Finite Group Databasebeta

← #29661947

Subgroups — #29661947

Order 15616.

784 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 171 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#21457033C212yesmaximal
2#21456341C211yesmaximal, characteristic
3#21445147C218yesmaximal, complemented
4#21444457C211yesmaximal, characteristic
5#21444434C212yesmaximal
6C61×#3556C211yesmaximal, characteristic
7#16840319C2211yescharacteristic
8#16840172C2214yes—
9#16840172C2214yes—
10#16840168—22no—
11#16840132C2214yes—
12#16840103—24no—
13#16840054—22no—
14#16840044—22no—
15#16839819C2212yes—
16#16839798—22no—
17C2×#14519688C2211yescharacteristic
18#16839504C2218yes—
19#16839504C2214yes—
20C61×#564C2214yes—
21C61×#557—22no—
22C61×#557—22no—
23C61×#557—22no—
24C61×#541—22no—
25C61×#541—24no—
26C61×#533C2212yes—
27C61×#533C2211yescharacteristic
28C22×#92418—21no—
29#14519777C2312yes—
30#14519769—21no—
31#14519756—22no—
32C4×Dic122—22no—
33#14519736—21no—
34#14519710—24no—
35#14519710—24no—
36#14519708—24no—
37#14519708—24no—
38C4⋊#92418—22no—
39#14519700—24no—
40#14519700—24no—
41C22⋊#92418—22no—
42#14519688—21no—
43#14519688C2312yes—
44#14519688C2314yes—
45C4⋊Q8×C61—24no—
46C4⋊Q8×C61—22no—
47C4⋊Q8×C61D412yes—
48C4⋊Q8×C61D412yes—
49C4⋊Q8×C61D412yes—
50#126×C61—28no—
51#125×C61—24no—
52#125×C61—22no—
53#125×C61—28no—
54#125×C61—28no—
55#125×C61—28no—
56#125×C61C2314yes—
57#125×C61C2312yes—
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61—22no—
60C22⋊Q8×C61—22no—
61C4×Q8×C61—24no—
62C4×Q8×C61—24no—
63C4×Q8×C61—24no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×Q8×C61—22no—
67#117×C61D412yes—
68#117×C61D414yes—
69C2×C42×C61C2311yescharacteristic
70C2×Q8×C61—22no—
71C2×Q8×C61—22no—
72C2×Q8×C61—22no—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D412yes—
75C2×Dic122—22no—
76C4⋊C4×C61—28no—
77C4⋊C4×C61—28no—
78C4⋊C4×C61—28no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—22no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—24no—
83C4⋊C4×C61—24no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61C2×D414yes—
86C4⋊C4×C61C2×D414yes—
87C4⋊C4×C61C2×D414yes—
88C22⋊C4×C61—22no—
89C22⋊C4×C61—24no—
90C42×C61—28no—
91C42×C61—24no—
92C42×C61—22no—
93C42×C61C2×D411yescharacteristic
94C42×C61C2×D412yes—
95C42×C61C2411yescharacteristic
96C22⋊Dic61—41no—
97C4⋊Dic61—22no—
98C4⋊Dic61—22no—
99#11758196—42no—
100C4×Dic61—41no—
101C2×#92418—24no—
102C2×#92418—22no—
103C2×#92418—24no—
104C23×C61#12011yescharacteristic
105Q8×C61—44no—
106Q8×C61—44no—
107Q8×C61—44no—
108C2×C4×C61—22no—
109C2×C4×C61—24no—
110C2×C4×C61—24no—
111C2×C4×C61—24no—
112C2×C4×C61—42no—
113C2×C4×C61#12012yes—
114C2×C4×C61#12011yescharacteristic
115C2×C4×C61C22×D411yescharacteristic
116C2×C4×C61C22×D412yes—
117C2×Dic61—42no—
118Dic122—44no—
119#92418—24no—
120#92418—44no—
121#19067—611nomaximal, Sylow
122C22×C61—21no—
123C22×C61—22no—
124C22×C61C2×#12011yescharacteristic
125C4×C61—21no—
126C4×C61—22no—
127C4×C61—22no—
128C4×C61—44no—
129C4×C61—44no—
130C4×C61—44no—
131C4×C61#45612yes—
132Dic61—82no—
133#3556D6111yescharacteristic
134#3237—612no—
135#3162—611no—
136#1663—618no—
137#1537—611no—
138#1531—612no—
139C2×C61—41no—
140C2×C61C2×#45611yescharacteristic
141C2×C61#302012yes—
142#564C2×D6114yes—
143#557—22no—
144#557—22no—
145#557—22no—
146#541—22no—
147#541—24no—
148#533—611no—
149#533C2×D6111yescharacteristic
150#533C2×D6112yes—
151#501—614no—
152#501—614no—
153#501—1222no—
154#491—614no—
155#485—1224no—
156#477—1222no—
157#475—1222no—
158#432—612no—
159#427—1222no—
160C2×#97—611no—
161#330—614no—
162#330—618no—
163C61#1906711yescharacteristic, complemented, Sylow
164C22×C8—1221no—
165C4⋊Q8—24no—
166C4⋊Q8—22no—
167C4⋊Q8—612no—
168C4⋊Q8C22⋊D6112yes—
169C4⋊Q8C22⋊D6112yes—
170C4⋊Q8C22⋊D6112yes—
171#126—28no—
172#125—24no—
173#125—22no—
174#125—28no—
175#125—28no—
176#125—28no—
177#125—1221no—
178#125C22×D6114yes—
179#125C22×D6112yes—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—1222no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—1222no—
191#117—1221no—
192#117C22⋊D6112yes—
193#117C22⋊D6114yes—
194C2×C42C22×D6111yescharacteristic
195#107—1224no—
196#107—1224no—
197#106—1224no—
198#106—1224no—
199C4⋊C8—1222no—
200#103—1224no—
201#103—1224no—
202C22⋊C8—1222no—
203#97—612no—
204#97—614no—
205#97—1221no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—1222no—
210C22×C4C2×C22⋊D6111yescharacteristic
211C22×C4C2×C22⋊D6112yes—
212C2×C8—1224no—
213C2×C8—1222no—
214C2×C8—1224no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—1222no—
225C4⋊C4—1222no—
226C4⋊C4—2442no—
227C4⋊C4C2×C22⋊D6114yes—
228C4⋊C4C2×C22⋊D6114yes—
229C4⋊C4C2×C22⋊D6114yes—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—2441no—
233C42—28no—
234C42—24no—
235C42—22no—
236C42—2441no—
237C42C2×C22⋊D6111yescharacteristic
238C42C2×C22⋊D6112yes—
239C42C23×D6111yescharacteristic
240C23#1451974611yescharacteristic
241Q8—44no—
242Q8—44no—
243Q8—44no—
244Q8—2444no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—2442no—
251C2×C4#1451974612yes—
252C2×C4#1451974611yescharacteristic
253C2×C4C22×C22⋊D6111yescharacteristic
254C2×C4C22×C22⋊D6112yes—
255C8—1224no—
256C8—2444no—
257C22—21no—
258C22—22no—
259C22C2×#1451974611yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—44no—
264C4—44no—
265C4—44no—
266C4—4882no—
267C4#1683994512yes—
268C2—41no—
269C2C2×#1683994511yescharacteristic
270C2#2145518312yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3556C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenter#3556D61
hypocenterC61#19067
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
3. #21445147C2semidirect
163. C61#19067semidirect

Maximal subgroups

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

subgroupindex
#214570332
#214563412
#214451472
#214444572
#214444342
C61×#35562
#1906761