Finite Group Databasebeta

← #29658304

Subgroups — #29658304

Order 15616.

808 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 179 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#21456967C212yesmaximal
2#21456201C211yesmaximal, characteristic
3#21444977C218yesmaximal, complemented
4#21444484C211yesmaximal, characteristic
5#21444240C212yesmaximal
6C61×#3556C211yesmaximal, characteristic
7C2×#14519777C2211yescharacteristic
8#16840165—22no—
9#16840132—22no—
10#16840114C2214yes—
11#16840112—22no—
12#16840095—24no—
13#16840057C2218yes—
14#16839815C2212yes—
15#16839770—22no—
16C4×#11758194C2211yescharacteristic
17#16839508C2214yes—
18Q8⋊#92418C2218yes—
19C61×#564—22no—
20C61×#557—22no—
21C61×#557—22no—
22C61×#557C2214yes—
23C61×#541—24no—
24C61×#541—22no—
25C61×#533C2212yes—
26C61×#533C2211yescharacteristic
27C22×Dic122—21no—
28C2×#11758194—21no—
29#14519777—22no—
30#14519777—21no—
31#14519777C2312yes—
32C2×C4⋊Dic61—22no—
33#14519710—28no—
34C4⋊#92418C2314yes—
35#14519701—28no—
36#14519700—28no—
37C22⋊#92418—22no—
38#14519688—21no—
39C4×#92418—22no—
40C4×#92418C2312yes—
41C4⋊Q8×C61—22no—
42C4⋊Q8×C61—24no—
43C4⋊Q8×C61D412yes—
44C4⋊Q8×C61D412yes—
45C4⋊Q8×C61C2312yes—
46#126×C61—28no—
47#125×C61—22no—
48#125×C61—28no—
49#125×C61—28no—
50#125×C61—24no—
51#125×C61—28no—
52#125×C61—22no—
53#125×C61D412yes—
54C22⋊Q8×C61—22no—
55C22⋊Q8×C61—22no—
56C22⋊Q8×C61D414yes—
57C4×Q8×C61—24no—
58C4×Q8×C61—24no—
59C4×Q8×C61—24no—
60C4×Q8×C61—22no—
61C4×Q8×C61—22no—
62C4×Q8×C61C2314yes—
63#117×C61—22no—
64#117×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66C2×Q8×C61—22no—
67C2×Q8×C61—22no—
68C2×Q8×C61C2×D414yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D412yes—
71C22×Dic61—41no—
72C2×Dic122—22no—
73C2×Dic122—44no—
74C2×Dic122—41no—
75C4⋊C4×C61—28no—
76C4⋊C4×C61—28no—
77C4⋊C4×C61—28no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—22no—
81C4⋊C4×C61—24no—
82C4⋊C4×C61—24no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61C2×D414yes—
86C4⋊C4×C61C2×D414yes—
87C22⋊C4×C61—24no—
88C22⋊C4×C61—22no—
89C42×C61—28no—
90C42×C61—22no—
91C42×C61—24no—
92C42×C61C2×D412yes—
93C42×C61C2×D411yescharacteristic
94C42×C61C2411yescharacteristic
95C4⋊Dic61—24no—
96C4⋊Dic61—42no—
97#11758194—42no—
98C2×#92418—24no—
99C2×#92418—22no—
100C23×C61#12011yescharacteristic
101Q8×C61—28no—
102Q8×C61—44no—
103Q8×C61—44no—
104C2×C4×C61—24no—
105C2×C4×C61—22no—
106C2×C4×C61—24no—
107C2×C4×C61—24no—
108C2×C4×C61—24no—
109C2×C4×C61#12012yes—
110C2×C4×C61#12011yescharacteristic
111C2×C4×C61C22×D411yescharacteristic
112C2×C4×C61C22×D412yes—
113C2×Dic61—42no—
114C2×Dic61—82no—
115Dic122—44no—
116Dic122—82no—
117#92418—42no—
118#92418—44no—
119#18732—611nomaximal, Sylow
120C22×C61—22no—
121C22×C61—21no—
122C22×C61C2×#12011yescharacteristic
123C4×C61—22no—
124C4×C61—21no—
125C4×C61—44no—
126C4×C61—44no—
127C4×C61—44no—
128C4×C61#45412yes—
129C4×C61#57814yes—
130Dic61—82no—
131#3556D6111yescharacteristic
132#3230—612no—
133#3139—611no—
134#1636—618no—
135#1544—611no—
136#1485—612no—
137C2×C61—41no—
138C2×C61#301112yes—
139C2×C61C2×#45411yescharacteristic
140#564—22no—
141#557—22no—
142#557—22no—
143#557C2×D6114yes—
144#541—24no—
145#541—22no—
146#533C2×D6112yes—
147#533C2×D6111yescharacteristic
148C2×C4⋊Q8—611no—
149#500—1222no—
150#491—1222no—
151#487—614no—
152#487—1222no—
153#484—1224no—
154#477—618no—
155#431—612no—
156#419—1222no—
157C4×M16—611no—
158#332—614no—
159Q8⋊C8—618no—
160C61#1873211yescharacteristic, complemented, Sylow
161C22×Q8—1221no—
162C2×M16—1221no—
163C4⋊Q8—22no—
164C4⋊Q8—24no—
165C4⋊Q8—612no—
166C4⋊Q8—1222no—
167C4⋊Q8—1221no—
168C4⋊Q8C22⋊D6112yes—
169C4⋊Q8C22⋊D6112yes—
170C4⋊Q8C22×D6112yes—
171#126—28no—
172#125—22no—
173#125—28no—
174#125—28no—
175#125—24no—
176#125—28no—
177#125—22no—
178#125C22⋊D6112yes—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8C22⋊D6114yes—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8C22×D6114yes—
188#117—22no—
189#117C22⋊D6112yes—
190C2×C4⋊C4—1222no—
191C2×C42C22×D6111yescharacteristic
192#107—1228no—
193C4⋊C8—614no—
194#103—1228no—
195#103—1228no—
196C22⋊C8—1222no—
197#97—1221no—
198C4×C8—612no—
199C4×C8—1222no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—1222no—
203C2×Q8—2444no—
204C2×Q8—2441no—
205C2×Q8C2×C22⋊D6114yes—
206C22×C4—2441no—
207C22×C4C2×C22⋊D6112yes—
208C22×C4C2×C22⋊D6111yescharacteristic
209M16—2442no—
210C2×C8—1222no—
211C2×C8—1224no—
212C4⋊C4—28no—
213C4⋊C4—28no—
214C4⋊C4—28no—
215C4⋊C4—22no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—1224no—
223C4⋊C4—2442no—
224C4⋊C4C2×C22⋊D6114yes—
225C4⋊C4C2×C22⋊D6114yes—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C42—28no—
229C42—24no—
230C42—22no—
231C42C2×C22⋊D6112yes—
232C42C2×C22⋊D6111yescharacteristic
233C42C23×D6111yescharacteristic
234C23#1451974611yescharacteristic
235Q8—28no—
236Q8—44no—
237Q8—44no—
238Q8—4882no—
239Q8—2444no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—4882no—
246C2×C4—2442no—
247C2×C4#1451974612yes—
248C2×C4#1451974611yescharacteristic
249C2×C4C22×C22⋊D6111yescharacteristic
250C2×C4C22×C22⋊D6112yes—
251C8—2442no—
252C8—2444no—
253C22—22no—
254C22—21no—
255C22C2×#1451974611yescharacteristic
256C4—22no—
257C4—21no—
258C4—44no—
259C4—44no—
260C4—44no—
261C4—4882no—
262C4#1683993512yes—
263C4#1684057714yes—
264C2—41no—
265C2#2145511412yes—
266C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3556C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenter#3556D61
hypocenterC61#18732
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. #21444977C2semidirect
160. C61#18732semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214562012
#214449772
#214444842
#214442402
C61×#35562
#1873261