Finite Group Databasebeta

← #27150042

Subgroups — #27150042

Order 12416.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690800C211yesmaximal, characteristic, complemented
2C22×C4×Dic97C211yesmaximal, characteristic
3#20260632C211yesmaximal, characteristic, complemented
4#20260630C211yesmaximal, characteristic, complemented
5#20260621C211yesmaximal, characteristic, complemented
6#20260620C211yesmaximal, characteristic, complemented
7#20260617C211yesmaximal, characteristic, complemented
8#20260616C211yesmaximal, characteristic, complemented
9#20260610C211yesmaximal, characteristic
10#20260608C211yesmaximal, characteristic
11#20260607C211yesmaximal, characteristic
12#20260599C211yesmaximal, characteristic, complemented
13#20260593C211yesmaximal, characteristic
14#20260593C211yesmaximal, characteristic
15#385×C97C211yesmaximal, characteristic
16C23×Dic97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18#15690800—24no—
19C2×C4⋊Dic97C2211yescharacteristic
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C4×Dic97—24no—
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46C23×C4×C97C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C4⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53C22×Dic97—22no—
54C22×Dic97—22no—
55C22×Dic97—22no—
56C22×Dic97—22no—
57C22×Dic97—22no—
58C22×Dic97Q814yes—
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C4⋊Dic97—22no—
68C4⋊Dic97—22no—
69#13095490—22no—
70#13095490—22no—
71#13095490—22no—
72#13095490—22no—
73#13095490—22no—
74#13095490—22no—
75#13095490—22no—
76#13095490—22no—
77C22⋊Dic97—22no—
78C22⋊Dic97—22no—
79C4×Dic97—22no—
80C4×Dic97—22no—
81C4×Dic97—22no—
82C4×Dic97D414yes—
83C24×C97C2311yescharacteristic
84C22×C4×C97—22no—
85C22×C4×C97—22no—
86C22×C4×C97—22no—
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—22no—
94C4⋊C4×C97—22no—
95C22⋊C4×C97—22no—
96C22⋊C4×C97—22no—
97C23×C97—21no—
98C23×C97—21no—
99C23×C97—21no—
100C23×C97—21no—
101C23×C97C2×Q812yes—
102C23×C97#4112yes—
103C23×C97#4112yes—
104C23×C97C2411yescharacteristic
105C2×C4×C97—28no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
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×C97C2×D412yes—
123C2×C4×C97#4112yes—
124C2×Dic97—28no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—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×Dic97C2×D412yes—
145C2×Dic97C2×D412yes—
146C2×Dic97C2×Q812yes—
147C2×Dic97C2×Q812yes—
148C2×Dic97#4112yes—
149C2×Dic97#4112yes—
150C22×C97—22no—
151C22×C97—22no—
152C22×C97—22no—
153C22×C97—22no—
154C22×C97—22no—
155C22×C97—22no—
156C22×C97#12514yescomplemented
157C22×C97C22×D411yescharacteristic
158C22×C97C22×Q811yescharacteristic
159C22×C97C2×#4111yescharacteristic
160C22×C97C2×#4111yescharacteristic
161C22×C97C2×#4111yescharacteristic
162C22×C97C2×#4111yescharacteristic
163C22×C97C2×#4111yescharacteristic
164C4×C97—22no—
165C4×C97—22no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97—41no—
170Dic97—22no—
171Dic97—22no—
172Dic97—22no—
173Dic97—41no—
174Dic97—41no—
175Dic97—41no—
176Dic97—41no—
177Dic97C22⋊Q814yes—
178C2×C97—24no—
179C2×C97C2×C22⋊Q811yescharacteristic
180C2×C97#52511yescharacteristic
181C2×C97#52511yescharacteristic
182C2×C97C2×#12511yescharacteristic
183C2×C97#52911yescharacteristic
184C2×C97#52911yescharacteristic
185C2×C97#53311yescharacteristic
186#2851—971nomaximal, Sylow
187C97#285111yescharacteristic, complemented, Sylow
188C2×#125—971no—
189C22×C42—971no—
190#389—971no—
191#389—971no—
192#387—971no—
193#387—971no—
194#387—971no—
195#387—971no—
196#385—971no—
197#385—971no—
198#385—971no—
199#385D9711yescharacteristic
200#382—971no—
201#380—971no—
202#380—971no—
203C23×C4—971no—
204C23×C4—971no—
205C23×C4C2×D9711yescharacteristic
206#125—1944no—
207C2×C4⋊C4—971no—
208C2×C4⋊C4—971no—
209C2×C4⋊C4—971no—
210C2×C4⋊C4—971no—
211C2×C4⋊C4—971no—
212C2×C4⋊C4—971no—
213C2×C4⋊C4—971no—
214C2×C4⋊C4—971no—
215C2×C4⋊C4—971no—
216C2×C4⋊C4—971no—
217C2×C4⋊C4C2×D9711yescharacteristic
218C2×C4⋊C4C2×D9711yescharacteristic
219C2×C22⋊C4—971no—
220C2×C22⋊C4—971no—
221C2×C22⋊C4C2×D9711yescharacteristic
222C2×C22⋊C4C2×D9711yescharacteristic
223C2×C42—971no—
224C2×C42—971no—
225C2×C42—971no—
226C2×C42—971no—
227C2×C42—1944no—
228#95—971no—
229#95—971no—
230#95—971no—
231#95—971no—
232#95—971no—
233#95—971no—
234#95—971no—
235#95—971no—
236#95—971no—
237#95—971no—
238#95C2×D9711yescharacteristic
239#95C2×D9711yescharacteristic
240C24C22×D9711yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—971no—
245C22×C4—974no—
246C22×C4—971no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—1942no—
254C22×C4—1942no—
255C22×C4—1942no—
256C22×C4—1942no—
257C22×C4—1942no—
258C22×C4C22×D9711yescharacteristic
259C22×C4C22×D9711yescharacteristic
260C22×C4C22×D9711yescharacteristic
261C22×C4C22×D9711yescharacteristic
262C22×C4C22×D9711yescharacteristic
263C22×C4C22×D9711yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—1942no—
267C4⋊C4—1942no—
268C4⋊C4—1942no—
269C4⋊C4—1942no—
270C4⋊C4—1942no—
271C4⋊C4—1942no—
272C4⋊C4—1942no—
273C4⋊C4—1942no—
274C4⋊C4—1942no—
275C4⋊C4—1942no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—1942no—
279C22⋊C4—1942no—
280C42—974no—
281C42—1942no—
282C42—1942no—
283C42—1942no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23Q8×D9712yes—
289C23#1309551012yes—
290C23#1309551112yes—
291C23C23×D9711yescharacteristic
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—972no—
314C2×C4—972no—
315C2×C4—1948no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1948no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4—1941no—
334C2×C4—1941no—
335C2×C4D4×D9712yes—
336C2×C4#1309551012yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1569080014yescomplemented
344C22C2×D4×D9711yescharacteristic
345C22C2×Q8×D9711yescharacteristic
346C22#41×D9711yescharacteristic
347C22#41×D9711yescharacteristic
348C22C2×#1309551011yescharacteristic
349C22C2×#1309551011yescharacteristic
350C22C2×#1309551111yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—974no—
358C4—3881no—
359C4—3881no—
360C4—3881no—
361C4—3881no—
362C4—1942no—
363C4—1942no—
364C4—1942no—
365C2—24no—
366C2C22⋊Q8×D9711yescharacteristic
367C2#2026124111yescharacteristic
368C2#2026124411yescharacteristic
369C2C2×#1569080011yescharacteristic
370C2#2026126211yescharacteristic
371C2#2026126511yescharacteristic
372C2#2026128711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#385×C97C2
prime radicalC24×C97C23
hypercenter#385D97
hypocenterC97#2851
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#15690800C2semidirect
3. #20260632C2semidirect
4. #20260630C2semidirect
5. #20260621C2semidirect
6. #20260620C2semidirect
7. #20260617C2semidirect
8. #20260616C2semidirect
12. #20260599C2semidirect
156. C22×C97#125semidirect
187. C97#2851semidirect
343. C22#15690800semidirect

Maximal subgroups

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

subgroupindex
C2×#156908002
C22×C4×Dic972
#202606322
#202606302
#202606212
#202606202
#202606172
#202606162
#202606102
#202606082
#202606072
#202605992
#202605932
#385×C972
#285197