Finite Group Databasebeta

← #27144263

Subgroups — #27144263

Order 12416.

688 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690792C211yesmaximal, characteristic
2C2×#15690770C211yesmaximal, characteristic
3C2×C42×D97C211yesmaximal, characteristic, complemented
4#20260623C211yesmaximal, characteristic, complemented
5#20260620C211yesmaximal, characteristic, complemented
6#20260619C211yesmaximal, characteristic, complemented
7#20260611C211yesmaximal, characteristic
8#20260606C211yesmaximal, characteristic, complemented
9#20260597C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
11#20260586C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690792C418yescomplemented
22#15690770—24no—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C42×D97—24no—
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C2×C22⋊D97—22no—
56C2×C22⋊D97—22no—
57C2×C22⋊D97C2×C414yes—
58C22×Dic97—22no—
59C22×Dic97—22no—
60C22×Dic97—22no—
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C2×C4×D97—22no—
69C2×C4×D97—22no—
70C2×C4×D97—22no—
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97C2×C414yescomplemented
76C2×C4×D97D414yes—
77C4⋊Dic97C2×C414yescomplemented
78#13095490—22no—
79#13095490—22no—
80#13095490C2×C414yescomplemented
81C22⋊Dic97—22no—
82C22⋊Dic97C2×C414yescomplemented
83#13095488—22no—
84#13095488—22no—
85#13095488C2×C414yescomplemented
86C4×Dic97—22no—
87C4×Dic97—22no—
88C4×Dic97—22no—
89C4×Dic97—22no—
90C4×Dic97—22no—
91C4×Dic97—22no—
92C24×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C22×C4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97—22no—
100C22⋊C4×C97—22no—
101C22⋊C4×C97—22no—
102C22⋊C4×C97C2×C414yescomplemented
103C42×C97—22no—
104C23×C97—21no—
105C23×C97—21no—
106C23×C97—21no—
107C23×C97—21no—
108C23×C97—21no—
109C23×C97—21no—
110C23×C97C22×C412yes—
111C23×C97C2411yescharacteristic
112C22×D97—21no—
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—21no—
116C22×D97C22×C412yes—
117C22×D97C2×D412yes—
118C22×D97#4112yes—
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×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97C22×C412yes—
134C2×C4×C97C22×C412yes—
135C2×C4×C97C2×D412yes—
136C2×C4×C97#4112yes—
137C22⋊D97—28no—
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—21no—
151C2×Dic97—21no—
152C2×Dic97—44no—
153C2×Dic97C22×C412yes—
154C2×Dic97C22×C412yes—
155C2×Dic97C22×C412yes—
156C2×Dic97C2×D412yes—
157C2×Dic97#4112yes—
158C2×Dic97#4112yes—
159C2×Dic97#4112yes—
160C4×D97—28no—
161C4×D97—28no—
162C4×D97—28no—
163C22×C97—22no—
164C22×C97—22no—
165C22×C97—22no—
166C22×C97—41no—
167C22×C97—41no—
168C22×C97—41no—
169C22×C97—41no—
170C22×C97C23×C411yescharacteristic
171C22×C97C22×D411yescharacteristic
172C22×C97C2×#4111yescharacteristic
173C22×C97C2×#4111yescharacteristic
174C22×C97C2×#4111yescharacteristic
175C22×C97C2×#4111yescharacteristic
176C22×C97C2×#4111yescharacteristic
177C2×D97—22no—
178C2×D97—22no—
179C2×D97—22no—
180C2×D97—22no—
181C2×D97—22no—
182C2×D97—22no—
183C2×D97C4×D414yes—
184C4×C97—22no—
185C4×C97—22no—
186C4×C97—22no—
187C4×C97—22no—
188C4×C97—41no—
189C4×C97—41no—
190Dic97—22no—
191Dic97—22no—
192Dic97—22no—
193Dic97—22no—
194Dic97—41no—
195Dic97—41no—
196Dic97#12314yescomplemented
197C2×C97—42no—
198C2×C97C2×C4×D411yescharacteristic
199C2×C97C4×#4111yescharacteristic
200C2×C97C4×#4111yescharacteristic
201C2×C97C2×#12311yescharacteristic
202C2×C97#52511yescharacteristic
203C2×C97#52911yescharacteristic
204C2×C97#52911yescharacteristic
205D97—24no—
206C4×#123—971nomaximal, Sylow
207C97C4×#12311yescharacteristic, complemented, Sylow
208C2×#123—971no—
209C2×C4×D4—971no—
210C22×C42—971no—
211#388—971no—
212#388D9711yescharacteristic, complemented
213#387—971no—
214#387—971no—
215#386—971no—
216#385—971no—
217#381—971no—
218C4×C4⋊C4—971no—
219C4×C4⋊C4—971no—
220C4×C22⋊C4—971no—
221C4×C22⋊C4—971no—
222C4×C22⋊C4—971no—
223C22×D4—971no—
224C23×C4—971no—
225C23×C4—971no—
226C23×C4—971no—
227C23×C4—971no—
228#123—978no—
229C4×D4—1944no—
230C2×C4⋊C4—971no—
231C2×C4⋊C4—971no—
232C2×C4⋊C4—971no—
233C2×C4⋊C4—971no—
234C2×C4⋊C4C2×D9711yescharacteristic
235C2×C22⋊C4—971no—
236C2×C22⋊C4—971no—
237C2×C22⋊C4—971no—
238C2×C22⋊C4—971no—
239C2×C22⋊C4—971no—
240C2×C22⋊C4C2×D9711yescharacteristic
241C2×C22⋊C4C2×D9711yescharacteristic
242C2×C22⋊C4C2×D9711yescharacteristic
243C2×C42—971no—
244C2×C42—971no—
245C2×C42—971no—
246C2×C42—971no—
247C2×C42—971no—
248C2×C42—971no—
249C2×C42—1944no—
250C2×C42C2×D9711yescharacteristic
251#95—971no—
252#95—971no—
253#95—971no—
254#95—971no—
255#95—971no—
256#95—971no—
257#95—971no—
258#95—971no—
259#95C2×D9711yescharacteristic
260#95C2×D9711yescharacteristic
261C24—971no—
262C24C22×D9711yescharacteristic
263C2×D4—974no—
264C2×D4—1942no—
265C2×D4—1942no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—971no—
271C22×C4—974no—
272C22×C4—974no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—1942no—
276C22×C4—1942no—
277C22×C4—1942no—
278C22×C4—1942no—
279C22×C4—1942no—
280C22×C4—1942no—
281C22×C4—1942no—
282C22×C4—1942no—
283C22×C4—1942no—
284C22×C4—1942no—
285C22×C4C22×D9711yescharacteristic
286C22×C4C22×D9711yescharacteristic
287C22×C4C22×D9711yescharacteristic
288C22×C4C22×D9711yescharacteristic
289C22×C4C22×D9711yescharacteristic
290C22×C4C22×D9711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—974no—
293C4⋊C4—974no—
294C4⋊C4—1942no—
295C4⋊C4—1942no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—974no—
299C22⋊C4—974no—
300C22⋊C4—1942no—
301C22⋊C4—1942no—
302C22⋊C4—1942no—
303C22⋊C4C4×D9714yescomplemented
304C42—22no—
305C42—1942no—
306C42—1942no—
307C42—1942no—
308C42—1942no—
309C42—1942no—
310C42—1942no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—972no—
318C23—972no—
319C23—972no—
320C23—1941no—
321C23—1941no—
322C23—1941no—
323C23—1941no—
324C23C2×C4×D9712yes—
325C23C23×D9711yescharacteristic
326D4—1948no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—972no—
342C2×C4—972no—
343C2×C4—972no—
344C2×C4—972no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—972no—
348C2×C4—3884no—
349C2×C4—1948no—
350C2×C4—1948no—
351C2×C4—1948no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4—1941no—
358C2×C4—1941no—
359C2×C4—1941no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4—1941no—
366C2×C4C2×C4×D9712yes—
367C2×C4C2×C4×D9712yes—
368C2×C4D4×D9712yes—
369C2×C4#1309551012yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—974no—
378C22—1942no—
379C22—1942no—
380C22—1942no—
381C22—1942no—
382C22—1942no—
383C22—1942no—
384C22C22×C4×D9711yescharacteristic
385C22C2×D4×D9711yescharacteristic
386C22#41×D9711yescharacteristic
387C22#41×D9711yescharacteristic
388C22#41×D9711yescharacteristic
389C22#41×D9711yescharacteristic
390C22C2×#1309551011yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—974no—
398C4—3881no—
399C4—3881no—
400C4—1942no—
401C4—1942no—
402C4—1942no—
403C4—1942no—
404C2—42no—
405C2—1944no—
406C2C4×D4×D9711yescharacteristic
407C2C4×#1309551011yescharacteristic
408C2#2026120311yescharacteristic
409C2#123×D9711yescharacteristic
410C2#2026124511yescharacteristic
411C2#2026126411yescharacteristic
412C2#2026126811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenter#388D97
hypocenterC97C4×#123
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. C2×C42×D97C2semidirect
4. #20260623C2semidirect
5. #20260620C2semidirect
6. #20260619C2semidirect
8. #20260606C2semidirect
9. #20260597C2semidirect
10. C4⋊C4×Dic97C2semidirect
11. #20260586C2semidirect
12. C22⋊C4×Dic97C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. #20260582C2semidirect
15. #388×C97C2semidirect
21. #15690792C4semidirect
75. C2×C4×D97C2×C4semidirect
77. C4⋊Dic97C2×C4semidirect
80. #13095490C2×C4semidirect
82. C22⋊Dic97C2×C4semidirect
85. #13095488C2×C4semidirect
102. C22⋊C4×C97C2×C4semidirect
196. Dic97#123semidirect
207. C97C4×#123semidirect
212. #388D97semidirect
303. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907922
C2×#156907702
C2×C42×D972
#202606232
#202606202
#202606192
#202606112
#202606062
#202605972
C4⋊C4×Dic972
#202605862
C22⋊C4×Dic972
#202605822
#388×C972
C4×#12397