Finite Group Databasebeta

← #29837690

Subgroups — #29837690

Order 15616.

528 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×Dic244C211yesmaximal, characteristic, complemented
2#21448042C213yesmaximal, complemented
3#21447966C213yesmaximal, complemented
4#21447856C213yesmaximal, complemented
5Q8×#11758185C211yesmaximal, characteristic, complemented
6C4×#14519721C213yesmaximal, complemented
7C61×#3391C211yesmaximal, characteristic
8Q8×Dic122—21no—
9C2×#14519721C2213yes—
10#16840167C2213yes—
11#16840058—23no—
12C4×Dic244C2213yes—
13#16839589C2213yes—
14C4⋊#11758185C2213yes—
15#16839565C2211yescharacteristic
16#16839563C2216yes—
17#16839563C2213yes—
18C4×#11758185C2213yes—
19C61×#550—21no—
20C61×#493C2213yes—
21C61×#459—23no—
22C8×Q8×C61C2211yescharacteristic
23C4×D8×C61C2213yes—
24C2×Dic244C2311yescharacteristic
25#14519777—23no—
26#14519776—43no—
27Q8×Dic61—41no—
28C4×Dic122—23no—
29#14519721—26no—
30C2×#11758185C2311yescharacteristic
31C2×#11758185C2313yes—
32#14519711—23no—
33#14519711C2313yes—
34#14519702—23no—
35C2×#41×C61—41no—
36C2×D8×C61—23no—
37C2×D8×C61C2313yes—
38#127×C61—23no—
39#121×C61—43no—
40C4×Q8×C61D411yescharacteristic
41C4×D4×C61—23no—
42#107×C61C2311yescharacteristic
43C4⋊C8×C61D413yes—
44#102×C61—23no—
45C4×C8×C61C2313yes—
46#41×C61—84no—
47C2×Q8×C61D811yescharacteristic
48C2×D4×C61—23no—
49C2×D4×C61—43no—
50C2×D4×C61—43no—
51C22×C4×C61—43no—
52C2×Dic122—21no—
53C2×Dic122—43no—
54D8×C61—23no—
55D8×C61—23no—
56D8×C61—83no—
57C2×C8×C61C2×D413yes—
58C2×C8×C61C2411yescharacteristic
59C4⋊C4×C61—21no—
60C4⋊C4×C61D813yes—
61C22⋊C4×C61—43no—
62C42×C61C2×D413yes—
63C4⋊Dic61—23no—
64C4⋊Dic61—43no—
65#11758196—43no—
66C4×Dic61—43no—
67Dic244—21no—
68Dic244#4112yescomplemented
69#11758185—23no—
70#11758185—21no—
71C23×C61—43no—
72Q8×C61SD3214yescomplemented
73D4×C61—43no—
74D4×C61—43no—
75D4×C61—86no—
76D4×C61—163no—
77C2×C4×C61—41no—
78C2×C4×C61—86no—
79C2×C4×C61C2×D813yes—
80C2×C4×C61C2×D813yes—
81C2×C4×C61C22×D411yescharacteristic
82C2×Dic61—41no—
83C2×Dic61—43no—
84Dic122—22no—
85Dic122—41no—
86Dic122—163no—
87C8×C61—23no—
88C8×C61C2×#4111yescharacteristic
89C8×C612−1+411yescharacteristic
90#33929—611nomaximal, Sylow
91C22×C61—83no—
92C22×C61—83no—
93C22×C61—83no—
94C22×C61C22×D811yescharacteristic
95C4×C61—23no—
96C4×C61—81no—
97C4×C61C2×SD3216yes—
98C4×C61#54811yescharacteristic
99C4×C61#57711yescharacteristic
100Dic61—42no—
101Dic61—83no—
102#3391D6111yescharacteristic
103Q8×Q16—611no—
104#2240—613no—
105#2226—613no—
106#2207—613no—
107Q8×C16—611no—
108C4×SD32—613no—
109C2×C61—163no—
110C2×C61#335811yescharacteristic
111C2×C61C22×SD3211yescharacteristic
112C2×C61#341511yescharacteristic
113Q82—1221no—
114#550—21no—
115C2×SD32—613no—
116#500—613no—
117#493C2×D6113yes—
118#477—1223no—
119#459—23no—
120C8×Q8C2×D6111yescharacteristic
121C4×Q16—613no—
122C4×D8C2×D6113yes—
123#367—613no—
124C4⋊C16—613no—
125#358—611no—
126#357—613no—
127#357—616no—
128C4×C16—613no—
129C61#3392911yescharacteristic, complemented, Sylow
130C2×#41—41no—
131C2×Q16—611no—
132C2×D8—23no—
133C2×D8C22×D6113yes—
134C4⋊Q8—1223no—
135C4⋊Q8—2443no—
136#127—23no—
137#121—43no—
138C4×Q8—1223no—
139C4×Q8—2441no—
140C4×Q8C22⋊D6111yescharacteristic
141C4×D4—23no—
142SD32—1226no—
143C2×C16—611no—
144C2×C16—613no—
145#107—613no—
146#107—1223no—
147#107C22×D6111yescharacteristic
148C4⋊C8C22⋊D6113yes—
149#103—1223no—
150#102—23no—
151C4×C8C22×D6113yes—
152#41—84no—
153C2×Q8—1221no—
154C2×Q8—2443no—
155C2×Q8#1175819911yescharacteristic
156C2×D4—23no—
157C2×D4—43no—
158C2×D4—43no—
159C22×C4—43no—
160Q16—612no—
161Q16—1221no—
162D8—23no—
163D8—23no—
164D8—83no—
165C2×C8C2×C22⋊D6113yes—
166C2×C8C23×D6111yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—1223no—
169C4⋊C4—2443no—
170C4⋊C4—2443no—
171C4⋊C4#1175819913yes—
172C22⋊C4—43no—
173C42—2443no—
174C42C2×C22⋊D6113yes—
175C16—1223no—
176C16—1221no—
177C23—43no—
178Q8—1222no—
179Q8—9763no—
180Q8—2441no—
181Q8#1451972114yescomplemented
182D4—43no—
183D4—43no—
184D4—86no—
185D4—163no—
186C2×C4—41no—
187C2×C4—86no—
188C2×C4—2443no—
189C2×C4—2441no—
190C2×C4C2×#1175819913yes—
191C2×C4C2×#1175819913yes—
192C2×C4C22×C22⋊D6111yescharacteristic
193C8—23no—
194C8#41×D6111yescharacteristic
195C8#1451982411yescharacteristic
196C22—83no—
197C22—83no—
198C22—83no—
199C22C22×#1175819911yescharacteristic
200C4—23no—
201C4—81no—
202C4—4883no—
203C4—2442no—
204C4C2×#1451972116yes—
205C4#1684041911yescharacteristic
206C4#1684057311yescharacteristic
207C2—163no—
208C2#2145845011yescharacteristic
209C2C22×#1451972111yescharacteristic
210C2#2145895811yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3391C2
socleC22×C61C22×D8
hypercenter#3391D61
hypocenterC61#33929
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. Q8×Dic244C2semidirect
2. #21448042C2semidirect
3. #21447966C2semidirect
4. #21447856C2semidirect
5. Q8×#11758185C2semidirect
6. C4×#14519721C2semidirect
68. Dic244#41semidirect
72. Q8×C61SD32semidirect
129. C61#33929semidirect
181. Q8#14519721semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2442
#214480422
#214479662
#214478562
Q8×#117581852
C4×#145197212
C61×#33912
#3392961