Finite Group Databasebeta

← #29647760

Subgroups — #29647760

Order 15616.

1202 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 395 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#21454900C211yesmaximal, characteristic, complemented
2#21448740C211yesmaximal, characteristic, complemented
3#21444379C216yesmaximal, complemented
4#21444314C216yesmaximal, complemented
5C61×#3497C211yesmaximal, characteristic
6C2×Q8×Dic61—21no—
7Q8×#92418—24no—
8#16839815C2216yes—
9#16839661—23no—
10#16839628C2213yes—
11C4×C4⋊Dic61C2213yes—
12#16839496C22112yes—
13#16839496C2214yes—
14C2×2−1+4×C61—21no—
15C61×#550C414yescomplemented
16D4×Q8×C61C414yescomplemented
17C61×#548—24no—
18C61×#537—23no—
19C61×#532C2216yes—
20C2×C4×Q8×C61C2211yescharacteristic
21Q8×Dic61—44no—
22C2×C4⋊Dic61—23no—
23C2×C4⋊Dic61—23no—
24C2×C4×Dic61—23no—
25C4⋊#92418—26no—
26C22⋊#92418C2314yes—
27C4×#92418—26no—
28#14519684—23no—
29#14519684C2311yescharacteristic
30#14519684C2313yes—
312−1+4×C61—44no—
322−1+4×C61—44no—
33C2×#41×C61—24no—
34C2×#41×C61—26no—
35C22×Q8×C61—24no—
36C22×Q8×C61D411yescharacteristic
37C4⋊Q8×C61C2×C416yes—
38#127×C61C2×C416yes—
39#124×C61—26no—
40C22⋊Q8×C61—212no—
41#121×C61—26no—
42#121×C61—212no—
43#121×C61C2314yes—
44C4×Q8×C61C2×C414yes—
45C4×Q8×C61D414yes—
46C4×D4×C61—26no—
47C4×D4×C61C2×C4112yes—
48C2×C4⋊C4×C61D413yes—
49C2×C42×C61C2313yes—
50#41×C61—416no—
51#41×C61—412no—
52#41×C61—412no—
53C2×Q8×C61—26no—
54C2×Q8×C61—416no—
55C2×Q8×C61C22⋊C414yes—
56C2×Q8×C61D814yes—
57C2×Q8×C61SD1614yes—
58C2×D4×C61—24no—
59C2×D4×C61—26no—
60C2×D4×C61—26no—
61C2×D4×C61C22×C414yes—
62C22×C4×C61—212no—
63C22×C4×C61C2×D413yes—
64C22×C4×C61C2×D413yes—
65C22×C4×C61C2411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—23no—
68C4⋊C4×C61—26no—
69C4⋊C4×C61C22⋊C4112yes—
70C4⋊C4×C61C22×C414yes—
71C22⋊C4×C61—212no—
72C22⋊C4×C61—24no—
73C42×C61C22×C416yes—
74C42×C61C2×D416yes—
75C4⋊Dic61—46no—
76C4⋊Dic61—46no—
77C4×Dic61—46no—
78C2×#92418—24no—
79C23×C61—24no—
80C23×C61C22×D411yescharacteristic
81Q8×C61—412no—
82Q8×C61#102116yes—
83D4×C61—44no—
84D4×C61—44no—
85D4×C61—44no—
86D4×C61—412no—
87D4×C61—86no—
88C2×C4×C61—24no—
89C2×C4×C61—43no—
90C2×C4×C61—41no—
91C2×C4×C61—424no—
92C2×C4×C61C2×C22⋊C416yes—
93C2×C4×C61C2×C22⋊C416yes—
94C2×C4×C61C2×D816yes—
95C2×C4×C61C2×SD1616yes—
96C2×C4×C61C23×C411yescharacteristic
97C2×C4×C61C22×D411yescharacteristic
98C2×Dic61—22no—
99C2×Dic61—41no—
100C2×Dic61—41no—
101C2×Dic61—43no—
102C2×Dic61—43no—
103C2×Dic61—43no—
104#92418—44no—
105#17618—611nomaximal, Sylow
106C22×C61—21no—
107C22×C61—21no—
108C22×C61—44no—
109C22×C61—44no—
110C22×C61—44no—
111C22×C61C22×C22⋊C411yescharacteristic
112C22×C61C22×D811yescharacteristic
113C22×C61C22×SD1611yescharacteristic
114C4×C61—21no—
115C4×C61—21no—
116C4×C61—26no—
117C4×C61—44no—
118C4×C61C2×#102112yes—
119Dic61—42no—
120Dic61—83no—
121#3497D6111yescharacteristic
122#2982—611no—
123#2351—611no—
124#1517—616no—
125#1502—616no—
126C2×C61—21no—
127C2×C61—21no—
128C2×C61—84no—
129C2×C61#288411yescharacteristic
130C2×C61C22×#10211yescharacteristic
131C2×C61#290111yescharacteristic
132C2×2−1+4—21no—
133#550Dic6114yescomplemented
134D4×Q8Dic6114yescomplemented
135#548—24no—
136#537—23no—
137#532C2×D6116yes—
138C2×C4×Q8—1221no—
139C2×C4×Q8C2×D6111yescharacteristic
140C8×Q8—1224no—
141#431—616no—
142#389—1223no—
143#382—613no—
144C4×C4⋊C4—613no—
145#327—614no—
146#327—6112no—
147C61#1761811yescharacteristic, complemented, Sylow
1482−1+4—44no—
1492−1+4—44no—
150C2×#41—24no—
151C2×#41—26no—
152C22×Q8—24no—
153C22×Q8C22⋊D6111yescharacteristic
154C4⋊Q8C2×Dic6116yes—
155#127C2×Dic6116yes—
156#124—26no—
157C22⋊Q8—212no—
158#121—26no—
159#121—212no—
160#121C22×D6114yes—
161C4×Q8—2444no—
162C4×Q8C2×Dic6114yes—
163C4×Q8C22⋊D6114yes—
164C4×D4—26no—
165C4×D4C2×Dic61112yes—
166C2×C4⋊C4—1223no—
167C2×C4⋊C4—1223no—
168C2×C4⋊C4C22⋊D6113yes—
169C2×C42—1223no—
170C2×C42C22×D6113yes—
171C4⋊C8—1226no—
172C22⋊C8—614no—
173C4×C8—1226no—
174#95—611no—
175#95—613no—
176#95—1223no—
177#41—416no—
178#41—412no—
179#41—412no—
180C2×Q8—26no—
181C2×Q8—416no—
182C2×Q8#1175819914yes—
183C2×Q8#1175820014yes—
184C2×Q8C22⋊Dic6114yes—
185C2×D4—24no—
186C2×D4—26no—
187C2×D4—26no—
188C2×D4C22×Dic6114yes—
189C22×C4—212no—
190C22×C4—1221no—
191C22×C4—1223no—
192C22×C4C2×C22⋊D6113yes—
193C22×C4C2×C22⋊D6113yes—
194C22×C4C23×D6111yescharacteristic
195C2×C8—1224no—
196C4⋊C4—26no—
197C4⋊C4—2446no—
198C4⋊C4—2446no—
199C4⋊C4C22⋊Dic61112yes—
200C4⋊C4C22×Dic6114yes—
201C22⋊C4—24no—
202C22⋊C4—212no—
203C42—2446no—
204C42C22×Dic6116yes—
205C42C2×C22⋊D6116yes—
206C23—24no—
207C23C22×C22⋊D6111yescharacteristic
208Q8—412no—
209Q8#14519698116yes—
210D4—44no—
211D4—44no—
212D4—44no—
213D4—412no—
214D4—86no—
215C2×C4—24no—
216C2×C4—43no—
217C2×C4—41no—
218C2×C4—424no—
219C2×C4—1222no—
220C2×C4—2443no—
221C2×C4—2443no—
222C2×C4—2443no—
223C2×C4—2441no—
224C2×C4—2441no—
225C2×C4C2×C22⋊Dic6116yes—
226C2×C4C2×C22⋊Dic6116yes—
227C2×C4C2×#1175819916yes—
228C2×C4C2×#1175820016yes—
229C2×C4C23×Dic6111yescharacteristic
230C2×C4C22×C22⋊D6111yescharacteristic
231C8—2444no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22C22×C22⋊Dic6111yescharacteristic
238C22C22×#1175819911yescharacteristic
239C22C22×#1175820011yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—44no—
244C4—4883no—
245C4—2442no—
246C4C2×#14519698112yes—
247C2—21no—
248C2—21no—
249C2—84no—
250C2#2145428311yescharacteristic
251C2C22×#1451969811yescharacteristic
252C2#2145436611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3497C2
socleC22×C61C22×C22⋊C4
cosocleC22×C4×C61C24
hypercenter#3497D61
hypocenterC61#17618
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21454900C2semidirect
2. #21448740C2semidirect
3. #21444379C2semidirect
4. #21444314C2semidirect
15. C61×#550C4semidirect
16. D4×Q8×C61C4semidirect
133. #550Dic61semidirect
134. D4×Q8Dic61semidirect
147. C61#17618semidirect

Maximal subgroups

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

subgroupindex
#214549002
#214487402
#214443792
#214443142
C61×#34972
#1761861