Finite Group Databasebeta

← #30599569

Subgroups — #30599569

Order 15616.

454 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 102 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#21462684C211yesmaximal, characteristic
2#21461941C211yesmaximal, characteristic
3#21461370C212yesmaximal, complemented
4#21460652C212yesmaximal, complemented
5#21445610C211yesmaximal, characteristic
6#16840796—22no—
7#16840796—22no—
8#16840778C2212yes—
9C2×C4⋊#92420C2211yescharacteristic
10#16840733—22no—
11C2×#14519834—22no—
12C2×C4×#92420C2211yescharacteristic
13#16840625—22no—
14#16840623—22no—
15C2×#14519739C2211yescharacteristic
16C2×#14519693C412yes—
17M16×Dic61C2212yes—
18#16839531C411yescharacteristic
19C61×#343C411yescharacteristic
20C2×#11758213—22no—
21C2×#11758213—22no—
22C22×#92420—21no—
23C22×#92420—21no—
24C4⋊#92420—22no—
25C4⋊#92420D412yes—
26C4⋊#92420D412yes—
27#14519834—24no—
28#14519834—44no—
29C22⋊#92420—22no—
30C4×#92420—22no—
31C4×#92420—22no—
32C22×D244C2×C411yescharacteristic
33C22×C4×D61D412yes—
34C2×#11758194C2×C412yes—
35#14519739—22no—
36#14519739—22no—
37C2×#11758198Q812yes—
38C2×C4×Dic61C2311yescharacteristic
39#14519693—24no—
40#14519687—22no—
41C8×Dic61—22no—
42C2×M16×C61C2×C412yes—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C23×D61C4⋊C412yes—
45C22×C4×C61C4⋊C412yes—
46C22×C4×C61C22×C411yescharacteristic
47C22×Dic61C2×D411yescharacteristic
48C22×Dic61C2×Q811yescharacteristic
49C2×D244—22no—
50C2×D244—22no—
51C2×D244—21no—
52C2×D244—21no—
53C2×C4×D61—22no—
54C2×C4×D61—22no—
55C2×C4×D61—42no—
56C2×C4×D61—42no—
57C2×#92420—22no—
58C2×#92420—22no—
59C2×#92420—22no—
60C2×#92420—41no—
61C2×#92420—41no—
62C2×#92420—42no—
63C2×#92420—41no—
64#11758213—82no—
65#11758213—82no—
66M16×C61—22no—
67M16×C61M1614yescomplemented
68C2×C8×C61—22no—
69C4⋊C4×C61—22no—
70#11758198—24no—
71C4×Dic61C2×D412yes—
72C4×Dic61#4112yes—
73#11758194—22no—
74#11758194—22no—
75C2×#92418—22no—
76C23×C61C2×C4⋊C411yescharacteristic
77C22×D61—22no—
78C22×D61—22no—
79C22×D61—22no—
80C22×D61—42no—
81C22×D61—42no—
82C2×C4×C61—22no—
83C2×C4×C61—42no—
84C2×C4×C61C4×D412yes—
85C2×C4×C61C2×M1612yes—
86C2×C4×C61#13111yescharacteristic
87C2×C4×C61#13111yescharacteristic
88C2×Dic61—21no—
89C2×Dic61—21no—
90C2×Dic61—41no—
91C2×Dic61#12011yescharacteristic
92C2×Dic61C22⋊Q812yes—
93C2×Dic61#12311yescharacteristic
94D244—42no—
95D244—42no—
96C4×D61—42no—
97C4×D61—42no—
98C4×D61—82no—
99#92420—42no—
100#92420—42no—
101#92420—42no—
102#92420—81no—
103C8×C61—24no—
104#92418—42no—
105#8837—611nomaximal, Sylow
106C22×C61—21no—
107C22×C61—21no—
108C22×C61#38511yescharacteristic
109C22×C61#42311yescharacteristic
110C22×C61#42411yescharacteristic
111C2×D61—44no—
112C2×D61—42no—
113C2×D61—42no—
114C2×D61—42no—
115C2×D61—82no—
116C4×C61—42no—
117C4×C61#43512yes—
118C4×C61#43612yes—
119Dic61—21no—
120Dic61—22no—
121Dic61#45311yescharacteristic
122Dic61#45511yescharacteristic
123#2975—611no—
124#1954—611no—
125C4×#100—611no—
126#1565—612no—
127#1278—612no—
128C2×C61—21no—
129C2×C61—21no—
130C2×C61#185111yescharacteristic
131C2×C61#185611yescharacteristic
132C2×C61#185811yescharacteristic
133D61—82no—
134C2×C4×D4—611no—
135#436—1222no—
136#436—1222no—
137#431—612no—
138C2×C4⋊C8—611no—
139C2×#100—612no—
140#407—1222no—
141C2×C22⋊C8—1222no—
142C4×M16—612no—
143C2×C4×C8—611no—
144#343—611no—
145#343#692311yescharacteristic
146#335—1222no—
147#334—1222no—
148C61#883711yescharacteristic, complemented, Sylow
149C22×D4—611no—
150C23×C4—612no—
151C2×M16—612no—
152C2×M16—1222no—
153C2×M16—1222no—
154C2×M16C2×#692312yes—
155C22×C8—1221no—
156C22×C8—1221no—
157C4×D4—1222no—
158C4×D4—1222no—
159C2×C4⋊C4C2×#692311yescharacteristic
160C2×C22⋊C4—612no—
161C2×C42—611no—
162C4⋊C8—612no—
163C4⋊C8—612no—
164C4⋊C8—1222no—
165#100—1224no—
166C22⋊C8—1222no—
167C22⋊C8—1224no—
168C22⋊C8—2444no—
169#97—1222no—
170C4×C8—1222no—
171C4×C8—1222no—
172C4×C8—1222no—
173C24—612no—
174C2×D4—1222no—
175C2×D4—1222no—
176C2×D4—1221no—
177C2×D4—1221no—
178C22×C4—611no—
179C22×C4—611no—
180C22×C4—1222no—
181C22×C4—1222no—
182C22×C4—2442no—
183C22×C4—2442no—
184C22×C4C4⋊#692312yes—
185C22×C4C22×#692311yescharacteristic
186M16—22no—
187M16—1222no—
188M16—1222no—
189M16—4882no—
190M16—4882no—
191M16#1175821314yescomplemented
192C2×C8—22no—
193C2×C8—1222no—
194C2×C8—1222no—
195C2×C8—1222no—
196C2×C8—1222no—
197C2×C8—2441no—
198C2×C8—2442no—
199C2×C8—2441no—
200C2×C8—2441no—
201C4⋊C4—22no—
202C22⋊C4—1224no—
203C42—612no—
204C42—612no—
205C23—1222no—
206C23—1222no—
207C23—1222no—
208C23—2442no—
209C23—2442no—
210C23C2×C4⋊#692311yescharacteristic
211D4—2442no—
212D4—2442no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—611no—
216C2×C4—612no—
217C2×C4—611no—
218C2×C4—1221no—
219C2×C4—1221no—
220C2×C4—4882no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4—2441no—
224C2×C4D4×#692312yes—
225C2×C4C2×#1175821312yes—
226C2×C4#1451986411yescharacteristic
227C2×C4#1451986511yescharacteristic
228C8—24no—
229C8—4881no—
230C8—2442no—
231C8—2442no—
232C8—2442no—
233C8—2442no—
234C22—21no—
235C22—21no—
236C22—4882no—
237C22—2444no—
238C22—2442no—
239C22—2442no—
240C22—2442no—
241C22#1684070911yescharacteristic
242C22#1684076611yescharacteristic
243C22#1684076911yescharacteristic
244C4—42no—
245C4—611no—
246C4—611no—
247C4—1221no—
248C4—1222no—
249C4#1684079112yes—
250C4#1684079612yes—
251C2—21no—
252C2—21no—
253C2—4882no—
254C2#2146176511yescharacteristic
255C2#2146177411yescharacteristic
256C2#2146177711yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#343C4
socleC22×C61#385
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#343#6923
hypocenterC61#8837
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #21461370C2semidirect
4. #21460652C2semidirect
67. M16×C61M16semidirect
148. C61#8837semidirect
191. M16#11758213semidirect

Maximal subgroups

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

subgroupindex
#214626842
#214619412
#214613702
#214606522
#214456102
#883761