—
131Q8×C97—42no—
132Q8×C97—42no—
133Q8×C97C2×D412yes—
134D4×C97—22no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—42no—
144C2×C4×C97—42no—
145C2×C4×C97C2×D411yescharacteristic
146C2×C4×C97C2×D411yescharacteristic
147C2×C4×C97C2×D411yescharacteristic
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2411yescharacteristic
150C22⋊D97—42no—
151C22⋊D97—42no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—42no—
156D388—21no—
157D388—21no—
158C4×D97—42no—
159Dic194—21no—
160Dic194—21no—
161Dic194—21no—
162Dic194—42no—
163Dic194C2×D412yes—
164#11715895—21no—
165#11715895—21no—
166#11715895—21no—
167#11715895C2×D412yes—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97#13712yescomplemented
174C22×C97C22×D411yescharacteristic
175C2×D97—41no—
176C2×D97—41no—
177C2×D97—41no—
178C4×C97—21no—
179C4×C97—21no—
180C4×C97—22no—
181C4×C97—21no—
182C4×C97—21no—
183C4×C97—41no—
184C4×C97—41no—
185C4×C97—41no—
186C4×C97C22×D411yescharacteristic
187C4×C97H(5,𝔽2)11yescharacteristic
188Dic97—22no—
189Dic97—41no—
190Dic97—41no—
191C2×C97—22no—
192C2×C97—41no—
193C2×C97D4211yescharacteristic
194C2×C97C2×#13711yescharacteristic
195C2×C97#57811yescharacteristic
196D97—81no—
197#3270—971nomaximal, Sylow
198C97#327011yescharacteristic, complemented, Sylow
199C2×#137—971no—
200C22×Q16—971no—
201D4×Q8D9711yescharacteristic, complemented
202#548—971no—
203#497—971no—
204#470—971no—
205#465—971no—
206#464—971no—
207#462—971no—
208#452—971no—
209#451—971no—
210#451—971no—
211#448—971no—
212#441—971no—
213#435—971no—
214C2×#41—971no—
215C22×Q8—971no—
216C22×Q8C2×D9711yescharacteristic
217C22×Q8C2×D9711yescharacteristic
218#137—1944no—
219C2×Q16—971no—
220C2×Q16—971no—
221C2×Q16—971no—
222C2×Q16—971no—
223C2×Q16—971no—
224C2×Q16—971no—
225C2×Q16—1944no—
226C2×SD16—971no—
227C2×SD16—971no—
228C2×SD16—971no—
229C2×M16—971no—
230C22×C8—971no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D9711yescharacteristic
233#124—971no—
234#124—1941no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—971no—
238C22⋊Q8—1941no—
239C22⋊Q8C2×D9711yescharacteristic
240C22⋊Q8C2×D9711yescharacteristic
241#121—971no—
242#121—1941no—
243C4×Q8—971no—
244C4×Q8C2×D9711yescharacteristic
245C4×D4—21no—
246C4×D4—1941no—
247C4×D4C2×D9711yescharacteristic
248#106—971no—
249C4⋊C8—971no—
250#103—971no—
251#103—971no—
252#103—971no—
253#103—971no—
254#103—971no—
255#102—971no—
256C22⋊C8—971no—
257C22⋊C8—971no—
258#97—971no—
259#41—3881no—
260#41—1942no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—971no—
268C2×Q8—971no—
269C2×Q8—3881no—
270C2×Q8—1942no—
271C2×Q8—1941no—
272C2×Q8C22×D9711yescharacteristic
273C2×Q8C22×D9711yescharacteristic
274C2×Q8C22×D9711yescharacteristic
275C2×D4—971no—
276C2×D4—1941no—
277C2×D4—1941no—
278C2×D4C22⋊D9711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1941no—
282C22×C4—1941no—
283C22×C4C22×D9711yescharacteristic
284C22×C4C22×D9711yescharacteristic
285Q16—974no—
286Q16—3881no—
287Q16—1942no—
288Q16—1942no—
289Q16—1942no—
290Q16—1942no—
291SD16—3881no—
292SD16—1942no—
293SD16—1942no—
294M16—1942no—
295C2×C8—971no—
296C2×C8—971no—
297C2×C8—971no—
298C2×C8—971no—
299C2×C8—1942no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—971no—
306C4⋊C4—1941no—
307C4⋊C4—1941no—
308C4⋊C4C22⋊D9711yescharacteristic
309C4⋊C4C22×D9711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1941no—
313C22⋊C4—1941no—
314C22⋊C4—1941no—
315C22⋊C4—1941no—
316C22⋊C4C22⋊D9711yescharacteristic
317C22⋊C4C22⋊D9711yescharacteristic
318C42—21no—
319C42—1941no—
320C42C22×D9711yescharacteristic
321C23—1941no—
322C23C2×C22⋊D9711yescharacteristic
323C23C2×C22⋊D9711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—972no—
332Q8—3882no—
333Q8—1941no—
334Q8—1941no—
335Q8—1941no—
336Q8D4×D9712yes—
337D4—22no—
338D4—3882no—
339D4—3882no—
340D4—1941no—
341D4—1941no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3882no—
353C2×C4—3882no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4C2×C22⋊D9711yescharacteristic
358C2×C4C2×C22⋊D9711yescharacteristic
359C2×C4C2×C22⋊D9711yescharacteristic
360C2×C4C2×C22⋊D9711yescharacteristic
361C2×C4C23×D9711yescharacteristic
362C8—972no—
363C8—1941no—
364C8—1941no—
365C8—1941no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—3881no—
372C22—3881no—
373C22—3881no—
374C22#1569084112yescomplemented
375C22C22×C22⋊D9711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3881no—
385C4—3881no—
386C4—1942no—
387C4C2×D4×D9711yescharacteristic
388C4#1569085511yescharacteristic
389C2—22no—
390C2—41no—
391C2—7761no—
392C2D4×C22⋊D9711yescharacteristic
393C2C2×#1569084111yescharacteristic
394C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD4×Q8×C97C2
socleC22×C97C22×D4
prime radical#20261394C2
hypercenterD4×Q8D97
hypocenterC97#3270
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
2. C22×#13095501C2semidirect
4. #20261127C2semidirect
5. #20260991C2semidirect
6. #20260966C2semidirect
7. #20260957C2semidirect
8. #20260942C2semidirect
9. #20260895C2semidirect
10. #20260891C2semidirect
11. #20260890C2semidirect
12. #20260874C2semidirect
13. #20260840C2semidirect
14. #20260807C2semidirect
15. D4×Q8×C97C2semidirect
78. #13095501D4semidirect
173. C22×C97#137semidirect
198. C97#3270semidirect
201. D4×Q8D97semidirect
374. C22#15690841semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C22×#130955012
#202613942
#202611272
#202609912
#202609662
#202609572
#202609422
#202608952
#202608912
#202608902
#202608742
#202608402
#202608072
D4×Q8×C972
#327097