Finite Group Databasebeta

← #27144335

Subgroups — #27144335

Order 12416.

512 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic
2#20260622C211yesmaximal, characteristic, complemented
3#20260621C211yesmaximal, characteristic, complemented
4#20260620C211yesmaximal, characteristic, complemented
5#20260619C211yesmaximal, characteristic, complemented
6#20260618C211yesmaximal, characteristic, complemented
7#20260617C211yesmaximal, characteristic, complemented
8#20260616C211yesmaximal, characteristic, complemented
9#20260597C211yesmaximal, characteristic
10#20260597C211yesmaximal, characteristic
11#20260594C211yesmaximal, characteristic
12#20260592C211yesmaximal, characteristic
13C4×#13095488C211yesmaximal, characteristic
14#20260582C211yesmaximal, characteristic
15#387×C97C211yesmaximal, characteristic, complemented
16C22×C4×D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C4⋊C4×D97C418yescomplemented
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
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×C4⋊C4×C97C2211yescharacteristic
47C2×C42×C97C2211yescharacteristic
48#95×C97C2211yescharacteristic
49#95×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C23×D97C2311yescharacteristic
53C22×Dic97C2311yescharacteristic
54C22×Dic97C2311yescharacteristic
55C22×Dic97C2311yescharacteristic
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C2×C4×D97—22no—
61C2×C4×D97—22no—
62C2×C4×D97—41no—
63C2×C4×D97—41no—
64C2×C4×D97—41no—
65C2×C4×D97—41no—
66C2×C4×D97C2×C414yes—
67C2×C4×D97C2×C414yes—
68C2×C4×D97C2×C414yes—
69C4⋊Dic97—41no—
70C4⋊Dic97C2×C414yes—
71#13095490—41no—
72#13095490—41no—
73#13095490C2×C414yes—
74#13095490C2×C414yes—
75#13095488—22no—
76#13095488—22no—
77#13095488—22no—
78#13095488—22no—
79C4×Dic97—41no—
80C4×Dic97—41no—
81C4×Dic97—41no—
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C22×C4×C97C2311yescharacteristic
86C22×C4×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C4⋊C4×C97—41no—
90C4⋊C4×C97C2×C414yes—
91C42×C97—41no—
92C23×C97C2411yescharacteristic
93C22×D97—21no—
94C22×D97—21no—
95C22×D97—21no—
96C22×D97—21no—
97C22×D97C22×C412yes—
98C22×D97#4112yes—
99C22×D97#4112yes—
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97—21no—
105C2×C4×C97—21no—
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×C97C22×C412yes—
119C2×C4×C97C22×C412yes—
120C2×C4×C97C22×C412yes—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97C22×C412yes—
140C2×Dic97C22×C412yes—
141C2×Dic97C22×C412yes—
142C4×D97—28no—
143C4×D97—44no—
144C4×D97—44no—
145C22×C97C23×C411yescharacteristic
146C22×C97C2×#4111yescharacteristic
147C22×C97C2×#4111yescharacteristic
148C22×C97H(5,𝔽2)11yescharacteristic
149C22×C97H(5,𝔽2)11yescharacteristic
150C22×C972−1+411yescharacteristic
151C22×C972−1+411yescharacteristic
152C2×D97—22no—
153C2×D97—22no—
154C2×D97—22no—
155C2×D97—22no—
156C2×D97—22no—
157C2×D97—22no—
158C2×D97#11714yes—
159C4×C97—22no—
160C4×C97—22no—
161C4×C97—22no—
162C4×C97—41no—
163C4×C97—41no—
164C4×C97—41no—
165C4×C97—41no—
166Dic97—22no—
167Dic97—22no—
168Dic97—22no—
169Dic97—41no—
170Dic97—41no—
171Dic97—41no—
172Dic97—41no—
173C2×C97C2×#11711yescharacteristic
174C2×C97#51811yescharacteristic
175C2×C97#51911yescharacteristic
176C2×C97#53911yescharacteristic
177C2×C97#53911yescharacteristic
178C2×C97#53911yescharacteristic
179C2×C97#53911yescharacteristic
180D97—24no—
181#2313—971nomaximal, Sylow
182C97#231311yescharacteristic, complemented, Sylow
183C22×C4⋊C4—971no—
184#387—971no—
185#387—971no—
186#387—971no—
187#387—971no—
188#387—971no—
189#387—971no—
190#387—971no—
191#387D9711yescharacteristic, complemented
192#381—971no—
193#381—971no—
194#380—971no—
195#380—971no—
196C4×C22⋊C4—971no—
197C4×C22⋊C4—971no—
198C23×C4—971no—
199C23×C4—971no—
200C23×C4—971no—
201C2×C4⋊C4—971no—
202C2×C4⋊C4—971no—
203C2×C4⋊C4—971no—
204C2×C4⋊C4—978no—
205C2×C4⋊C4—971no—
206C2×C4⋊C4—971no—
207C2×C4⋊C4—971no—
208C2×C4⋊C4C2×D9711yescharacteristic
209C2×C4⋊C4C2×D9711yescharacteristic
210C2×C22⋊C4—971no—
211C2×C22⋊C4—971no—
212C2×C22⋊C4—971no—
213C2×C22⋊C4—971no—
214C2×C42—971no—
215C2×C42—971no—
216C2×C42—971no—
217C2×C42C2×D9711yescharacteristic
218#95—971no—
219#95—971no—
220#95—971no—
221#95—971no—
222#95—971no—
223#95—971no—
224#95—971no—
225#95—971no—
226#95—971no—
227#95—971no—
228#95—971no—
229#95—971no—
230#95C2×D9711yescharacteristic
231#95C2×D9711yescharacteristic
232#95C2×D9711yescharacteristic
233#95C2×D9711yescharacteristic
234C24—971no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—974no—
239C22×C4—974no—
240C22×C4—974no—
241C22×C4—971no—
242C22×C4—971no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—3881no—
246C22×C4—3881no—
247C22×C4—3881no—
248C22×C4—3881no—
249C22×C4—1942no—
250C22×C4—1942no—
251C22×C4C22×D9711yescharacteristic
252C22×C4C22×D9711yescharacteristic
253C22×C4C22×D9711yescharacteristic
254C22×C4C22×D9711yescharacteristic
255C22×C4C22×D9711yescharacteristic
256C22×C4C22×D9711yescharacteristic
257C22×C4C22×D9711yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—974no—
260C4⋊C4—974no—
261C4⋊C4—974no—
262C4⋊C4—3881no—
263C4⋊C4—3881no—
264C4⋊C4—3881no—
265C4⋊C4C4×D9714yes—
266C22⋊C4—1942no—
267C22⋊C4—1942no—
268C22⋊C4—1942no—
269C22⋊C4—1942no—
270C42—41no—
271C42—3881no—
272C42—3881no—
273C42—3881no—
274C23—972no—
275C23—972no—
276C23—972no—
277C23—1941no—
278C23—1941no—
279C23—1941no—
280C23—1941no—
281C23C23×D9711yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—972no—
301C2×C4—972no—
302C2×C4—972no—
303C2×C4—3884no—
304C2×C4—3884no—
305C2×C4—1948no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4—1941no—
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×C4C2×C4×D9712yes—
325C2×C4C2×C4×D9712yes—
326C2×C4C2×C4×D9712yes—
327C22—974no—
328C22—1942no—
329C22—1942no—
330C22—1942no—
331C22—1942no—
332C22—1942no—
333C22—1942no—
334C22C22×C4×D9711yescharacteristic
335C22#41×D9711yescharacteristic
336C22#41×D9711yescharacteristic
337C22#1569085511yescharacteristic
338C22#1569085611yescharacteristic
339C22#1569085711yescharacteristic
340C22#1569085811yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3881no—
349C4—3881no—
350C4—3881no—
351C4—3881no—
352C4—1942no—
353C4—1942no—
354C4—1942no—
355C2—1944no—
356C2#117×D9711yescharacteristic
357C2#2026120711yescharacteristic
358C2#2026121111yescharacteristic
359C2#2026132611yescharacteristic
360C2#2026132811yescharacteristic
361C2#2026133111yescharacteristic
362C2#2026133311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#387×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#387D97
hypocenterC97#2313
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #20260622C2semidirect
3. #20260621C2semidirect
4. #20260620C2semidirect
5. #20260619C2semidirect
6. #20260618C2semidirect
7. #20260617C2semidirect
8. #20260616C2semidirect
15. #387×C97C2semidirect
25. C4⋊C4×D97C4semidirect
182. C97#2313semidirect
191. #387D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
#202606222
#202606212
#202606202
#202606192
#202606182
#202606172
#202606162
#202605972
#202605942
#202605922
C4×#130954882
#202605822
#387×C972
#231397