Finite Group Databasebeta

← #29572555

Subgroups — #29572555

Order 15616.

656 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 131 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#21444545C212yesmaximal, complemented
2#21444446C212yesmaximal, complemented
3#21444153C212yesmaximal
4C2×C61×#541C211yesmaximal, characteristic
5#16839819C2214yes—
6#16839511—24no—
7#16839504—24no—
8#16839488—22no—
9#16839486—22no—
10C61×#541—24no—
11C61×#541—24no—
12C2×C4⋊Q8×C61C2211yescharacteristic
13C2×#126×C61—22no—
14C2×#125×C61—22no—
15C2×#125×C61C2211yescharacteristic
16C2×C22⋊Q8×C61C412yes—
17C2×C4×Q8×C61C412yes—
18C2×#117×C61C2211yescharacteristic
19C4⋊#92418—24no—
20C22⋊#92418—24no—
21#14519688—24no—
22C22×Q8×C61C2×C412yes—
23C23×C4×C61C2×C411yescharacteristic
24C4⋊Q8×C61—22no—
25C4⋊Q8×C61D414yes—
26#126×C61—216no—
27#125×C61—216no—
28#125×C61—22no—
29#125×C61D414yes—
30C22⋊Q8×C61—24no—
31C22⋊Q8×C61—24no—
32C4×Q8×C61—24no—
33C4×Q8×C61—24no—
34#117×C61—22no—
35#117×C61—22no—
36C2×C4⋊C4×C61—22no—
37C2×C4⋊C4×C61—22no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61C2×C412yes—
41C2×C4⋊C4×C61D412yes—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61D412yes—
44C2×C42×C61—22no—
45C2×C42×C61C2×C411yescharacteristic
46C2×C42×C61C2311yescharacteristic
47C24×C61C22⋊C411yescharacteristic
48C2×Q8×C61—22no—
49C2×Q8×C61—44no—
50C2×Q8×C61C22⋊C414yes—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—42no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D412yes—
60C4⋊C4×C61—28no—
61C4⋊C4×C61—28no—
62C4⋊C4×C61—22no—
63C4⋊C4×C61—28no—
64C4⋊C4×C61—28no—
65C4⋊C4×C61—24no—
66C4⋊C4×C61C22⋊C414yes—
67C22⋊C4×C61—28no—
68C22⋊C4×C61—24no—
69C42×C61—28no—
70C42×C61—21no—
71C42×C61—21no—
72C42×C61C2×D412yes—
73C42×C61C2×D412yes—
74C2×#92418—44no—
75C23×C61—21no—
76C23×C61—22no—
77C23×C61—42no—
78C23×C61C2×C22⋊C411yescharacteristic
79Q8×C61—44no—
80Q8×C61—44no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—24no—
84C2×C4×C61—24no—
85C2×C4×C61—21no—
86C2×C4×C61—22no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61—44no—
91C2×C4×C61—44no—
92C2×C4×C61—42no—
93C2×C4×C61C2×C22⋊C411yescharacteristic
94C2×C4×C61C2×C22⋊C411yescharacteristic
95C2×C4×C61#12014yes—
96#92418—84no—
97#8527—611nomaximal, Sylow
98C22×C61—22no—
99C22×C61—42no—
100C22×C61—41no—
101C22×C61—82no—
102C22×C61#37911yescharacteristic
103C22×C61#41312yes—
104C4×C61—24no—
105C4×C61—21no—
106C4×C61—21no—
107C4×C61—42no—
108C4×C61—44no—
109C4×C61—41no—
110C4×C61—44no—
111C2×#541D6111yescharacteristic
112#1558—612no—
113#1534—612no—
114#1464—612no—
115C2×C61—22no—
116C2×C61—81no—
117C2×C61#178511yescharacteristic
118C2×C61#179612yes—
119#541—24no—
120#541—24no—
121C2×C4⋊Q8C2×D6111yescharacteristic
122C2×#126—22no—
123C2×#125—22no—
124C2×#125C2×D6111yescharacteristic
125C2×C22⋊Q8Dic6112yes—
126C2×C4×Q8Dic6112yes—
127C2×#117C2×D6111yescharacteristic
128#432—614no—
129#333—1224no—
130#330—1224no—
131#324—1222no—
132#323—1222no—
133C61#852711yescharacteristic, complemented, Sylow
134C22×Q8C2×Dic6112yes—
135C23×C4C2×Dic6111yescharacteristic
136C4⋊Q8—22no—
137C4⋊Q8C22⋊D6114yes—
138#126—216no—
139#125—216no—
140#125—22no—
141#125C22⋊D6114yes—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C2×Dic6112yes—
153C2×C4⋊C4C22⋊D6112yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C22⋊D6112yes—
156C2×C42—22no—
157C2×C42C2×Dic6111yescharacteristic
158C2×C42C22×D6111yescharacteristic
159C4⋊C8—1224no—
160C22⋊C8—1224no—
161#97—1224no—
162C24C22⋊Dic6111yescharacteristic
163C2×Q8—22no—
164C2×Q8—44no—
165C2×Q8C22⋊Dic6114yes—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4C22⋊Dic6111yescharacteristic
172C22×C4C22⋊Dic6112yes—
173C22×C4C22×Dic6111yescharacteristic
174C22×C4C2×C22⋊D6112yes—
175C2×C8—2444no—
176C4⋊C4—28no—
177C4⋊C4—28no—
178C4⋊C4—22no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4C22⋊Dic6114yes—
183C22⋊C4—28no—
184C22⋊C4—24no—
185C42—28no—
186C42—21no—
187C42—21no—
188C42C2×C22⋊D6112yes—
189C42C2×C22⋊D6112yes—
190C23—21no—
191C23—22no—
192C23—42no—
193C23C2×C22⋊Dic6111yescharacteristic
194Q8—44no—
195Q8—44no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4C2×C22⋊Dic6111yescharacteristic
209C2×C4C2×C22⋊Dic6111yescharacteristic
210C2×C4#1451974614yes—
211C8—4884no—
212C22—22no—
213C22—41no—
214C22—42no—
215C22—82no—
216C22#1683961811yescharacteristic
217C22#1683974612yes—
218C4—21no—
219C4—21no—
220C4—24no—
221C4—41no—
222C4—44no—
223C4—42no—
224C4—44no—
225C2—22no—
226C2—81no—
227C2#2144574711yescharacteristic
228C2#2144579412yes—
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#541C2
socleC22×C61#379
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenterC2×#541D61
hypocenterC61#8527
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. #21444545C2semidirect
2. #21444446C2semidirect
133. C61#8527semidirect

Maximal subgroups

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

subgroupindex
#214445452
#214444462
#214441532
C2×C61×#5412
#852761