Finite Group Databasebeta

← #344×D97 #27141476

Subgroups — #344×D97 #27141476

Order 12416.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D97C211yesmaximal, characteristic
2C2×M16×D97C211yesmaximal, characteristic
3#117×D97C211yesmaximal, characteristic
4#20260507C211yesmaximal, characteristic, complemented
5#20260506C211yesmaximal, characteristic, complemented
6#20260505C211yesmaximal, characteristic, complemented
7#344×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C2×#13095495C2211yescharacteristic
11C2×#13095494—21no—
12C2×#13095494C412yescomplemented
13M16×D97—21no—
14M16×D97—21no—
15M16×D97—21no—
16M16×D97—21no—
17M16×D97—21no—
18M16×D97—21no—
19M16×D97C412yescomplemented
20M16×D97C412yescomplemented
21C2×C8×D97—21no—
22C2×C8×D97C412yescomplemented
23#15690769C2211yescharacteristic
24#15690768—21no—
25#15690767—21no—
26#15690766C412yes—
27C4⋊C4×D97—21no—
28C22⋊C4×D97C412yes—
29C42×D97—21no—
30C2×M16×C97C2211yescharacteristic
31C2×M16×C97C2211yescharacteristic
32#117×C97C2211yescharacteristic
33C23×D97C2×C411yescharacteristic
34C22×Dic97C2×C411yescharacteristic
35C2×C4×D97—22no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97C2×C411yescharacteristic
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97C2×C411yescharacteristic
40C2×C4×D97D411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C2×C4×D97D411yescharacteristic
43C2×C4×D97D411yescharacteristic
44C2×C4×D97D411yescharacteristic
45C2×C4×D97D411yescharacteristic
46C2×C4×D97Q811yescharacteristic
47C2×C4×D97Q811yescharacteristic
48#13095495—21no—
49#13095495—21no—
50#13095495—21no—
51#13095495C2×C412yescomplemented
52#13095494—22no—
53#13095494—22no—
54#13095494—22no—
55#13095494—22no—
56C2×#11715895—21no—
57C2×#11715895C2×C412yescomplemented
58C8×D97—22no—
59C8×D97—22no—
60C8×D97—22no—
61C8×D97—22no—
62C4⋊Dic97—21no—
63#13095490—22no—
64C22⋊Dic97C2×C412yes—
65#13095488—22no—
66C4×Dic97—21no—
67C4×Dic97—22no—
68C22×C4×C97C2311yescharacteristic
69M16×C97—21no—
70M16×C97—21no—
71M16×C97—21no—
72M16×C97C2×C412yescomplemented
73C2×C8×C97—21no—
74C2×C8×C97C2×C412yescomplemented
75C4⋊C4×C97—21no—
76C22⋊C4×C97C2×C412yes—
77C42×C97—21no—
78C23×C97C22×C411yescharacteristic
79C22×D97—42no—
80C22×D97C4211yescharacteristic
81C22×D97C4211yescharacteristic
82C22×D97C22⋊C411yescharacteristic
83C22×D97C22⋊C411yescharacteristic
84C22×D97C4⋊C411yescharacteristic
85C22×D97C4⋊C411yescharacteristic
86C2×C4×C97—22no—
87C2×C4×C97C22×C411yescharacteristic
88C2×C4×C97C22×C411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×Q811yescharacteristic
93C2×Dic97—22no—
94C2×Dic97C4211yescharacteristic
95C2×Dic97C4211yescharacteristic
96C2×Dic97C22⋊C411yescharacteristic
97C2×Dic97C22⋊C411yescharacteristic
98C2×Dic97C4⋊C411yescharacteristic
99C2×Dic97C4⋊C411yescharacteristic
100C4×D97—44no—
101C4×D97C22⋊C411yescharacteristic
102C4×D97C22⋊C411yescharacteristic
103C4×D97C22⋊C411yescharacteristic
104C4×D97C22⋊C411yescharacteristic
105C4×D97C22⋊C411yescharacteristic
106C4×D97C22⋊C411yescharacteristic
107C4×D97C22⋊C411yescharacteristic
108C4×D97C22⋊C411yescharacteristic
109C4×D97C4⋊C411yescharacteristic
110C4×D97C4⋊C411yescharacteristic
111C4×D97C4⋊C411yescharacteristic
112C4×D97C4⋊C411yescharacteristic
113C4×D97C4⋊C411yescharacteristic
114C4×D97C4⋊C411yescharacteristic
115C4×D97C4⋊C411yescharacteristic
116C4×D97C4⋊C411yescharacteristic
117C8×C97—22no—
118C8×C97—22no—
119#11715895—22no—
120#11715895—22no—
121C22×C97—41no—
122C22×C97C2×C4211yescharacteristic
123C22×C97C2×C22⋊C411yescharacteristic
124C22×C97C2×C4⋊C411yescharacteristic
125C2×D97—22no—
126C2×D97—22no—
127C2×D97—22no—
128C2×D97—41no—
129C2×D97—41no—
130C2×D97—41no—
131C2×D97#9511yescharacteristic
132C2×D97#9511yescharacteristic
133C2×D97#9511yescharacteristic
134C2×D97#9511yescharacteristic
135C4×C97—42no—
136C4×C97C2×C22⋊C411yescharacteristic
137C4×C97C2×C22⋊C411yescharacteristic
138C4×C97C2×C4⋊C411yescharacteristic
139C4×C97C2×C4⋊C411yescharacteristic
140Dic97—42no—
141Dic97#9511yescharacteristic
142Dic97#9511yescharacteristic
143Dic97#9511yescharacteristic
144Dic97#9511yescharacteristic
145C2×C97—21no—
146C2×C97—21no—
147C2×C97—21no—
148C2×C97C2×#9511yescharacteristic
149D97—21no—
150D97—21no—
151D97—21no—
152D97#34412yesdirect product factor
153C2×#344—971nomaximal, Sylow
154C97C2×#34411yescharacteristic, complemented, Sylow
155C22×M16—971no—
156C22×M16—971no—
157C2×#117—971no—
158#344—971no—
159#344—971no—
160#344—971no—
161#344D9711yescharacteristic, direct product factor
162C23×C4—971no—
163C2×M16—971no—
164C2×M16—972no—
165C2×M16—971no—
166C2×M16—972no—
167C2×M16—972no—
168C2×M16—1941no—
169C2×M16—1941no—
170C2×M16—1941no—
171C2×M16—1941no—
172C2×M16—1941no—
173C2×M16—1941no—
174C2×M16—1941no—
175C2×M16C2×D9711yescharacteristic
176C2×M16C2×D9711yescharacteristic
177C22×C8—972no—
178C22×C8—1941no—
179#117—971no—
180#117—972no—
181#117—1941no—
182#117—1941no—
183#117C2×D9711yescharacteristic
184C2×C4⋊C4—1941no—
185C2×C22⋊C4—972no—
186C2×C42—1941no—
187C24—971no—
188C22×C4—971no—
189C22×C4—971no—
190C22×C4—971no—
191C22×C4—971no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—971no—
195C22×C4—971no—
196C22×C4—971no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—1942no—
202C22×C4C22×D9711yescharacteristic
203M16—21no—
204M16—21no—
205M16—21no—
206M16—972no—
207M16—1942no—
208M16—1942no—
209M16—1942no—
210M16—1942no—
211M16—1941no—
212M16—1941no—
213M16—1941no—
214M16C4×D9712yescomplemented
215C2×C8—21no—
216C2×C8—972no—
217C2×C8—1942no—
218C2×C8—1942no—
219C2×C8—1942no—
220C2×C8—1942no—
221C2×C8—1941no—
222C2×C8C4×D9712yescomplemented
223C4⋊C4—21no—
224C4⋊C4—1942no—
225C4⋊C4—1941no—
226C22⋊C4—972no—
227C22⋊C4—1942no—
228C22⋊C4C4×D9712yes—
229C42—21no—
230C42—1942no—
231C42—1941no—
232C23—971no—
233C23—971no—
234C23—971no—
235C23—971no—
236C23—971no—
237C23—971no—
238C23—3882no—
239C23C2×C4×D9711yescharacteristic
240C2×C4—22no—
241C2×C4—971no—
242C2×C4—971no—
243C2×C4—971no—
244C2×C4—971no—
245C2×C4—971no—
246C2×C4—971no—
247C2×C4—971no—
248C2×C4—971no—
249C2×C4—971no—
250C2×C4—971no—
251C2×C4—971no—
252C2×C4—971no—
253C2×C4—971no—
254C2×C4—971no—
255C2×C4—971no—
256C2×C4—971no—
257C2×C4—971no—
258C2×C4—971no—
259C2×C4—971no—
260C2×C4—971no—
261C2×C4—971no—
262C2×C4—971no—
263C2×C4—3884no—
264C2×C4—1942no—
265C2×C4C2×C4×D9711yescharacteristic
266C2×C4C2×C4×D9711yescharacteristic
267C2×C4D4×D9711yescharacteristic
268C2×C4D4×D9711yescharacteristic
269C2×C4D4×D9711yescharacteristic
270C2×C4Q8×D9711yescharacteristic
271C8—22no—
272C8—22no—
273C8—1942no—
274C8—1942no—
275C22—41no—
276C22—971no—
277C22—971no—
278C22—971no—
279C22—971no—
280C22—3881no—
281C22—3881no—
282C22—3881no—
283C22—1942no—
284C22—1942no—
285C22—1942no—
286C22C42×D9711yescharacteristic
287C22C22⋊C4×D9711yescharacteristic
288C22C4⋊C4×D9711yescharacteristic
289C4—42no—
290C4—971no—
291C4—971no—
292C4—971no—
293C4—971no—
294C4—3882no—
295C4C22⋊C4×D9711yescharacteristic
296C4C22⋊C4×D9711yescharacteristic
297C4C4⋊C4×D9711yescharacteristic
298C4C4⋊C4×D9711yescharacteristic
299C2—21no—
300C2—21no—
301C2—21no—
302C2—972no—
303C2—1941no—
304C2—1941no—
305C2—1941no—
306C2#95×D9711yescharacteristic
subgroupG/H
centerC4C22⋊C4×D97
commutatorC22×C97C2×C42
FrattiniC22×C4C22×D97
Fitting#344×C97C2
socleC2×C97C2×#95
prime radicalC23×D97C2×C4
cosocleC22×C4×C97C23
hypercenter#344D97
hypocenterC97C2×#344
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #20260507C2semidirect
5. #20260506C2semidirect
6. #20260505C2semidirect
7. #344×C97C2semidirect
12. C2×#13095494C4semidirect
19. M16×D97C4semidirect
20. M16×D97C4semidirect
22. C2×C8×D97C4semidirect
51. #13095495C2×C4semidirect
57. C2×#11715895C2×C4semidirect
72. M16×C97C2×C4semidirect
74. C2×C8×C97C2×C4semidirect
152. D97#344direct (N × H)
154. C97C2×#344semidirect
161. #344D97direct (N × H)
214. M16C4×D97semidirect
222. C2×C8C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D972
#117×D972
#202605072
#202605062
#202605052
#344×C972
C2×#34497