Finite Group Databasebeta

← #27144817

Subgroups — #27144817

Order 12416.

548 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×#15690787C211yesmaximal, characteristic
2#20260640C211yesmaximal, characteristic, complemented
3#20260639C211yesmaximal, characteristic, complemented
4#20260620C211yesmaximal, characteristic, complemented
5#20260617C211yesmaximal, characteristic, complemented
6#20260616C211yesmaximal, characteristic, complemented
7#20260616C211yesmaximal, characteristic, complemented
8#20260607C211yesmaximal, characteristic
9C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
10#20260586C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic97C211yesmaximal, characteristic
12C22⋊C4×Dic97C211yesmaximal, characteristic
13C4×C22⋊Dic97C211yesmaximal, characteristic
14C2×#15690717C211yesmaximal, characteristic
15#385×C97C211yesmaximal, characteristic
16C22×Dic194C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C23×Dic97C2211yescharacteristic
19#15690787C418yescomplemented
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C4×Dic97C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
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
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
52C2×Dic194—41no—
53C2×Dic194—41no—
54C2×Dic194C2×C414yescomplemented
55C22×Dic97—22no—
56C22×Dic97—22no—
57C22×Dic97—41no—
58C22×Dic97—41no—
59C22×Dic97—41no—
60C22×Dic97C2×C414yes—
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C4⋊Dic97C2×C414yescomplemented
70#13095490—22no—
71#13095490—41no—
72#13095490C2×C414yescomplemented
73#13095490C2×C414yescomplemented
74C22⋊Dic97C2×C414yes—
75C22⋊Dic97Q814yescomplemented
76C4×Dic97—22no—
77C4×Dic97—22no—
78C4×Dic97—41no—
79C4×Dic97—41no—
80C4×Dic97—41no—
81C4×Dic97—41no—
82C24×C97C2311yescharacteristic
83C22×C4×C97—22no—
84C22×C4×C97—22no—
85C22×C4×C97—41no—
86C22×C4×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C4⋊C4×C97—22no—
93C4⋊C4×C97—41no—
94C22⋊C4×C97—22no—
95C22⋊C4×C97C2×C414yes—
96C23×C97—21no—
97C23×C97—21no—
98C23×C97—21no—
99C23×C97—21no—
100C23×C97C22×C412yes—
101C23×C97C2×Q812yes—
102C23×C97#4112yes—
103C23×C97C2411yescharacteristic
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×C97—21no—
119C2×C4×C97—44no—
120C2×C4×C97C22×C412yes—
121C2×C4×C97C22×C412yes—
122C2×C4×C97#4112yes—
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×Dic97—21no—
140C2×Dic97—44no—
141C2×Dic97—44no—
142C2×Dic97C22×C412yes—
143C2×Dic97C22×C412yes—
144C2×Dic97C22×C412yes—
145C2×Dic97C22×C412yes—
146C2×Dic97C2×Q812yes—
147C2×Dic97C2×Q812yes—
148C2×Dic97#4112yes—
149Dic194—44no—
150C22×C97—22no—
151C22×C97—22no—
152C22×C97—22no—
153C22×C97—22no—
154C22×C97—22no—
155C22×C97—22no—
156C22×C97C4×Q814yes—
157C22×C97C23×C411yescharacteristic
158C22×C97C22×Q811yescharacteristic
159C22×C97C2×#4111yescharacteristic
160C22×C97C2×#4111yescharacteristic
161C22×C97C2×#4111yescharacteristic
162C22×C97H(5,𝔽2)11yescharacteristic
163C22×C97H(5,𝔽2)11yescharacteristic
164C4×C97—22no—
165C4×C97—22no—
166C4×C97—22no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97—41no—
170Dic97—22no—
171Dic97—22no—
172Dic97—22no—
173Dic97—22no—
174Dic97—22no—
175Dic97—22no—
176Dic97—22no—
177Dic97—41no—
178C2×C97—24no—
179C2×C97C2×C4×Q811yescharacteristic
180C2×C97C4×#4111yescharacteristic
181C2×C97#51811yescharacteristic
182C2×C97#55111yescharacteristic
183C2×C97#55111yescharacteristic
184C2×C97#55411yescharacteristic
185C2×C97#55411yescharacteristic
186#2368—971nomaximal, Sylow
187C97#236811yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—971no—
189#391—971no—
190#391—971no—
191#387—971no—
192#387—971no—
193#387—971no—
194#387—971no—
195#385—971no—
196#385D9711yescharacteristic
197C4×C4⋊C4—971no—
198C4×C4⋊C4—971no—
199C4×C22⋊C4—971no—
200C4×C22⋊C4—971no—
201C4×C22⋊C4—971no—
202C2×#95—971no—
203C22×Q8—971no—
204C23×C4—971no—
205C23×C4—971no—
206C23×C4C2×D9711yescharacteristic
207C22⋊Q8—978no—
208C2×C4⋊C4—971no—
209C2×C4⋊C4—971no—
210C2×C4⋊C4—971no—
211C2×C4⋊C4—971no—
212C2×C4⋊C4—971no—
213C2×C4⋊C4C2×D9711yescharacteristic
214C2×C4⋊C4C2×D9711yescharacteristic
215C2×C22⋊C4—971no—
216C2×C22⋊C4—971no—
217C2×C22⋊C4C2×D9711yescharacteristic
218C2×C22⋊C4C2×D9711yescharacteristic
219C2×C42—971no—
220C2×C42—971no—
221C2×C42—971no—
222C2×C42—971no—
223C2×C42—971no—
224C2×C42—971no—
225#95—971no—
226#95—971no—
227#95—971no—
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#95C2×D9711yescharacteristic
238#95C2×D9711yescharacteristic
239C24C22×D9711yescharacteristic
240C2×Q8—974no—
241C2×Q8—3881no—
242C2×Q8—3881no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—974no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—3881no—
256C22×C4—3881no—
257C22×C4—3881no—
258C22×C4—1942no—
259C22×C4—1942no—
260C22×C4C22×D9711yescharacteristic
261C22×C4C22×D9711yescharacteristic
262C22×C4C22×D9711yescharacteristic
263C22×C4C22×D9711yescharacteristic
264C22×C4C22×D9711yescharacteristic
265C22×C4C22×D9711yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—974no—
269C4⋊C4—974no—
270C4⋊C4—974no—
271C4⋊C4—3881no—
272C4⋊C4—1942no—
273C22⋊C4—22no—
274C22⋊C4—974no—
275C22⋊C4—974no—
276C22⋊C4C4×D9714yes—
277C42—3881no—
278C42—3881no—
279C42—3881no—
280C42—3881no—
281C42—1942no—
282C42—1942no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23C2×C4×D9712yes—
288C23Q8×D9712yes—
289C23#1309551012yes—
290C23C23×D9711yescharacteristic
291Q8—3884no—
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—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—44no—
308C2×C4—972no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—972no—
314C2×C4—972no—
315C2×C4—3884no—
316C2×C4—3884no—
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—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4—1941no—
334C2×C4C2×C4×D9712yes—
335C2×C4C2×C4×D9712yes—
336C2×C4#1309551012yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1569077514yes—
344C22C22×C4×D9711yescharacteristic
345C22C2×Q8×D9711yescharacteristic
346C22#41×D9711yescharacteristic
347C22C2×#1309551011yescharacteristic
348C22C2×#1309551011yescharacteristic
349C22#1569085511yescharacteristic
350C22#1569085611yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3881no—
358C4—1942no—
359C4—1942no—
360C4—1942no—
361C4—1942no—
362C4—1942no—
363C4—1942no—
364C4—1942no—
365C2—24no—
366C2C2×#1569077511yescharacteristic
367C2C4×#1309551011yescharacteristic
368C2#2026120711yescharacteristic
369C2#2026140811yescharacteristic
370C2#2026141611yescharacteristic
371C2#2026143911yescharacteristic
372C2#2026144311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#385×C97C2
socle, cosocleC23×C97C24
prime radicalC24×C97C23
hypercenter#385D97
hypocenterC97#2368
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. #20260620C2semidirect
5. #20260617C2semidirect
6. #20260616C2semidirect
7. #20260616C2semidirect
9. C4⋊C4×Dic97C2semidirect
10. #20260586C2semidirect
19. #15690787C4semidirect
54. C2×Dic194C2×C4semidirect
69. C4⋊Dic97C2×C4semidirect
72. #13095490C2×C4semidirect
73. #13095490C2×C4semidirect
75. C22⋊Dic97Q8semidirect
187. C97#2368semidirect

Maximal subgroups

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

subgroupindex
C2×#156907872
#202606402
#202606392
#202606202
#202606172
#202606162
#202606072
C4⋊C4×Dic972
#202605862
C22⋊C4×Dic972
C4×C22⋊Dic972
C2×#156907172
#385×C972
#236897