Finite Group Databasebeta

← #29656315

Subgroups — #29656315

Order 15616.

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

normal subgroups only

This group has 918 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
1C2×#16840492C211yesmaximal, characteristic
2#21456199C211yesmaximal, characteristic
3#21455950C212yesmaximal
4#21445041C218yesmaximal, complemented
5#21444265C212yesmaximal
6C61×#1558C211yesmaximal, characteristic
7#16840492—24no—
8#16840492—22no—
9#16840492C2214yes—
10#16840319C2211yescharacteristic
11C2×#14519776—22no—
12C2×#14519775—22no—
13C2×C4×Dic122C2212yes—
14C2×#14519742—22no—
15#16840130—22no—
16#16840112C2214yes—
17#16840093—22no—
18#16840048C2214yes—
19#16839973C2214yes—
20#16839920—22no—
21#16839870—24no—
22#16839771—22no—
23C4×#11758194C2211yescharacteristic
24#16839492C2218yes—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#432C2212yes—
27C61×#333C2214yes—
28C22×Dic122—22no—
29C22×Dic122D412yes—
30C2×Dic244—24no—
31C2×#11758194—21no—
32#14519777C2312yes—
33#14519776—28no—
34#14519776—28no—
35#14519775—28no—
36#14519775—28no—
37#14519769—21no—
38#14519756—22no—
39C4×Dic122—22no—
40C4×Dic122—22no—
41C4×Dic122—22no—
42C4×Dic122D414yes—
43C4×Dic122C2314yes—
44#14519742—28no—
45#14519742—28no—
46#14519736—21no—
47C2×C4⋊Dic61D412yes—
48C2×#11758196—24no—
49C2×#11758196—22no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61—22no—
52#14519709—24no—
53#14519702—24no—
54#14519702—24no—
55#14519701—28no—
56#14519688—21no—
57C4×#92418C2312yes—
58C22×Q8×C61D412yes—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61D412yes—
61C4⋊Q8×C61C2312yes—
62C2×C4⋊C4×C61—22no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61C2314yes—
65C22⋊C8×C61—22no—
66#97×C61—22no—
67C2×Q8×C61—22no—
68C2×Q8×C61—44no—
69C2×Q8×C61C2×D414yes—
70C22×C4×C61—22no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D412yes—
73C22×Dic61—22no—
74C22×Dic61—22no—
75C2×Dic122—22no—
76C2×Dic122—28no—
77C2×Dic122—22no—
78C2×Dic122—28no—
79C2×Dic122—84no—
80C2×Dic122C2×D414yes—
81C2×C8×C61—24no—
82C4⋊C4×C61—24no—
83C4⋊C4×C61—24no—
84C42×C61C2×D411yescharacteristic
85C42×C61C2×D412yes—
86C42×C61C2411yescharacteristic
87C22⋊Dic61—41no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90C4⋊Dic61—22no—
91C4⋊Dic61C2×D414yes—
92#11758196—216no—
93#11758196—24no—
94#11758196—24no—
95#11758196—42no—
96C4×Dic61—28no—
97C4×Dic61—24no—
98C4×Dic61—24no—
99C4×Dic61—41no—
100#11758194—42no—
101C2×#92418—22no—
102Dic244—48no—
103C23×C61#12011yescharacteristic
104Q8×C61—44no—
105Q8×C61—82no—
106C2×C4×C61—22no—
107C2×C4×C61—24no—
108C2×C4×C61—24no—
109C2×C4×C61—44no—
110C2×C4×C61#12012yes—
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61C22×D411yescharacteristic
113C2×C4×C61C22×D412yes—
114C2×Dic61—28no—
115C2×Dic61—24no—
116C2×Dic61—24no—
117C2×Dic61—44no—
118C2×Dic61—42no—
119Dic122—28no—
120Dic122—24no—
121Dic122—44no—
122Dic122—88no—
123Dic122#13718yes—
124C8×C61—44no—
125#92418—42no—
126#18551—611nomaximal, Sylow
127C22×C61—22no—
128C22×C61#45512yes—
129C22×C61C2×#12011yescharacteristic
130C4×C61—22no—
131C4×C61—21no—
132C4×C61—21no—
133C4×C61—44no—
134C4×C61C2×#13714yes—
135Dic61—28no—
136Dic61—44no—
137Dic61—82no—
138C2×Q82—611no—
139#3139—611no—
140#3113—612no—
141#1645—618no—
142#1558D6111yescharacteristic
143#1491—612no—
144C2×C61—22no—
145C2×C61#300412yes—
146C2×C61C2×#45511yescharacteristic
147Q82—614no—
148Q82—1224no—
149Q82—1222no—
150#533—611no—
151C2×C4⋊Q8—1222no—
152C2×C4⋊Q8—1222no—
153C2×C4⋊Q8C2×D6111yescharacteristic
154C2×C4×Q8—612no—
155C2×C4×Q8—1222no—
156#491—1222no—
157#487—614no—
158#483—1222no—
159#475—614no—
160#462—614no—
161#451—1222no—
162#441—1224no—
163#432C2×D6112yes—
164#419—1222no—
165C4×M16—611no—
166#333C2×D6114yes—
167Q8⋊C8—618no—
168C61#1855111yescharacteristic, complemented, Sylow
169C22×Q8—612no—
170C22×Q8—1222no—
171C22×Q8D24412yes—
172C2×Q16—1224no—
173C2×M16—1221no—
174C4⋊Q8—22no—
175C4⋊Q8—612no—
176C4⋊Q8—1228no—
177C4⋊Q8—1228no—
178C4⋊Q8—1228no—
179C4⋊Q8—1228no—
180C4⋊Q8D24412yes—
181C4⋊Q8C22×D6112yes—
182#125—1221no—
183C22⋊Q8—1222no—
184C4×Q8—614no—
185C4×Q8—614no—
186C4×Q8—1228no—
187C4×Q8—1228no—
188C4×Q8—1222no—
189C4×Q8—1222no—
190C4×Q8—1222no—
191#117—1221no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—612no—
194C2×C4⋊C4—1224no—
195C2×C4⋊C4—1222no—
196C2×C42—1222no—
197C2×C42—1222no—
198C2×C42C22×D6111yescharacteristic
199#106—1224no—
200C4⋊C8C22×D6114yes—
201#103—1224no—
202#103—1224no—
203#103—1228no—
204C22⋊C8—22no—
205#97—22no—
206#97—1221no—
207C4×C8—612no—
208C2×Q8—22no—
209C2×Q8—44no—
210C2×Q8—614no—
211C2×Q8—1228no—
212C2×Q8—1228no—
213C2×Q8—1222no—
214C2×Q8—1222no—
215C2×Q8—4884no—
216C2×Q8C2×D24414yes—
217C22×C4—22no—
218C22×C4—1222no—
219C22×C4—1222no—
220C22×C4C2×D24411yescharacteristic
221C22×C4D4×D6112yes—
222Q16—2448no—
223M16—2442no—
224C2×C8—24no—
225C2×C8—1222no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—614no—
229C4⋊C4—12216no—
230C4⋊C4—1224no—
231C4⋊C4—1222no—
232C4⋊C4—1224no—
233C4⋊C4—1222no—
234C4⋊C4—1222no—
235C4⋊C4—2442no—
236C22⋊C4—2441no—
237C42—1228no—
238C42—1224no—
239C42—1224no—
240C42—2441no—
241C42C2×D24411yescharacteristic
242C42D4×D6112yes—
243C42C23×D6111yescharacteristic
244C23#1451974711yescharacteristic
245Q8—44no—
246Q8—82no—
247Q8—618no—
248Q8—1228no—
249Q8—1224no—
250Q8—4888no—
251Q8—2444no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—44no—
256C2×C4—1228no—
257C2×C4—1224no—
258C2×C4—1224no—
259C2×C4—2442no—
260C2×C4—2444no—
261C2×C4#1451974712yes—
262C2×C4#1451974711yescharacteristic
263C2×C4C2×D4×D6112yes—
264C2×C4C22×D24411yescharacteristic
265C8—44no—
266C8—2442no—
267C22—22no—
268C22#1683994212yes—
269C22C2×#1451974711yescharacteristic
270C4—21no—
271C4—21no—
272C4—22no—
273C4—44no—
274C4—1228no—
275C4—4882no—
276C4—2444no—
277C4#137×D6114yes—
278C2—22no—
279C2#2145505512yes—
280C2C2×#1683994211yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1558C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenter#1558D61
hypocenterC61#18551
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
4. #21445041C2semidirect
168. C61#18551semidirect

Maximal subgroups

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

subgroupindex
C2×#168404922
#214561992
#214559502
#214450412
#214442652
C61×#15582
#1855161