Finite Group Databasebeta

← #29837654

Subgroups — #29837654

Order 15616.

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

normal subgroups only

This group has 562 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×D488C211yesmaximal, characteristic, complemented
2#21448042C213yesmaximal, complemented
3#21447956C213yesmaximal, complemented
4#21447851C213yesmaximal, complemented
5Q8×#11758185C211yesmaximal, characteristic, complemented
6C4×#14519718C213yesmaximal, complemented
7C61×#3391C211yesmaximal, characteristic, complemented
8Q8×D244—21no—
9C2×#14519718C2213yes—
10#16840167C2213yes—
11#16840043—23no—
12C4×D488C2213yes—
13#16839587C2213yes—
14C4⋊#11758185C2213yes—
15#16839563C2216yes—
16#16839563C2213yes—
17#16839562C2211yescharacteristic
18C4×#11758185C2213yes—
19C61×#550—21no—
20C61×#493C2213yes—
21C61×#459—23no—
22C8×Q8×C61C2211yescharacteristic
23C4×D8×C61C2213yes—
24C2×Q8×D61—41no—
25C2×D488C2311yescharacteristic
26#14519777—23no—
27#14519757—43no—
28C4×D244—23no—
29#14519718—26no—
30C2×#11758185C2311yescharacteristic
31C2×#11758185C2313yes—
32#14519711—23no—
33#14519711C2313yes—
34#14519699—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—
52Q8×D61—84no—
53C2×D244—21no—
54C2×C4×D61—43no—
55C2×Dic122—43no—
56D8×C61—23no—
57D8×C61—23no—
58D8×C61—83no—
59C2×C8×C61C2×D413yes—
60C2×C8×C61C2411yescharacteristic
61C4⋊C4×C61—21no—
62C4⋊C4×C61D813yes—
63C22⋊C4×C61—43no—
64C42×C61C2×D413yes—
65#11758198—43no—
66C4⋊Dic61—23no—
67C4⋊Dic61—43no—
68D488—21no—
69D488#4112yescomplemented
70#11758185—23no—
71#11758185—21no—
72C23×C61—43no—
73C22×D61—41no—
74Q8×C61D1614yescomplemented
75D4×C61—43no—
76D4×C61—43no—
77D4×C61—86no—
78D4×C61—163no—
79C2×C4×C61—41no—
80C2×C4×C61—86no—
81C2×C4×C61C2×D813yes—
82C2×C4×C61C2×D813yes—
83C2×C4×C61C22×D411yescharacteristic
84C2×Dic61—43no—
85D244—22no—
86D244—41no—
87C4×D61—86no—
88Dic122—86no—
89C8×C61—23no—
90C8×C61C2×#4111yescharacteristic
91C8×C612−1+411yescharacteristic
92#33925—611nomaximal, Sylow
93C22×C61—83no—
94C22×C61—83no—
95C22×C61—83no—
96C22×C61C22×D811yescharacteristic
97C2×D61—42no—
98C2×D61—81no—
99C2×D61—81no—
100C4×C61—23no—
101C4×C61—81no—
102C4×C61C2×D1616yes—
103C4×C61#54811yescharacteristic
104C4×C61#57711yescharacteristic
105Dic61—83no—
106#3391D6111yescharacteristic, complemented
107Q8×D8—611no—
108#2240—613no—
109#2224—613no—
110#2206—613no—
111Q8×C16—611no—
112C4×D16—613no—
113C2×C61—163no—
114C2×C61#335811yescharacteristic
115C2×C61C22×D1611yescharacteristic
116C2×C61#341611yescharacteristic
117D61—82no—
118#550—21no—
119D4×Q8—1221no—
120C2×D16—613no—
121#500—613no—
122#493C2×D6113yes—
123#474—1223no—
124#459—23no—
125C8×Q8C2×D6111yescharacteristic
126C4×D8—613no—
127C4×D8C2×D6113yes—
128#366—613no—
129C4⋊C16—613no—
130#357—613no—
131#357—616no—
132#357—611no—
133C4×C16—613no—
134C61#3392511yescharacteristic, complemented, Sylow
135C2×#41—41no—
136C22×Q8—2441no—
137C2×D8—23no—
138C2×D8—611no—
139C2×D8C22×D6113yes—
140C4⋊Q8—1223no—
141#127—23no—
142C22⋊Q8—2443no—
143#121—43no—
144C4×Q8C22⋊D6111yescharacteristic
145C4×D4—23no—
146C4×D4—1223no—
147D16—1226no—
148C2×C16—611no—
149C2×C16—613no—
150#107—613no—
151#107—1223no—
152#107C22×D6111yescharacteristic
153C4⋊C8C22⋊D6113yes—
154#102—23no—
155#102—1223no—
156C4×C8C22×D6113yes—
157#41—84no—
158C2×Q8—4884no—
159C2×Q8—2443no—
160C2×Q8#1175819911yescharacteristic
161C2×D4—23no—
162C2×D4—43no—
163C2×D4—43no—
164C2×D4—1221no—
165C22×C4—43no—
166C22×C4—2443no—
167D8—23no—
168D8—23no—
169D8—83no—
170D8—612no—
171D8—1221no—
172C2×C8C2×C22⋊D6113yes—
173C2×C8C23×D6111yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—1223no—
176C4⋊C4—2443no—
177C4⋊C4#1175819913yes—
178C22⋊C4—43no—
179C22⋊C4—2443no—
180C42C2×C22⋊D6113yes—
181C16—1223no—
182C16—1221no—
183C23—43no—
184C23—2441no—
185Q8—4886no—
186Q8#1451971814yescomplemented
187D4—43no—
188D4—43no—
189D4—86no—
190D4—163no—
191D4—1222no—
192D4—2441no—
193C2×C4—41no—
194C2×C4—86no—
195C2×C4—4886no—
196C2×C4—2443no—
197C2×C4C2×#1175819913yes—
198C2×C4C2×#1175819913yes—
199C2×C4C22×C22⋊D6111yescharacteristic
200C8—23no—
201C8#41×D6111yescharacteristic
202C8#1451982411yescharacteristic
203C22—83no—
204C22—83no—
205C22—83no—
206C22—4881no—
207C22—4881no—
208C22—2442no—
209C22C22×#1175819911yescharacteristic
210C4—23no—
211C4—81no—
212C4—4883no—
213C4C2×#1451971816yes—
214C4#1684041911yescharacteristic
215C4#1684057311yescharacteristic
216C2—163no—
217C2—4882no—
218C2#2145845011yescharacteristic
219C2C22×#1451971811yescharacteristic
220C2#2145896611yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3391C2
socleC22×C61C22×D8
hypercenter#3391D61
hypocenterC61#33925
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×D488C2semidirect
2. #21448042C2semidirect
3. #21447956C2semidirect
4. #21447851C2semidirect
5. Q8×#11758185C2semidirect
6. C4×#14519718C2semidirect
7. C61×#3391C2semidirect
69. D488#41semidirect
74. Q8×C61D16semidirect
106. #3391D61semidirect
134. C61#33925semidirect
186. Q8#14519718semidirect

Maximal subgroups

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

subgroupindex
Q8×D4882
#214480422
#214479562
#214478512
Q8×#117581852
C4×#145197182
C61×#33912
#3392561