Finite Group Databasebeta

← #27144366

Subgroups — #27144366

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690770C211yesmaximal, characteristic
2#20260628C211yesmaximal, characteristic, complemented
3#20260619C211yesmaximal, characteristic, complemented
4#20260617C211yesmaximal, characteristic, complemented
5#20260611C211yesmaximal, characteristic
6#20260611C211yesmaximal, characteristic
7#20260609C211yesmaximal, characteristic, complemented
8#20260606C211yesmaximal, characteristic, complemented
9#20260602C211yesmaximal, characteristic, complemented
10#20260600C211yesmaximal, characteristic, complemented
11#20260596C211yesmaximal, characteristic, complemented
12#20260593C211yesmaximal, characteristic, complemented
13C4×#13095488C211yesmaximal, characteristic, complemented
14#95×D97C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690770C418yescomplemented
22C2×C4⋊Dic97C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33#15690717C2211yescharacteristic
34#15690717C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45C2×C4⋊C4×C97C2211yescharacteristic
46C2×C22⋊C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C23×D97C2311yescharacteristic
53C2×C22⋊D97—22no—
54C2×C22⋊D97—22no—
55C2×C22⋊D97C2×C414yes—
56C22×Dic97—41no—
57C22×Dic97—41no—
58C22×Dic97C2×C414yescomplemented
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C2×C4×D97—22no—
67C2×C4×D97—22no—
68C2×C4×D97—41no—
69C2×C4×D97—41no—
70C2×C4×D97—41no—
71C2×C4×D97—41no—
72C2×C4×D97—41no—
73C2×C4×D97—41no—
74C2×C4×D97C2×C414yes—
75C4⋊Dic97—41no—
76#13095490—41no—
77#13095490—41no—
78#13095490C2×C414yescomplemented
79C22⋊Dic97—41no—
80C22⋊Dic97—41no—
81#13095488—22no—
82#13095488—22no—
83#13095488C2×C414yes—
84C4×Dic97—41no—
85C4×Dic97C2×C414yescomplemented
86C24×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—41no—
94C22⋊C4×C97—41no—
95C22⋊C4×C97—41no—
96C22⋊C4×C97C2×C414yescomplemented
97C42×C97—41no—
98C23×C97—21no—
99C23×C97—21no—
100C23×C97—21no—
101C23×C97—21no—
102C23×C97—21no—
103C23×C97—21no—
104C23×C97C22×C412yes—
105C23×C97C2411yescharacteristic
106C22×D97—21no—
107C22×D97—21no—
108C22×D97—21no—
109C22×D97—21no—
110C22×D97C22×C412yes—
111C22×D97#4112yes—
112C22×D97#4112yes—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×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—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97C22×C412yes—
130C2×C4×C97C22×C412yes—
131C22⋊D97—28no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150C2×Dic97—44no—
151C2×Dic97C22×C412yes—
152C2×Dic97C22×C412yes—
153C2×Dic97C22×C412yes—
154C4×D97—44no—
155C4×D97—44no—
156C4×D97—44no—
157C22×C97—22no—
158C22×C97—22no—
159C22×C97—22no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97C23×C411yescharacteristic
165C22×C97C2×#4111yescharacteristic
166C22×C97C2×#4111yescharacteristic
167C22×C97H(5,𝔽2)11yescharacteristic
168C22×C97H(5,𝔽2)11yescharacteristic
169C22×C97H(5,𝔽2)11yescharacteristic
170C22×C972−1+411yescharacteristic
171C2×D97—22no—
172C2×D97—22no—
173C2×D97—22no—
174C2×D97—22no—
175C2×D97—22no—
176C2×D97—22no—
177C2×D97#11714yes—
178C4×C97—22no—
179C4×C97—22no—
180C4×C97—41no—
181C4×C97—41no—
182C4×C97—41no—
183C4×C97—41no—
184Dic97—22no—
185Dic97—22no—
186Dic97—22no—
187Dic97—41no—
188Dic97—41no—
189Dic97—41no—
190Dic97—41no—
191C2×C97—42no—
192C2×C97C2×#11711yescharacteristic
193C2×C97#51811yescharacteristic
194C2×C97#52011yescharacteristic
195C2×C97#53811yescharacteristic
196C2×C97#53811yescharacteristic
197C2×C97#53911yescharacteristic
198C2×C97#53911yescharacteristic
199D97—24no—
200#2318—971nomaximal, Sylow
201C97#231811yescharacteristic, complemented, Sylow
202C2×C4×D4—971no—
203#388—971no—
204#388D9711yescharacteristic, complemented
205#387—971no—
206#387—971no—
207#386—971no—
208#386—971no—
209#385—971no—
210#385—971no—
211#383—971no—
212#382—971no—
213#381—971no—
214#380—971no—
215C4×C22⋊C4—971no—
216C2×#95—971no—
217C22×D4—971no—
218C23×C4—971no—
219C23×C4—971no—
220C23×C4—971no—
221C23×C4—971no—
222C4×D4—978no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—971no—
226C2×C4⋊C4—971no—
227C2×C4⋊C4C2×D9711yescharacteristic
228C2×C22⋊C4—971no—
229C2×C22⋊C4—971no—
230C2×C22⋊C4—971no—
231C2×C22⋊C4—971no—
232C2×C22⋊C4—971no—
233C2×C22⋊C4C2×D9711yescharacteristic
234C2×C22⋊C4C2×D9711yescharacteristic
235C2×C22⋊C4C2×D9711yescharacteristic
236C2×C42—971no—
237C2×C42—971no—
238C2×C42C2×D9711yescharacteristic
239#95—971no—
240#95—971no—
241#95—971no—
242#95—971no—
243#95—971no—
244#95—971no—
245#95—971no—
246#95—971no—
247#95—971no—
248#95—971no—
249#95—971no—
250#95—971no—
251#95C2×D9711yescharacteristic
252#95C2×D9711yescharacteristic
253C24—971no—
254C24C22×D9711yescharacteristic
255C2×D4—974no—
256C2×D4—1942no—
257C2×D4—1942no—
258C22×C4—974no—
259C22×C4—971no—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—974no—
263C22×C4—971no—
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—3881no—
268C22×C4—3881no—
269C22×C4—3881no—
270C22×C4—3881no—
271C22×C4—3881no—
272C22×C4—3881no—
273C22×C4—3881no—
274C22×C4—3881no—
275C22×C4—1942no—
276C22×C4—1942no—
277C22×C4C22×D9711yescharacteristic
278C22×C4C22×D9711yescharacteristic
279C22×C4C22×D9711yescharacteristic
280C22×C4C22×D9711yescharacteristic
281C22×C4C22×D9711yescharacteristic
282C22×C4C22×D9711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—974no—
285C4⋊C4—3881no—
286C4⋊C4—3881no—
287C4⋊C4—3881no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—974no—
291C22⋊C4—3881no—
292C22⋊C4—3881no—
293C22⋊C4—1942no—
294C22⋊C4—1942no—
295C22⋊C4C4×D9714yescomplemented
296C42—41no—
297C42—974no—
298C42—3881no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—972no—
306C23—972no—
307C23—972no—
308C23—1941no—
309C23—1941no—
310C23—1941no—
311C23—1941no—
312C23C2×C4×D9712yes—
313C23C23×D9711yescharacteristic
314D4—1948no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—972no—
332C2×C4—972no—
333C2×C4—972no—
334C2×C4—3884no—
335C2×C4—3884no—
336C2×C4—3884no—
337C2×C4—3884no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4—1941no—
345C2×C4—1941no—
346C2×C4—1941no—
347C2×C4—1941no—
348C2×C4—1941no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4C2×C4×D9712yes—
357C2×C4C2×C4×D9712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—974no—
366C22—1942no—
367C22—1942no—
368C22—1942no—
369C22—1942no—
370C22—1942no—
371C22—1942no—
372C22C22×C4×D9711yescharacteristic
373C22#41×D9711yescharacteristic
374C22#41×D9711yescharacteristic
375C22#1569085511yescharacteristic
376C22#1569085511yescharacteristic
377C22#1569085611yescharacteristic
378C22#1569085711yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3881no—
386C4—3881no—
387C4—3881no—
388C4—3881no—
389C4—1942no—
390C4—1942no—
391C4—1942no—
392C2—42no—
393C2—1944no—
394C2#117×D9711yescharacteristic
395C2#2026120711yescharacteristic
396C2#2026121411yescharacteristic
397C2#2026131711yescharacteristic
398C2#2026131911yescharacteristic
399C2#2026132611yescharacteristic
400C2#2026132811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenter#388D97
hypocenterC97#2318
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20260628C2semidirect
3. #20260619C2semidirect
4. #20260617C2semidirect
7. #20260609C2semidirect
8. #20260606C2semidirect
9. #20260602C2semidirect
10. #20260600C2semidirect
11. #20260596C2semidirect
12. #20260593C2semidirect
13. C4×#13095488C2semidirect
14. #95×D97C2semidirect
15. #388×C97C2semidirect
21. #15690770C4semidirect
58. C22×Dic97C2×C4semidirect
78. #13095490C2×C4semidirect
85. C4×Dic97C2×C4semidirect
96. C22⋊C4×C97C2×C4semidirect
201. C97#2318semidirect
204. #388D97semidirect
295. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907702
#202606282
#202606192
#202606172
#202606112
#202606092
#202606062
#202606022
#202606002
#202605962
#202605932
C4×#130954882
#95×D972
#388×C972
#231897