Finite Group Databasebeta

← #24906552

Subgroups — #24906552

Order 10624.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#19128049C211yesmaximal, characteristic, complemented
2#19127896C211yesmaximal, characteristic, complemented
3#19127896C211yesmaximal, characteristic, complemented
4#19127447C211yesmaximal, characteristic, complemented
5D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
7#547×C83C211yesmaximal, characteristic, complemented
8C2×#13050575—21no—
9C2×#13050575—21no—
10#14581789C2211yescharacteristic
11C4⋊(C83⋊C8)C2211yescharacteristic
12C4⋊(C83⋊C8)C2211yescharacteristic
13#14581712—21no—
14#14581712—21no—
15#14581712—22no—
16C4×C83⋊C8C2211yescharacteristic
17C2×#41×C83—21no—
18C4⋊Q8×C83C2211yescharacteristic
19#123×C83—22no—
20C22⋊Q8×C83—21no—
21#121×C83—21no—
22C4×D4×C83C2211yescharacteristic
23C4×D4×C83C2211yescharacteristic
24C2×C4⋊C4×C83—21no—
25C2×D332—21no—
26C2×D332—21no—
27C2×D332—21no—
28#13050575—42no—
29#13050575—42no—
30C2×C83⋊C8—21no—
31C2×C83⋊C8—21no—
32C2×C83⋊C8—21no—
33#41×C83—42no—
34#41×C83—42no—
35C2×Q8×C83—21no—
36C2×D4×C83—22no—
37C2×D4×C83D411yescharacteristic
38C2×D4×C83D411yescharacteristic
39C22×C4×C83—22no—
40C22×C4×C83—21no—
41C22×C4×C83—21no—
42C4⋊C4×C83—22no—
43C4⋊C4×C83—21no—
44C4⋊C4×C83D411yescharacteristic
45C4⋊C4×C83D411yescharacteristic
46C4⋊C4×C83D412yes—
47C22⋊C4×C83—22no—
48C22⋊C4×C83—21no—
49C22⋊C4×C83—21no—
50C42×C83C2311yescharacteristic
51C23×C83—21no—
52C23×C83—21no—
53C22×D83—41no—
54Q8×C83—41no—
55Q8×C83—41no—
56D4×C83—21no—
57D4×C83—21no—
58D4×C83—21no—
59D4×C83—44no—
60D4×C83D812yescomplemented
61C2×C4×C83—21no—
62C2×C4×C83—21no—
63C2×C4×C83—22no—
64C2×C4×C83—44no—
65C2×C4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72D332—42no—
73D332—41no—
74D332—41no—
75D332—41no—
76D332—41no—
77C83⋊C8—41no—
78C83⋊C8—41no—
79C83⋊C8—41no—
80C22×C83—22no—
81C22×C83—41no—
82C22×C83—41no—
83C22×C83—41no—
84C22×C83—41no—
85C22×C83—41no—
86C22×C83#12011yescharacteristic
87C2×D83—81no—
88C2×D83—81no—
89C2×D83—81no—
90C4×C83—21no—
91C4×C83—41no—
92C4×C83—42no—
93C4×C83—41no—
94C4×C83C2×D811yescharacteristic
95C4×C83#13511yescharacteristic
96C4×C83AGL(1,ℤ/8)11yescharacteristic
97C4×C83AGL(1,ℤ/8)11yescharacteristic
98C2×C83—42no—
99C2×C83—81no—
100C2×C83#44711yescharacteristic
101C2×C83#44911yescharacteristic
102C2×C83#45411yescharacteristic
103D83—161no—
104#1625—831nomaximal, Sylow
105C83#162511yescharacteristic, complemented, Sylow
106#547D8311yescharacteristic, complemented
107#486—831no—
108#459—831no—
109#459—831no—
110#332—831no—
111D4⋊C8—831no—
112D4⋊C8—831no—
113C2×#41—21no—
114C2×D8—1661no—
115C2×D8—1661no—
116C4⋊Q8C2×D8311yescharacteristic
117#127—831no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D8311yescharacteristic
122C4×D4C2×D8311yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—831no—
125C4⋊C8—831no—
126#102—1661no—
127#102—1661no—
128#102—1662no—
129C4×C8—831no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—1661no—
135C2×D4—1661no—
136C2×D4—1661no—
137C2×D4C22⋊D8311yescharacteristic
138C2×D4C22⋊D8311yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—3322no—
143D8—3322no—
144C2×C8—1661no—
145C2×C8—1661no—
146C2×C8—1661no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D8311yescharacteristic
150C4⋊C4C22⋊D8311yescharacteristic
151C4⋊C4C22⋊D8312yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D8311yescharacteristic
156C23—21no—
157C23—21no—
158C23—3321no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—3322no—
166D4—3321no—
167D4—3321no—
168D4—3321no—
169D4—3321no—
170D4#1305057512yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D8311yescharacteristic
180C2×C4C2×C22⋊D8311yescharacteristic
181C2×C4C2×C22⋊D8311yescharacteristic
182C8—3321no—
183C8—3321no—
184C8—3321no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—6641no—
192C22—6641no—
193C22—6641no—
194C22#1458176111yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×#1305057511yescharacteristic
200C4#1458181211yescharacteristic
201C4#1458181711yescharacteristic
202C4#1458181811yescharacteristic
203C2—42no—
204C2—81no—
205C2—13281no—
206C2#1912783511yescharacteristic
207C2#1912784611yescharacteristic
208C2#1912787411yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#547×C83C2
socleC22×C83#120
hypercenter#547D83
hypocenterC83#1625
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #19128049C2semidirect
2. #19127896C2semidirect
3. #19127896C2semidirect
4. #19127447C2semidirect
5. D4⋊(C83⋊C8)C2semidirect
6. D4⋊(C83⋊C8)C2semidirect
7. #547×C83C2semidirect
60. D4×C83D8semidirect
105. C83#1625semidirect
106. #547D83semidirect
170. D4#13050575semidirect

Maximal subgroups

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

subgroupindex
#191280492
#191278962
#191274472
D4⋊(C83⋊C8)2
#547×C832
#162583