Finite Group Databasebeta

← #29579437

Subgroups — #29579437

Order 15616.

374 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 107 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#21445570C211yesmaximal, characteristic
2#21444582C211yesmaximal, characteristic
3#21444434C212yesmaximal
4#21444331C212yesmaximal
5C61×#3170C211yesmaximal, characteristic
6#16839819C2212yes—
7C2×#14519688C2211yescharacteristic
8#16839608C2211yescharacteristic
9#16839516—21no—
10#16839516—21no—
11#16839515—21no—
12#16839513—21no—
13#16839504—24no—
14#16839499—24no—
15C61×#533C2212yes—
16C61×#500C411yescharacteristic, complemented
17C61×#494C411yescharacteristic, complemented
18C61×#487C412yescomplemented
19C61×#467—21no—
20C61×#467—21no—
21C4×Q16×C61—21no—
22C4×Q16×C61—21no—
23C2×C4×C8×C61C2211yescharacteristic
24C22×#92418—21no—
25C2×C4×Dic61—21no—
26C4⋊#92418—22no—
27C22⋊#92418—22no—
28#14519688—21no—
29#14519688—22no—
30#14519688D412yes—
31#14519684—21no—
32#14519684D412yes—
33C2×Q16×C61—21no—
34C2×Q16×C61—21no—
35C22×C8×C61D411yescharacteristic
36C22×C8×C61D411yescharacteristic
37C4⋊Q8×C61C2×C412yes—
38C4⋊Q8×C61C2×C412yes—
39#125×C61—22no—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—22no—
42C4×Q8×C61—22no—
43C4×Q8×C61—22no—
44#117×C61—22no—
45C2×C42×C61C2311yescharacteristic
46#107×C61—21no—
47#107×C61—21no—
48#103×C61—22no—
49#103×C61—22no—
50C4×C8×C61C2×C411yescharacteristic
51C4×C8×C61C2×C411yescharacteristic
52C4×C8×C61D411yescharacteristic
53C4×C8×C61D411yescharacteristic
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61#4111yescharacteristic
59C22×Dic61—22no—
60Q16×C61—42no—
61Q16×C61—42no—
62C2×C8×C61—41no—
63C2×C8×C61—41no—
64C2×C8×C61C22⋊C411yescharacteristic
65C2×C8×C61C22⋊C411yescharacteristic
66C2×C8×C61C22⋊C411yescharacteristic
67C2×C8×C61C22⋊C411yescharacteristic
68C4⋊C4×C61—22no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—22no—
72C4⋊C4×C61—42no—
73C4⋊C4×C61—42no—
74C22⋊C4×C61—42no—
75C42×C61—42no—
76C42×C61C22×C411yescharacteristic
77C42×C61C2×D411yescharacteristic
78C42×C61C2×D411yescharacteristic
79C42×C61#4111yescharacteristic
80C4×Dic61—41no—
81C4×Dic61—41no—
82C2×#92418—22no—
83C2×#92418—41no—
84C2×#92418—41no—
85C2×#92418—42no—
86C23×C61#12111yescharacteristic
87Q8×C61—42no—
88Q8×C61—42no—
89Q8×C61—82no—
90C2×C4×C61—22no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—41no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61C2×C22⋊C411yescharacteristic
97C2×C4×C61C4×D411yescharacteristic
98C2×C4×C61C4×D411yescharacteristic
99C2×C4×C61#12111yescharacteristic
100C2×C4×C61#12411yescharacteristic
101C2×C4×C61#12711yescharacteristic
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C8×C61—21no—
106C8×C61—21no—
107C8×C61—21no—
108C8×C61—21no—
109#92418—42no—
110#92418—82no—
111#9765—611nomaximal, Sylow
112C22×C61—21no—
113C22×C61—21no—
114C22×C61#39011yescharacteristic
115C22×C61#47312yes—
116C4×C61—21no—
117C4×C61—21no—
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—21no—
121C4×C61—82no—
122C4×C61—82no—
123C4×C61#42111yescharacteristic
124C4×C61#42111yescharacteristic
125Dic61—82no—
126#3170D6111yescharacteristic
127#1746—611no—
128#1568—611no—
129#1531—612no—
130#1506—612no—
131C2×C61—22no—
132C2×C61#196311yescharacteristic
133C2×C61#197611yescharacteristic
134C2×C61#197911yescharacteristic
135#533C2×D6112yes—
136#500Dic6111yescharacteristic, complemented
137#494Dic6111yescharacteristic, complemented
138#487Dic6112yescomplemented
139#467—21no—
140#467—21no—
141C4×Q16—21no—
142C4×Q16—21no—
143#432—612no—
144C2×#97—611no—
145C2×C4×C8C2×D6111yescharacteristic
146#376—611no—
147#336—1221no—
148#336—1221no—
149#335—1221no—
150#334—1221no—
151#330—1224no—
152#328—1224no—
153C61#976511yescharacteristic, complemented, Sylow
154C2×Q16—21no—
155C2×Q16—21no—
156C22×C8—1221no—
157C22×C8C22⋊D6111yescharacteristic
158C22×C8C22⋊D6111yescharacteristic
159C4⋊Q8C2×Dic6112yes—
160C4⋊Q8C2×Dic6112yes—
161#125—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166#117—22no—
167C2×C42—1221no—
168C2×C42C22×D6111yescharacteristic
169#107—21no—
170#107—21no—
171C4⋊C8—1222no—
172#103—22no—
173#103—22no—
174C22⋊C8—1222no—
175#97—612no—
176#97—1222no—
177#97—1221no—
178C4×C8C2×Dic6111yescharacteristic
179C4×C8C2×Dic6111yescharacteristic
180C4×C8C22⋊D6111yescharacteristic
181C4×C8C22⋊D6111yescharacteristic
182#95—612no—
183#95—1221no—
184C2×Q8—22no—
185C2×Q8—22no—
186C22×C4—1222no—
187C22×C4D4×D6111yescharacteristic
188C22×C4#1175822411yescharacteristic
189C22×C4C2×C22⋊D6111yescharacteristic
190Q16—42no—
191Q16—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—1222no—
195C2×C8—2442no—
196C2×C8—2441no—
197C2×C8—2441no—
198C2×C8C22⋊Dic6111yescharacteristic
199C2×C8C22⋊Dic6111yescharacteristic
200C2×C8C22⋊Dic6111yescharacteristic
201C2×C8C22⋊Dic6111yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C22⋊C4—42no—
209C42—42no—
210C42—2441no—
211C42—2441no—
212C42D4×D6111yescharacteristic
213C42#1175822411yescharacteristic
214C42C22×Dic6111yescharacteristic
215C42C2×C22⋊D6111yescharacteristic
216C23#1451974811yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—82no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—2442no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4C2×C22⋊Dic6111yescharacteristic
230C2×C4D4×Dic6111yescharacteristic
231C2×C4D4×Dic6111yescharacteristic
232C2×C4#1451974811yescharacteristic
233C2×C4#1451976411yescharacteristic
234C2×C4#1451977311yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—21no—
239C8—4882no—
240C8—2442no—
241C22—21no—
242C22—21no—
243C22#1683966511yescharacteristic
244C22#1684003412yes—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—82no—
251C4—82no—
252C4—4882no—
253C4#1683977911yescharacteristic
254C4#1683977911yescharacteristic
255C2—22no—
256C2#2144665211yescharacteristic
257C2#2144671811yescharacteristic
258C2#2144673511yescharacteristic
subgroupG/H
centerC22#16839665
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3170C2
socleC22×C61#390
prime radicalC23×C61#121
cosocleC2×C42×C61C23
hypercenter#3170D61
hypocenterC61#9765
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
16. C61×#500C4semidirect
17. C61×#494C4semidirect
18. C61×#487C4semidirect
136. #500Dic61semidirect
137. #494Dic61semidirect
138. #487Dic61semidirect
153. C61#9765semidirect

Maximal subgroups

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

subgroupindex
#214455702
#214445822
#214444342
#214443312
C61×#31702
#976561