Finite Group Databasebeta

← #49120336

Subgroups — #49120336

Order 36224.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#31827175C211yesmaximal, characteristic, complemented
2#31826838C211yesmaximal, characteristic, complemented
3#31826665C211yesmaximal, characteristic, complemented
4#31826661C211yesmaximal, characteristic
5#31826213C211yesmaximal, characteristic, complemented
6#31826194C211yesmaximal, characteristic, complemented
7Q8⋊C8×C283C211yesmaximal, characteristic, complemented
8C2×#17985381—22no—
9C2×#17985379—21no—
10C2×D2264—21no—
11#22583610C2211yescharacteristic
12#22583607C2211yescharacteristic
13#22583602—21no—
14#22583588—22no—
15#22583586—21no—
16C4×D1132—21no—
17C4×D1132C2211yescharacteristic
18C4⋊C4×D283—21no—
19C4⋊(C283⋊C8)C2211yescharacteristic
20#22583532—21no—
21#22583530—21no—
22C4×Q8×C283C2211yescharacteristic
23C4⋊C8×C283C2211yescharacteristic
24C4×C8×C283C2211yescharacteristic
25C2×Dic566D412yes—
26C2×D1132—21no—
27C2×D1132—21no—
28C2×D1132—21no—
29C2×D1132D411yescharacteristic
30C2×C4×D283—22no—
31C2×C4×D283—21no—
32#17985381—44no—
33#17985379—42no—
34D2264—42no—
35C2×C283⋊C8—21no—
36C4⋊Dic283—21no—
37C4⋊Dic283—21no—
38C4⋊Dic283D411yescharacteristic
39Dic283⋊C4—22no—
40Dic283⋊C4—22no—
41(C2×D283)⋊C4—22no—
42(C2×D283)⋊C4—21no—
43C2×Q8×C283D411yescharacteristic
44C2×C8×C283—21no—
45C2×C8×C283—21no—
46C4⋊C4×C283—21no—
47C4⋊C4×C283D411yescharacteristic
48C42×C283—21no—
49C42×C283C2311yescharacteristic
50C22×D283—21no—
51C22×D283—41no—
52Dic566—42no—
53D1132—21no—
54D1132—42no—
55D1132—41no—
56D1132—41no—
57D1132—41no—
58D1132—41no—
59D1132SD1612yescomplemented
60C2×Dic283—21no—
61C2×Dic283—22no—
62C4×D283—41no—
63C4×D283—41no—
64C4×D283—82no—
65C283⋊C8—41no—
66Q8×C283—21no—
67Q8×C283—21no—
68C2×C4×C283—21no—
69C2×C4×C283—21no—
70C2×C4×C283C2×D411yescharacteristic
71C2×C4×C283C2×D411yescharacteristic
72C2×C4×C283C2×D411yescharacteristic
73C8×C283—22no—
74C8×C283—41no—
75C22×C283#12011yescharacteristic
76C2×D283—22no—
77C2×D283—41no—
78C2×D283—41no—
79C2×D283—81no—
80C2×D283—81no—
81C2×D283—81no—
82C4×C283—21no—
83C4×C283—41no—
84C4×C283—41no—
85C4×C283C2×SD1611yescharacteristic
86C4×C283#13511yescharacteristic
87C4×C283AGL(1,ℤ/8)11yescharacteristic
88C4×C283#13711yescharacteristic
89Dic283—41no—
90Dic283—42no—
91C2×C283#44811yescharacteristic
92C2×C283#44911yescharacteristic
93C2×C283#45311yescharacteristic
94D283—42no—
95D283—161no—
96C283#162211yescharacteristic, complemented, Sylow
97#1622—2831nomaximal, Sylow
98#554—2831no—
99#492—2831no—
100#460—2831no—
101#459—2831no—
102#333—2831no—
103Q8⋊C8D28311yescharacteristic, complemented
104D4⋊C8—2831no—
105C2×SD16—5662no—
106C2×SD16—5661no—
107C2×D8—5661no—
108C4⋊Q8—2831no—
109#127—2831no—
110#125—5661no—
111C22⋊Q8—5662no—
112C22⋊Q8—5661no—
113C4×Q8C2×D28311yescharacteristic
114C4×D4—2831no—
115C4×D4—5661no—
116C2×C4⋊C4—5661no—
117C4⋊C8—2831no—
118C4⋊C8C2×D28311yescharacteristic
119#102—5661no—
120#102—5661no—
121C4×C8C2×D28311yescharacteristic
122C2×Q8—2832no—
123C2×Q8D113211yescharacteristic
124C2×D4—2831no—
125C2×D4—5661no—
126C2×D4—5661no—
127C2×D4—5661no—
128C22×C4—5662no—
129C22×C4—5661no—
130SD16—11324no—
131SD16—11322no—
132D8—11322no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—5661no—
136C4⋊C4—21no—
137C4⋊C4—2831no—
138C4⋊C4—5662no—
139C4⋊C4—5662no—
140C4⋊C4—5661no—
141C4⋊C4—5661no—
142C4⋊C4D113211yescharacteristic
143C22⋊C4—5662no—
144C22⋊C4—5661no—
145C42—21no—
146C42C22×D28311yescharacteristic
147C23—5661no—
148C23—11321no—
149Q8—21no—
150Q8—21no—
151Q8—11322no—
152D4—2832no—
153D4—5661no—
154D4—11322no—
155D4—11321no—
156D4—11321no—
157D4—11321no—
158D4—11321no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—5661no—
162C2×C4—5662no—
163C2×C4—11321no—
164C2×C4—11321no—
165C2×C4—22642no—
166C2×C4D4×D28311yescharacteristic
167C2×C4D4×D28311yescharacteristic
168C2×C4C2×D113211yescharacteristic
169C8—22no—
170C8—41no—
171C8—11321no—
172C22—5662no—
173C22—11321no—
174C22—11321no—
175C22—22641no—
176C22—22641no—
177C22—22641no—
178C22#2258358011yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—11321no—
183C4—11322no—
184C4SD16×D28311yescharacteristic
185C4#2258363211yescharacteristic
186C4#2258363811yescharacteristic
187C4#2258364211yescharacteristic
188C2—11322no—
189C2—45281no—
190C2#3182660511yescharacteristic
191C2#3182661211yescharacteristic
192C2#3182663411yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
FittingQ8⋊C8×C283C2
socleC22×C283#120
prime radical#31826661C2
hypercenterQ8⋊C8D283
hypocenterC283#1622
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #31827175C2semidirect
2. #31826838C2semidirect
3. #31826665C2semidirect
5. #31826213C2semidirect
6. #31826194C2semidirect
7. Q8⋊C8×C283C2semidirect
59. D1132SD16semidirect
96. C283#1622semidirect
103. Q8⋊C8D283semidirect

Maximal subgroups

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

subgroupindex
#318271752
#318268382
#318266652
#318266612
#318262132
#318261942
Q8⋊C8×C2832
#1622283