Finite Group Databasebeta

← #27142719

Subgroups — #27142719

Order 12416.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690787C211yesmaximal, characteristic
2#20260593C211yesmaximal, characteristic
3C22⋊C4×Dic97C211yesmaximal, characteristic
4#20260502C211yesmaximal, characteristic
5#20260501C211yesmaximal, characteristic
6#20260501C211yesmaximal, characteristic
7#342×C97C211yesmaximal, characteristic
8C22×Dic194C411yescharacteristic, complemented
9C23×Dic97C2211yescharacteristic
10#15690787—21no—
11#15690787—21no—
12#15690787—21no—
13#15690787—21no—
14C2×C4⋊Dic97C411yescharacteristic, complemented
15C2×#13095490—21no—
16C2×#13095490C411yescharacteristic, complemented
17C2×#13095490C411yescharacteristic, complemented
18C2×C22⋊Dic97C2211yescharacteristic
19C2×C22⋊Dic97C2211yescharacteristic
20C2×C22⋊Dic97C2211yescharacteristic
21C2×C4×Dic97—21no—
22#15690717—21no—
23#15690717—21no—
24C2×C22⋊C4×C97C2211yescharacteristic
25C2×C22⋊C4×C97C2211yescharacteristic
26C2×C22⋊C4×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—42no—
30C22×Dic97—21no—
31C22×Dic97—21no—
32C22×Dic97—42no—
33C22×Dic97C2×C411yescharacteristic
34C22×Dic97C2×C411yescharacteristic
35C22×Dic97C2×C411yescharacteristic
36C22×Dic97C2×C411yescharacteristic
37C22×Dic97D411yescharacteristic
38C22×Dic97D411yescharacteristic
39C22×Dic97D411yescharacteristic
40C22×Dic97D411yescharacteristic
41C4⋊Dic97—21no—
42C4⋊Dic97—21no—
43#13095490—21no—
44#13095490—21no—
45#13095490—21no—
46#13095490—21no—
47#13095490—42no—
48C22⋊Dic97—21no—
49C22⋊Dic97—21no—
50C22⋊Dic97—21no—
51C22⋊Dic97—21no—
52C22⋊Dic97—21no—
53C22⋊Dic97Q812yescomplemented
54C4×Dic97—42no—
55C24×C97C2311yescharacteristic
56C22×C4×C97—21no—
57C22×C4×C97—21no—
58C22×C4×C97C2×C411yescharacteristic
59C22×C4×C97C2×C411yescharacteristic
60C22⋊C4×C97—21no—
61C22⋊C4×C97—21no—
62C22⋊C4×C97—21no—
63C22⋊C4×C97—21no—
64C22⋊C4×C97—21no—
65C22⋊C4×C97Q812yescomplemented
66C23×C97—41no—
67C23×C97—41no—
68C23×C97C22×C411yescharacteristic
69C23×C97C2×D411yescharacteristic
70C23×C97C2×D411yescharacteristic
71C23×C97C2×Q811yescharacteristic
72C23×C97C2×Q811yescharacteristic
73C23×C97#4111yescharacteristic
74C23×C97#4111yescharacteristic
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—41no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97—41no—
82C2×C4×C97—41no—
83C2×C4×C97—41no—
84C2×C4×C97—41no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97—22no—
91C2×Dic97—41no—
92C2×Dic97—41no—
93C2×Dic97—41no—
94C2×Dic97—41no—
95C2×Dic97—41no—
96C2×Dic97—41no—
97C2×Dic97—41no—
98C2×Dic97—41no—
99C2×Dic97—41no—
100C2×Dic97—41no—
101C2×Dic97C22⋊C411yescharacteristic
102C2×Dic97C22⋊C411yescharacteristic
103C2×Dic97C22⋊C411yescharacteristic
104C2×Dic97C22⋊C411yescharacteristic
105Dic194—42no—
106Dic194—42no—
107C22×C97—21no—
108C22×C97—21no—
109C22×C97—21no—
110C22×C97—21no—
111C22×C97—21no—
112C22×C97—21no—
113C22×C97—41no—
114C22×C97—41no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97C2×C22⋊C411yescharacteristic
118C22×C97C4×Q811yescharacteristic
119C22×C97C4×Q811yescharacteristic
120C22×C97C22⋊Q811yescharacteristic
121C22×C97C22⋊Q811yescharacteristic
122C22×C97#12411yescharacteristic
123C22×C97C4⋊Q811yescharacteristic
124C4×C97—42no—
125C4×C97—41no—
126C4×C97—41no—
127C4×C97—81no—
128Dic97—21no—
129Dic97—21no—
130Dic97—21no—
131Dic97—42no—
132Dic97—41no—
133Dic97—41no—
134Dic97—81no—
135Dic97#9912yes—
136C2×C97—21no—
137C2×C97—21no—
138C2×C97—21no—
139C2×C97—21no—
140C2×C97—21no—
141C2×C97—21no—
142C2×C97#39111yescharacteristic
143C2×C97C2×#9911yescharacteristic
144C2×C97#41011yescharacteristic
145#1996—971nomaximal, Sylow
146C97#199611yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—971no—
148#380—971no—
149C4×C22⋊C4—971no—
150#342—971no—
151#342—971no—
152#342—971no—
153#342D9711yescharacteristic
154C22×Q8—971no—
155C23×C4—971no—
156C22⋊Q8—1941no—
157C22⋊Q8—1941no—
158C22⋊Q8—1941no—
159C22⋊Q8—1941no—
160C2×C4⋊C4—971no—
161C2×C4⋊C4—971no—
162C2×C4⋊C4—971no—
163C2×C4⋊C4—1941no—
164C2×C22⋊C4—971no—
165C2×C22⋊C4—971no—
166C2×C22⋊C4—971no—
167C2×C22⋊C4C2×D9711yescharacteristic
168C2×C22⋊C4C2×D9711yescharacteristic
169C2×C22⋊C4C2×D9711yescharacteristic
170C2×C42—1941no—
171#95—1941no—
172#95—1941no—
173C24C22×D9711yescharacteristic
174C2×Q8—3882no—
175C2×Q8—1941no—
176C2×Q8—1941no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—971no—
180C22×C4—971no—
181C22×C4—971no—
182C22×C4—971no—
183C22×C4—971no—
184C22×C4—971no—
185C22×C4—971no—
186C22×C4—971no—
187C22×C4—3882no—
188C22×C4—1941no—
189C22×C4—1941no—
190C22×C4C4×D9711yescharacteristic
191C22×C4C4×D9711yescharacteristic
192C4⋊C4—3882no—
193C4⋊C4—1941no—
194C4⋊C4—1941no—
195C4⋊C4—1941no—
196C4⋊C4—1941no—
197C4⋊C4—1941no—
198C4⋊C4—1941no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—972no—
205C22⋊C4—1941no—
206C22⋊C4—1941no—
207C22⋊C4—1941no—
208C22⋊C4—1941no—
209C22⋊C4—1941no—
210C22⋊C4Dic19412yescomplemented
211C42—3882no—
212C23—41no—
213C23—41no—
214C23C2×C4×D9711yescharacteristic
215C23D4×D9711yescharacteristic
216C23D4×D9711yescharacteristic
217C23Q8×D9711yescharacteristic
218C23C2×Dic19411yescharacteristic
219C23#1309551011yescharacteristic
220C23#1309551111yescharacteristic
221Q8—3882no—
222Q8—3882no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—971no—
234C2×C4—971no—
235C2×C4—971no—
236C2×C4—971no—
237C2×C4—3881no—
238C2×C4—3881no—
239C2×C4—3881no—
240C2×C4—3881no—
241C2×C4—3881no—
242C2×C4—3881no—
243C2×C4—3881no—
244C2×C4—3881no—
245C2×C4—3881no—
246C2×C4—3881no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1941no—
251C2×C4—1941no—
252C2×C4—1942no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22C22⋊C4×D9711yescharacteristic
264C22#1569077511yescharacteristic
265C22C4×Dic19411yescharacteristic
266C22#1569078711yescharacteristic
267C22#1569078811yescharacteristic
268C22#1569079611yescharacteristic
269C22#1569080911yescharacteristic
270C4—42no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—7761no—
275C4—972no—
276C4—3882no—
277C4—3881no—
278C4—3881no—
279C4—1941no—
280C4—1941no—
281C4—1941no—
282C2—21no—
283C2—21no—
284C2—21no—
285C2—21no—
286C2—21no—
287C2—21no—
288C2#2026063911yescharacteristic
289C2#99×D9711yescharacteristic
290C2#2026070711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC24C22×D97
Fitting#342×C97C2
socleC22×C97C2×C22⋊C4
prime radical, cosocleC24×C97C23
hypercenter#342D97
hypocenterC97#1996
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
8. C22×Dic194C4semidirect
14. C2×C4⋊Dic97C4semidirect
16. C2×#13095490C4semidirect
17. C2×#13095490C4semidirect
53. C22⋊Dic97Q8semidirect
65. C22⋊C4×C97Q8semidirect
146. C97#1996semidirect
210. C22⋊C4Dic194semidirect

Maximal subgroups

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

subgroupindex
C2×#156907872
#202605932
C22⋊C4×Dic972
#202605022
#202605012
#342×C972
#199697