Finite Group Databasebeta

← #389×D97 #27144183

Subgroups — #389×D97 #27144183

Order 12416.

800 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic
2C2×C4⋊C4×D97C212yesmaximal
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4C2×C42×D97C211yesmaximal, characteristic
5#20260634C211yesmaximal, characteristic, complemented
6#20260633C211yesmaximal, characteristic, complemented
7#20260632C212yesmaximal, complemented
8#20260631C211yesmaximal, characteristic, complemented
9#20260630C212yesmaximal, complemented
10#95×D97C212yesmaximal
11#389×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97C2212yes—
13C22×C4×D97C2212yes—
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16C22×C4×D97C2211yescharacteristic
17C2×C4⋊Dic97C2212yes—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490C2212yes—
21C2×#13095490C2212yes—
22C2×#13095490C2212yes—
23C2×#13095490C2212yes—
24C4⋊C4×D97—28no—
25C4⋊C4×D97—24no—
26C4⋊C4×D97C418yescomplemented
27C2×C4×Dic97C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30C42×D97—24no—
31#15690717C2212yes—
32#15690717C2212yes—
33#15690717C2212yes—
34C2×C4⋊C4×C97C2212yes—
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C4⋊C4×C97C2211yescharacteristic
37C2×C42×C97C2211yescharacteristic
38#95×C97C2212yes—
39C23×D97C2311yescharacteristic
40C22×Dic97C2312yes—
41C22×Dic97C2312yes—
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2311yescharacteristic
44C22×Dic97C2311yescharacteristic
45C2×C4×D97—22no—
46C2×C4×D97—22no—
47C2×C4×D97—24no—
48C2×C4×D97—24no—
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97—24no—
52C2×C4×D97—24no—
53C2×C4×D97—24no—
54C2×C4×D97C2×C418yes—
55C2×C4×D97C2×C414yes—
56C2×C4×D97D414yes—
57C2×C4×D97D414yes—
58C2×C4×D97Q814yes—
59C2×C4×D97Q814yes—
60C4⋊Dic97—22no—
61C4⋊Dic97—24no—
62C4⋊Dic97C2×C414yescomplemented
63#13095490—24no—
64#13095490—24no—
65#13095490—24no—
66#13095490C2×C418yescomplemented
67C4×Dic97—22no—
68C4×Dic97—22no—
69C4×Dic97—22no—
70C22×C4×C97C2312yes—
71C22×C4×C97C2312yes—
72C22×C4×C97C2311yescharacteristic
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2311yescharacteristic
75C4⋊C4×C97—24no—
76C4⋊C4×C97—22no—
77C4⋊C4×C97C2×C414yescomplemented
78C42×C97—22no—
79C23×C97C2411yescharacteristic
80C22×D97C22×C412yes—
81C22×D97C2×D412yes—
82C22×D97C2×D412yes—
83C22×D97C2×Q812yes—
84C22×D97C2×Q812yes—
85C22×D97#4112yes—
86C22×D97#4112yes—
87C2×C4×C97—21no—
88C2×C4×C97—21no—
89C2×C4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97—21no—
92C2×C4×C97—21no—
93C2×C4×C97—22no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97C22×C414yes—
97C2×C4×C97C22×C412yes—
98C2×C4×C97C2×D412yes—
99C2×C4×C97C2×D412yes—
100C2×C4×C97C2×Q812yes—
101C2×C4×C97C2×Q812yes—
102C2×Dic97—21no—
103C2×Dic97—21no—
104C2×Dic97—22no—
105C2×Dic97—22no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97—22no—
111C2×Dic97C22×C414yes—
112C2×Dic97C22×C412yes—
113C2×Dic97C2×D412yes—
114C2×Dic97C2×D412yes—
115C2×Dic97C2×Q812yes—
116C2×Dic97C2×Q812yes—
117C4×D97—216no—
118C4×D97—28no—
119C4×D97—28no—
120C4×D97—48no—
121C4×D97C4⋊C4116yescomplemented
122C22×C97C23×C411yescharacteristic
123C22×C97C22×D411yescharacteristic
124C22×C97C22×D411yescharacteristic
125C22×C97C22×Q811yescharacteristic
126C22×C97C22×Q811yescharacteristic
127C22×C97C2×#4111yescharacteristic
128C22×C97C2×#4111yescharacteristic
129C2×D97C2×C4⋊C414yes—
130C2×D97C4×D414yes—
131C2×D97C4×Q814yes—
132C2×D97#12114yes—
133C2×D97C22⋊Q814yes—
134C2×D97#12514yes—
135C2×D97C4⋊Q814yes—
136C4×C97—24no—
137C4×C97—22no—
138C4×C97—22no—
139C4×C97—42no—
140C4×C97C2×C4⋊C414yescomplemented
141Dic97—24no—
142Dic97—22no—
143Dic97—22no—
144Dic97—42no—
145Dic97C2×C4⋊C414yescomplemented
146C2×C97C22×C4⋊C411yescharacteristic
147C2×C97C2×C4×D411yescharacteristic
148C2×C97C2×C4×Q811yescharacteristic
149C2×C97C2×#12111yescharacteristic
150C2×C97C2×C22⋊Q811yescharacteristic
151C2×C97C2×#12511yescharacteristic
152C2×C97C2×C4⋊Q811yescharacteristic
153D97#38918yesdirect product factor
154C2×#389—971nomaximal, Sylow
155C97C2×#38911yescharacteristic, complemented, Sylow
156C22×C4⋊C4—971no—
157C22×C4⋊C4—971no—
158C22×C4⋊C4—972no—
159C22×C42—971no—
160#389—972no—
161#389—971no—
162#389—971no—
163#389—971no—
164#389—972no—
165#389D9711yescharacteristic, direct product factor
166C2×#95—972no—
167C23×C4—971no—
168C23×C4—972no—
169C23×C4—971no—
170C23×C4—971no—
171C23×C4—972no—
172C2×C4⋊C4—972no—
173C2×C4⋊C4—971no—
174C2×C4⋊C4—978no—
175C2×C4⋊C4—972no—
176C2×C4⋊C4—972no—
177C2×C4⋊C4—972no—
178C2×C4⋊C4—972no—
179C2×C4⋊C4—971no—
180C2×C4⋊C4—1944no—
181C2×C4⋊C4—1948no—
182C2×C4⋊C4C2×D9711yescharacteristic
183C2×C4⋊C4C2×D9712yes—
184C2×C4⋊C4C2×D9711yescharacteristic
185C2×C42—971no—
186C2×C42—971no—
187C2×C42—971no—
188C2×C42—1944no—
189C2×C42C2×D9711yescharacteristic
190#95—972no—
191#95—972no—
192#95—972no—
193#95C2×D9712yes—
194C24—971no—
195C22×C4—971no—
196C22×C4—974no—
197C22×C4—974no—
198C22×C4—974no—
199C22×C4—971no—
200C22×C4—972no—
201C22×C4—971no—
202C22×C4—978no—
203C22×C4—974no—
204C22×C4—974no—
205C22×C4—972no—
206C22×C4—1944no—
207C22×C4—1944no—
208C22×C4—1944no—
209C22×C4—1942no—
210C22×C4—1942no—
211C22×C4—1944no—
212C22×C4—1944no—
213C22×C4—1942no—
214C22×C4—1942no—
215C22×C4C22×D9712yes—
216C22×C4C22×D9712yes—
217C22×C4C22×D9711yescharacteristic
218C22×C4C22×D9711yescharacteristic
219C22×C4C22×D9711yescharacteristic
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—978no—
223C4⋊C4—974no—
224C4⋊C4—1944no—
225C4⋊C4—1944no—
226C4⋊C4—1944no—
227C4⋊C4—1944no—
228C4⋊C4—1942no—
229C4⋊C4C4×D9714yescomplemented
230C42—22no—
231C42—1942no—
232C42—1942no—
233C42—1942no—
234C23—972no—
235C23—972no—
236C23—972no—
237C23—972no—
238C23—972no—
239C23—972no—
240C23—972no—
241C23C23×D9711yescharacteristic
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—972no—
252C2×C4—972no—
253C2×C4—972no—
254C2×C4—9716no—
255C2×C4—974no—
256C2×C4—972no—
257C2×C4—972no—
258C2×C4—3888no—
259C2×C4—19416no—
260C2×C4—1948no—
261C2×C4—1948no—
262C2×C4—1942no—
263C2×C4—1942no—
264C2×C4—1942no—
265C2×C4—1941no—
266C2×C4—1941no—
267C2×C4—1942no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1941no—
271C2×C4C2×C4×D9714yes—
272C2×C4C2×C4×D9712yes—
273C2×C4D4×D9712yes—
274C2×C4D4×D9712yes—
275C2×C4Q8×D9712yes—
276C2×C4Q8×D9712yes—
277C22—974no—
278C22—974no—
279C22—974no—
280C22—974no—
281C22—974no—
282C22—974no—
283C22—974no—
284C22C22×C4×D9711yescharacteristic
285C22C2×D4×D9711yescharacteristic
286C22C2×D4×D9711yescharacteristic
287C22C2×Q8×D9711yescharacteristic
288C22C2×Q8×D9711yescharacteristic
289C22#41×D9711yescharacteristic
290C22#41×D9711yescharacteristic
291C4—24no—
292C4—22no—
293C4—22no—
294C4—42no—
295C4—974no—
296C4—3882no—
297C4—1944no—
298C4—1942no—
299C4—1942no—
300C4C4⋊C4×D9714yescomplemented
301C2—978no—
302C2C2×C4⋊C4×D9711yescharacteristic
303C2C4×D4×D9711yescharacteristic
304C2C4×Q8×D9711yescharacteristic
305C2#121×D9711yescharacteristic
306C2C22⋊Q8×D9711yescharacteristic
307C2#125×D9711yescharacteristic
308C2C4⋊Q8×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#389×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#389D97
hypocenterC97C2×#389
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
5. #20260634C2semidirect
6. #20260633C2semidirect
7. #20260632C2semidirect
8. #20260631C2semidirect
9. #20260630C2semidirect
11. #389×C97C2semidirect
26. C4⋊C4×D97C4semidirect
62. C4⋊Dic97C2×C4semidirect
66. #13095490C2×C4semidirect
77. C4⋊C4×C97C2×C4semidirect
121. C4×D97C4⋊C4semidirect
140. C4×C97C2×C4⋊C4semidirect
145. Dic97C2×C4⋊C4semidirect
153. D97#389direct (N × H)
155. C97C2×#389semidirect
165. #389D97direct (N × H)
229. C4⋊C4C4×D97semidirect
300. C4C4⋊C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
C2×C42×D972
#202606342
#202606332
#202606322
#202606312
#202606302
#95×D972
#389×C972
C2×#38997