Finite Group Databasebeta

← #29596015

Subgroups — #29596015

Order 15616.

468 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#21448388C211yesmaximal, characteristic
2C2×#16839560C212yesmaximal
3#21445813C212yesmaximal
4#21443946C211yesmaximal, characteristic
5C61×#1394C211yesmaximal, characteristic
6C2×#14519775C2211yescharacteristic
7C2×Q8×Dic61C2211yescharacteristic
8C2×#14519742C411yescharacteristic
9#16839785—22no—
10C2×#14519696C2212yes—
11#16839668C411yescharacteristic
12#16839659—22no—
13#16839631C411yescharacteristic
14#16839613—22no—
15C42×Dic61C411yescharacteristic
16#16839560—24no—
17#16839530—22no—
18C2×#101×C61C2212yes—
19C61×#391C2211yescharacteristic
20C22×Dic122C2×C411yescharacteristic
21C2×#11758194—22no—
22#14519775—22no—
23#14519775D412yes—
24#14519775Q812yes—
25Q8×Dic61—41no—
26Q8×Dic61—41no—
27#14519742—41no—
28#14519742—41no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×C4⋊Dic61D412yes—
31C2×#11758196—22no—
32C2×#11758196—21no—
33C2×#11758196—22no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61D412yes—
40#14519706—42no—
41#14519706—42no—
42#14519696—22no—
43#14519696—22no—
44#14519684—22no—
45C22×Q8×C61C2311yescharacteristic
46C2×M16×C61—22no—
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C42×C61C2×C411yescharacteristic
49#101×C61—22no—
50#101×C61—22no—
51#95×C61—22no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61C2×D411yescharacteristic
55C2×Q8×C61C2×Q811yescharacteristic
56C2×Q8×C61#4112yes—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D412yes—
61C22×Dic61—21no—
62C22×Dic61—21no—
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C22⋊C412yes—
65C22×Dic61C22⋊C411yescharacteristic
66C2×Dic122—21no—
67C2×Dic122—41no—
68C2×Dic122—41no—
69C2×Dic122C4⋊C412yes—
70M16×C61—42no—
71M16×C61—42no—
72C2×C8×C61—42no—
73C4⋊C4×C61—41no—
74C42×C61—41no—
75C4⋊Dic61—41no—
76C4⋊Dic61—42no—
77#11758196—24no—
78#11758196—24no—
79#11758196—44no—
80#11758196—42no—
81C4×Dic61—24no—
82C4×Dic61—21no—
83C4×Dic61—22no—
84C4×Dic61—22no—
85C4×Dic61—21no—
86C4×Dic61—21no—
87C4×Dic61—42no—
88C4×Dic61—41no—
89C4×Dic61—41no—
90C4×Dic61C4⋊C412yes—
91#11758194—42no—
92#11758194—42no—
93C2×#92418—42no—
94C23×C61C2×C22⋊C411yescharacteristic
95Q8×C61—41no—
96Q8×C61—41no—
97Q8×C61—42no—
98C2×C4×C61—22no—
99C2×C4×C61—21no—
100C2×C4×C61—22no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61—41no—
107C2×C4×C61C2×C4⋊C411yescharacteristic
108C2×C4×C61#11711yescharacteristic
109C2×C4×C61#12112yes—
110C2×C4×C61C22⋊Q812yes—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—21no—
117C2×Dic61—22no—
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121C2×Dic61—42no—
122C2×Dic61#9911yescharacteristic
123C2×Dic61#9911yescharacteristic
124Dic122—42no—
125Dic122—42no—
126C8×C61—82no—
127#92418—82no—
128#12802—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61#38011yescharacteristic
132C22×C61C2×#9911yescharacteristic
133C22×C61#41011yescharacteristic
134C4×C61—22no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—22no—
138C4×C61—42no—
139C4×C61—81no—
140Dic61—24no—
141Dic61—21no—
142Dic61—42no—
143Dic61—41no—
144Dic61—42no—
145Dic61#35612yescomplemented
146C4×C4⋊Q8—611no—
147C2×#356—612no—
148#1801—612no—
149#1394—611no—
150#1394D6111yescharacteristic
151C2×C61—21no—
152C2×C61—21no—
153C2×C61#179911yescharacteristic
154C2×C61C2×#35611yescharacteristic
155C2×C61#213211yescharacteristic
156C2×C4⋊Q8—611no—
157C2×C4×Q8—611no—
158C2×C4×Q8—611no—
159#423—1222no—
160C2×#101—612no—
161C2×#101C2×D6112yes—
162#391—611no—
163#391C2×D6111yescharacteristic
164#389—1222no—
165#384—611no—
166C4×C4⋊C4—1222no—
167C43—611no—
168#356—1224no—
169#343—1222no—
170C61#1280211yescharacteristic, complemented, Sylow
171C22×Q8—611no—
172C22×Q8C22×D6111yescharacteristic
173C2×M16—22no—
174C2×M16—1222no—
175C4⋊Q8—612no—
176C4⋊Q8—612no—
177C4⋊Q8—1222no—
178C4×Q8—2441no—
179C4×Q8—2441no—
180C4×Q8—2441no—
181C4×Q8—2441no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4—612no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4—1222no—
187C2×C4⋊C4C4×D6111yescharacteristic
188C2×C42—611no—
189C2×C42—611no—
190C2×C42—612no—
191C2×C42—1221no—
192C2×C42—1221no—
193C2×C42—1221no—
194C2×C42C4×D6111yescharacteristic
195C4⋊C8—2442no—
196C4⋊C8—2442no—
197#101—22no—
198#101—22no—
199#101—1222no—
200#101—1222no—
201#95—22no—
202#95—1222no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—612no—
206C2×Q8—1221no—
207C2×Q8—2441no—
208C2×Q8—2441no—
209C2×Q8#1175822212yes—
210C2×Q8D4×D6111yescharacteristic
211C2×Q8Q8×D6111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—611no—
215C22×C4—612no—
216C22×C4—611no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4C2×C4×D6111yescharacteristic
220C22×C4D4×D6112yes—
221M16—42no—
222M16—42no—
223M16—2442no—
224M16—2442no—
225C2×C8—42no—
226C2×C8—2442no—
227C4⋊C4—41no—
228C4⋊C4—1224no—
229C4⋊C4—1224no—
230C4⋊C4—2441no—
231C4⋊C4—2442no—
232C4⋊C4—2444no—
233C4⋊C4—2442no—
234C42—41no—
235C42—612no—
236C42—1222no—
237C42—1222no—
238C42—1221no—
239C42—1221no—
240C42—1224no—
241C42—1221no—
242C42—2441no—
243C42—2441no—
244C42—2442no—
245C23C22⋊C4×D6111yescharacteristic
246Q8—42no—
247Q8—41no—
248Q8—41no—
249Q8—2442no—
250Q8—2442no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—611no—
261C2×C4—611no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—1221no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4—2442no—
273C2×C4C4⋊C4×D6111yescharacteristic
274C2×C4#1451973411yescharacteristic
275C2×C4#1451975012yes—
276C2×C4#1451975512yes—
277C8—82no—
278C8—4882no—
279C22—21no—
280C22—21no—
281C22#1683962111yescharacteristic
282C22#99×D6111yescharacteristic
283C22#1683973411yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—42no—
289C4—81no—
290C4—612no—
291C4—1224no—
292C4—1221no—
293C4—2441no—
294C4—2442no—
295C4—2442no—
296C2—21no—
297C2—21no—
298C2#2144580611yescharacteristic
299C2#356×D6111yescharacteristic
300C2#2144750111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC23×C61C2×C22⋊C4
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61#12802
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
145. Dic61#356semidirect
170. C61#12802semidirect

Maximal subgroups

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

subgroupindex
#214483882
C2×#168395602
#214458132
#214439462
C61×#13942
#1280261