Finite Group Databasebeta

← #27144550

Subgroups — #27144550

Order 12416.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690796C211yesmaximal, characteristic
2#20260640C211yesmaximal, characteristic, complemented
3#20260639C211yesmaximal, characteristic, complemented
4#20260628C211yesmaximal, characteristic, complemented
5#20260625C211yesmaximal, characteristic, complemented
6#20260625C211yesmaximal, characteristic, complemented
7#20260623C211yesmaximal, characteristic, complemented
8#20260614C211yesmaximal, characteristic
9#20260611C211yesmaximal, characteristic
10#20260609C211yesmaximal, characteristic, complemented
11#20260602C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#385×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19#15690796C418yescomplemented
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×C22⋊Dic97C2211yescharacteristic
23C2×C22⋊Dic97C2211yescharacteristic
24C2×C22⋊Dic97C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45C23×C4×C97C2211yescharacteristic
46C2×C4⋊C4×C97C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C23×D97C2311yescharacteristic
53C2×Dic194—41no—
54C2×Dic194—41no—
55C2×Dic194C2×C414yescomplemented
56C2×C22⋊D97—41no—
57C2×C22⋊D97—41no—
58C2×C22⋊D97C2×C414yes—
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C2×C4×D97—22no—
67C2×C4×D97—41no—
68C2×C4×D97—41no—
69C4⋊Dic97—22no—
70#13095490—41no—
71C22⋊Dic97—22no—
72C22⋊Dic97—41no—
73C22⋊Dic97C2×C414yescomplemented
74#13095488—22no—
75#13095488—41no—
76#13095488—41no—
77#13095488C2×C414yescomplemented
78#13095488C2×C414yescomplemented
79C4×Dic97—41no—
80C4×Dic97—41no—
81C4×Dic97—41no—
82C4×Dic97—41no—
83C4×Dic97C2×C414yescomplemented
84C24×C97C2311yescharacteristic
85C22×C4×C97—22no—
86C22×C4×C97—41no—
87C22×C4×C97—41no—
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C4⋊C4×C97—22no—
95C4⋊C4×C97—41no—
96C22⋊C4×C97—41no—
97C22⋊C4×C97C2×C414yescomplemented
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×D97—21no—
111C22×D97—21no—
112C22×D97C22×C412yes—
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—44no—
129C2×C4×C97C22×C412yes—
130C2×C4×C97C22×C412yes—
131C2×C4×C97#4112yes—
132C22⋊D97—44no—
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×Dic97C22×C412yes—
151C2×Dic97C22×C412yes—
152C2×Dic97C22×C412yes—
153C2×Dic97#4112yes—
154C4×D97—44no—
155Dic194—44no—
156C22×C97—22no—
157C22×C97—22no—
158C22×C97—22no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97C23×C411yescharacteristic
164C22×C97C2×#4111yescharacteristic
165C22×C97C2×#4111yescharacteristic
166C22×C97H(5,𝔽2)11yescharacteristic
167C22×C97H(5,𝔽2)11yescharacteristic
168C22×C97H(5,𝔽2)11yescharacteristic
169C22×C972−1+411yescharacteristic
170C2×D97—22no—
171C2×D97—22no—
172C2×D97—22no—
173C2×D97—41no—
174C2×D97—41no—
175C2×D97—41no—
176C2×D97—41no—
177C4×C97—22no—
178C4×C97—22no—
179C4×C97—22no—
180C4×C97—41no—
181C4×C97—41no—
182C4×C97—41no—
183Dic97—22no—
184Dic97—22no—
185Dic97—22no—
186Dic97—22no—
187Dic97—41no—
188Dic97—41no—
189Dic97—41no—
190C2×C97—42no—
191C2×C97C4×#4111yescharacteristic
192C2×C97#51811yescharacteristic
193C2×C97#52011yescharacteristic
194C2×C97#53811yescharacteristic
195C2×C97#53811yescharacteristic
196C2×C97#54211yescharacteristic
197C2×C97#54211yescharacteristic
198D97—42no—
199#2337—971nomaximal, Sylow
200C97#233711yescharacteristic, complemented, Sylow
201C2×#124—971no—
202#391—971no—
203#391—971no—
204#388—971no—
205#388—971no—
206#388—971no—
207#388—971no—
208#386—971no—
209#386—971no—
210#385—971no—
211#385D9711yescharacteristic, complemented
212#383—971no—
213C4×C4⋊C4—971no—
214C4×C22⋊C4—971no—
215C4×C22⋊C4—971no—
216C22×Q8—971no—
217C22×D4—971no—
218C23×C4—971no—
219C23×C4C2×D9711yescharacteristic
220#124—978no—
221C2×C4⋊C4—971no—
222C2×C4⋊C4—971no—
223C2×C4⋊C4C2×D9711yescharacteristic
224C2×C4⋊C4C2×D9711yescharacteristic
225C2×C22⋊C4—971no—
226C2×C22⋊C4—971no—
227C2×C22⋊C4—971no—
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×C42—971no—
236C2×C42—971no—
237C2×C42—971no—
238C2×C42—971no—
239C2×C42—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#95C2×D9711yescharacteristic
251#95C2×D9711yescharacteristic
252C24—971no—
253C24C22×D9711yescharacteristic
254C2×Q8—974no—
255C2×Q8—3881no—
256C2×Q8—3881no—
257C2×D4—974no—
258C2×D4—3881no—
259C2×D4—3881no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—971no—
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—3881no—
271C22×C4—3881no—
272C22×C4—1942no—
273C22×C4C22×D9711yescharacteristic
274C22×C4C22×D9711yescharacteristic
275C22×C4C22×D9711yescharacteristic
276C22×C4C22×D9711yescharacteristic
277C22×C4C22×D9711yescharacteristic
278C22×C4C22×D9711yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—3881no—
282C4⋊C4—1942no—
283C22⋊C4—41no—
284C22⋊C4—974no—
285C22⋊C4—974no—
286C22⋊C4—974no—
287C22⋊C4—3881no—
288C22⋊C4—3881no—
289C22⋊C4—3881no—
290C22⋊C4—1942no—
291C22⋊C4—1942no—
292C22⋊C4C4×D9714yescomplemented
293C42—974no—
294C42—3881no—
295C42—3881no—
296C42—3881no—
297C42—3881no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—972no—
305C23—1941no—
306C23—1941no—
307C23—1941no—
308C23—1941no—
309C23—1941no—
310C23—1941no—
311C23C2×C4×D9712yes—
312C23C23×D9711yescharacteristic
313Q8—3884no—
314D4—3884no—
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—44no—
331C2×C4—972no—
332C2×C4—972no—
333C2×C4—972no—
334C2×C4—972no—
335C2×C4—3884no—
336C2×C4—1941no—
337C2×C4—1941no—
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×C4C2×C4×D9712yes—
354C2×C4C2×C4×D9712yes—
355C2×C4#1309551012yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3881no—
364C22—3881no—
365C22—3881no—
366C22—3881no—
367C22—1942no—
368C22—1942no—
369C22—1942no—
370C22C22×C4×D9711yescharacteristic
371C22#41×D9711yescharacteristic
372C22C2×#1309551011yescharacteristic
373C22#1569085511yescharacteristic
374C22#1569085511yescharacteristic
375C22#1569085611yescharacteristic
376C22#1569085811yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3881no—
384C4—3881no—
385C4—3881no—
386C4—1942no—
387C4—1942no—
388C4—1942no—
389C4—1942no—
390C2—42no—
391C2—3882no—
392C2C4×#1309551011yescharacteristic
393C2#2026120711yescharacteristic
394C2#2026121611yescharacteristic
395C2#2026132011yescharacteristic
396C2#2026132111yescharacteristic
397C2#2026135511yescharacteristic
398C2#2026135811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#385×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenter#385D97
hypocenterC97#2337
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #20260640C2semidirect
3. #20260639C2semidirect
4. #20260628C2semidirect
5. #20260625C2semidirect
6. #20260625C2semidirect
7. #20260623C2semidirect
10. #20260609C2semidirect
11. #20260602C2semidirect
12. C4⋊C4×Dic97C2semidirect
13. C4×C22⋊Dic97C2semidirect
14. #20260582C2semidirect
15. #385×C97C2semidirect
19. #15690796C4semidirect
55. C2×Dic194C2×C4semidirect
73. C22⋊Dic97C2×C4semidirect
77. #13095488C2×C4semidirect
78. #13095488C2×C4semidirect
83. C4×Dic97C2×C4semidirect
97. C22⋊C4×C97C2×C4semidirect
200. C97#2337semidirect
211. #385D97semidirect
292. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907962
#202606402
#202606392
#202606282
#202606252
#202606232
#202606142
#202606112
#202606092
#202606022
C4⋊C4×Dic972
C4×C22⋊Dic972
#202605822
#385×C972
#233797