Finite Group Databasebeta

← #29589214

Subgroups — #29589214

Order 15616.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16839814C211yesmaximal, characteristic
2#21447619C212yesmaximal, complemented
3#21443759C211yesmaximal, characteristic, complemented
4#21443742C212yesmaximal
5C61×#1322C211yesmaximal, characteristic, complemented
6C2×C42×D61C411yescharacteristic
7#16839814—22no—
8#16839814—22no—
9C2×C4⋊#92418C411yescharacteristic
10C2×#14519689C411yescharacteristic
11C2×#14519689C2211yescharacteristic
12C2×C8×Dic61C411yescharacteristic
13C2×C8×Dic61C2211yescharacteristic
14#16839580—22no—
15C22⋊#11758185C2212yes—
16#16839537—22no—
17C2×C4⋊C8×C61C2211yescharacteristic
18C61×C22⋊C16C2212yes—
19C22×C4×D61C811yescharacteristic
20C22×C4×D61C811yescharacteristic
21C22×C4×D61C2×C411yescharacteristic
22C22×#92418C2×C411yescharacteristic
23C22×#92418C2×C411yescharacteristic
24C2×C4×Dic61C811yescharacteristic
25C2×C4×Dic61C811yescharacteristic
26C2×C4×Dic61C2×C411yescharacteristic
27C42×D61—21no—
28C42×D61—21no—
29C42×D61—21no—
30C42×D61—21no—
31C2×#11758185—22no—
32C4⋊#92418—22no—
33#14519689—22no—
34#14519689—22no—
35C8×Dic61—21no—
36C8×Dic61—21no—
37C8×Dic61D412yes—
38C8×Dic61D412yes—
39C22×C8×C61C2×C411yescharacteristic
40C22×C8×C61C2311yescharacteristic
41C2×C42×C61C2×C411yescharacteristic
42C2×C16×C61—22no—
43C4⋊C8×C61—22no—
44C23×D61C2×C811yescharacteristic
45C22×C4×C61C2×C811yescharacteristic
46C22×C4×C61C2×C811yescharacteristic
47C22×C4×C61C22×C411yescharacteristic
48C22×Dic61C2×C811yescharacteristic
49C22×Dic61C2×C811yescharacteristic
50C22×Dic61C2×C811yescharacteristic
51C2×C4×D61—22no—
52C2×C4×D61—22no—
53C2×C4×D61—21no—
54C2×C4×D61—21no—
55C2×C4×D61—21no—
56C2×C4×D61—21no—
57C2×C4×D61—21no—
58C2×C4×D61—21no—
59C2×C4×D61—42no—
60C2×C4×D61—41no—
61C2×C4×D61—41no—
62C2×C8×C61—21no—
63C2×C8×C61—41no—
64C2×C8×C61—41no—
65C2×C8×C61C2×D412yes—
66C42×C61—21no—
67C42×C61—21no—
68C4×Dic61—21no—
69C4×Dic61—21no—
70C4×Dic61—21no—
71C4×Dic61—21no—
72C4×Dic61C22⋊C411yescharacteristic
73C4×Dic61C22⋊C411yescharacteristic
74C4×Dic61M1611yescharacteristic
75C4×Dic61M1611yescharacteristic
76C2×#92418—21no—
77C2×#92418—41no—
78C2×#92418—41no—
79C2×#92418C22⋊C412yes—
80C16×C61—42no—
81#11758185—42no—
82C23×C61C22×C811yescharacteristic
83C22×D61—21no—
84C22×D61—21no—
85C22×D61—41no—
86C22×D61—41no—
87C22×D61M3212yes—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—41no—
94C2×C4×C61M3212yes—
95C2×C4×C61C2×C22⋊C411yescharacteristic
96C2×C4×C61C2×M1611yescharacteristic
97C2×Dic61—21no—
98C2×Dic61—21no—
99C2×Dic61—22no—
100C2×Dic61—21no—
101C2×Dic61—41no—
102C2×Dic61C22⋊C811yescharacteristic
103C2×Dic61C22⋊C811yescharacteristic
104C2×Dic61C22⋊C811yescharacteristic
105C2×Dic61C22⋊C811yescharacteristic
106C4×D61—24no—
107C4×D61—24no—
108C4×D61—42no—
109C4×D61—42no—
110C4×D61—41no—
111C4×D61—41no—
112C4×D61—42no—
113C8×C61—22no—
114C8×C61—41no—
115#92418—22no—
116#92418—41no—
117#11460—611nomaximal, Sylow
118C22×C61—21no—
119C22×C61—21no—
120C22×C61C2×C22⋊C811yescharacteristic
121C22×C61C2×M3211yescharacteristic
122C22×C61C2×M3211yescharacteristic
123C2×D61—22no—
124C2×D61—42no—
125C2×D61—41no—
126C2×D61—41no—
127C2×D61—42no—
128C4×C61—22no—
129C4×C61—21no—
130C4×C61—41no—
131C4×C61#41011yescharacteristic
132C4×C61#41311yescharacteristic
133Dic61—21no—
134Dic61—21no—
135Dic61—21no—
136Dic61—41no—
137Dic61#34912yes—
138C2×#431—611no—
139#2163—612no—
140#1322—611no—
141#1322D6111yescharacteristic, complemented
142#1314—612no—
143C2×C61—21no—
144C2×C61—21no—
145C2×C61#145811yescharacteristic
146C2×C61#211211yescharacteristic
147C2×C61C2×#34911yescharacteristic
148D61—42no—
149C22×C42—611no—
150#431—1222no—
151#431—1222no—
152C2×C4⋊C8—611no—
153C2×C4⋊C8C2×D6111yescharacteristic
154C2×C22⋊C8—611no—
155C2×C22⋊C8—611no—
156C2×C4×C8—611no—
157C2×C4×C8—611no—
158C4⋊C16—1222no—
159C22⋊C16—612no—
160C22⋊C16C2×D6112yes—
161#346—1222no—
162C61#1146011yescharacteristic, complemented, Sylow
163C23×C4—611no—
164C23×C4—611no—
165C23×C4—611no—
166C22×C8—611no—
167C22×C8—611no—
168C22×C8C4×D6111yescharacteristic
169C22×C8C22×D6111yescharacteristic
170C2×C42—611no—
171C2×C42—611no—
172C2×C42—611no—
173C2×C42—1221no—
174C2×C42—1221no—
175C2×C42—1221no—
176C2×C42—1221no—
177C2×C42C4×D6111yescharacteristic
178C2×C16—22no—
179C2×C16—1222no—
180C4⋊C8—22no—
181C4⋊C8—1222no—
182C22⋊C8—1222no—
183C22⋊C8—1222no—
184C4×C8—612no—
185C4×C8—612no—
186C4×C8—1221no—
187C4×C8—1221no—
188C24—611no—
189C22×C4—611no—
190C22×C4—611no—
191C22×C4—611no—
192C22×C4—1221no—
193C22×C4—1221no—
194C22×C4—1221no—
195C22×C4—1221no—
196C22×C4—1222no—
197C22×C4—1222no—
198C22×C4—1221no—
199C22×C4—1221no—
200C22×C4—2441no—
201C22×C4—2441no—
202C22×C4—2442no—
203C22×C4C8×D6111yescharacteristic
204C22×C4C8×D6111yescharacteristic
205C22×C4C2×C4×D6111yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—612no—
210C2×C8—1221no—
211C2×C8—2441no—
212C2×C8—2441no—
213C2×C8D4×D6112yes—
214C42—21no—
215C42—21no—
216C42—611no—
217C42—611no—
218C42—611no—
219C42—611no—
220C42—1221no—
221C42—1221no—
222C42—1221no—
223C42—1221no—
224C16—42no—
225C16—2442no—
226C23—612no—
227C23—1221no—
228C23—1221no—
229C23—2441no—
230C23—2441no—
231C23C2×C8×D6111yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—611no—
239C2×C4—611no—
240C2×C4—611no—
241C2×C4—611no—
242C2×C4—1224no—
243C2×C4—1224no—
244C2×C4—1222no—
245C2×C4—1221no—
246C2×C4—1221no—
247C2×C4—1221no—
248C2×C4—2442no—
249C2×C4—2442no—
250C2×C4—2442no—
251C2×C4—2441no—
252C2×C4—2441no—
253C2×C4—2441no—
254C2×C4#1451971612yes—
255C2×C4C22⋊C4×D6111yescharacteristic
256C2×C4M16×D6111yescharacteristic
257C8—22no—
258C8—41no—
259C8—1222no—
260C8—2441no—
261C22—21no—
262C22—21no—
263C22—1222no—
264C22—2442no—
265C22—2442no—
266C22—2441no—
267C22—2441no—
268C22C22⋊C8×D6111yescharacteristic
269C22M32×D6111yescharacteristic
270C22C2×#1451971611yescharacteristic
271C4—22no—
272C4—21no—
273C4—41no—
274C4—612no—
275C4—1221no—
276C4—1221no—
277C4—1221no—
278C4—2441no—
279C4#1683973411yescharacteristic
280C4#1683974511yescharacteristic
281C2—21no—
282C2—21no—
283C2—2442no—
284C2#2144413211yescharacteristic
285C2#2144742811yescharacteristic
286C2#349×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC23×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1322C2
socleC22×C61C2×C22⋊C8
prime radicalC23×D61C2×C8
cosocleC22×C8×C61C23
hypercenter#1322D61
hypocenterC61#11460
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #21447619C2semidirect
3. #21443759C2semidirect
5. C61×#1322C2semidirect
141. #1322D61semidirect
162. C61#11460semidirect

Maximal subgroups

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

subgroupindex
C2×#168398142
#214476192
#214437592
#214437422
C61×#13222
#1146061