Finite Group Databasebeta

← #41013817

Subgroups — #41013817

Order 27008.

342 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 93 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#27213913C211yesmaximal, characteristic, complemented
2#27213130C211yesmaximal, characteristic, complemented
3#27213110C212yesmaximal, complemented
4#27212916C212yesmaximal, complemented
5#525×C211C211yesmaximal, characteristic
6#20269984—24no—
7C2×#16809946—22no—
8C2×#16809946C2212yes—
9C2×#16809946C2212yes—
10#20269896—21no—
11#20269894—21no—
12C22⋊(C211⋊C8)—24no—
13C2×#41×C211C2211yescharacteristic
14C23×C4×C211C2211yescharacteristic
15#123×C211C412yes—
16C22⋊Q8×C211—21no—
17#121×C211—21no—
18#120×C211C412yes—
19C4×D4×C211—22no—
20#117×C211C2211yescharacteristic
21#16809946—24no—
22#16809946—22no—
23#16809946—22no—
24#16809946—22no—
25#16809946—22no—
26C2×C211⋊C8—24no—
27C2×C211⋊C8—22no—
28C2×C211⋊C8D414yescomplemented
29C24×C211C2×C411yescharacteristic
30#41×C211—22no—
31#41×C211—22no—
32C2×Q8×C211D411yescharacteristic
33C2×D4×C211—22no—
34C2×D4×C211C2×C412yes—
35C2×D4×C211D411yescharacteristic
36C22×C4×C211—21no—
37C22×C4×C211—21no—
38C22×C4×C211—22no—
39C22×C4×C211—22no—
40C22×C4×C211C2×C411yescharacteristic
41C22×C4×C211D412yes—
42C22×C4×C211C2311yescharacteristic
43C4⋊C4×C211—22no—
44C4⋊C4×C211—21no—
45C22⋊C4×C211—22no—
46C22⋊C4×C211—22no—
47C22⋊C4×C211C2×C412yes—
48C42×C211—21no—
49C211⋊C8—24no—
50C211⋊C8—42no—
51C23×C211—21no—
52C23×C211—22no—
53C23×C211—42no—
54C23×C211C22⋊C412yes—
55C23×C211C22×C411yescharacteristic
56Q8×C211—22no—
57D4×C211—24no—
58D4×C211—22no—
59D4×C211—44no—
60C2×C4×C211—22no—
61C2×C4×C211—22no—
62C2×C4×C211—22no—
63C2×C4×C211—21no—
64C2×C4×C211—22no—
65C2×C4×C211—41no—
66C2×C4×C211—42no—
67C2×C4×C211—41no—
68C2×C4×C211C22⋊C412yes—
69C2×C4×C211C2×D412yes—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211#4112yes—
73C22×C211—22no—
74C22×C211—24no—
75C22×C211—22no—
76C22×C211—41no—
77C22×C211—42no—
78C22×C211—41no—
79C22×C211—42no—
80C22×C211—41no—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211C4×D412yes—
83C4×C211—22no—
84C4×C211—42no—
85C4×C211—41no—
86C4×C211#12011yescharacteristic
87C4×C211#12112yes—
88C4×C211#12311yescharacteristic
89C2×C211—21no—
90C2×C211—22no—
91C2×C211—42no—
92C2×C211—42no—
93C2×C211#38611yescharacteristic
94C211#189111yescharacteristic, complemented, Sylow
95#1891—2111nomaximal, Sylow
96#568—2111no—
97#525D21111yescharacteristic
98#413—2111no—
99#407—2112no—
100#344—2112no—
101C2×#41C2×D21111yescharacteristic
102C23×C4C2×D21111yescharacteristic
103#131—4224no—
104C2×M16—4222no—
105C2×M16—2112no—
106C2×M16—2112no—
107#123Dic21112yes—
108C22⋊Q8—21no—
109#121—21no—
110#120Dic21112yes—
111C4×D4—22no—
112#117C2×D21111yescharacteristic
113#101—4221no—
114#100—4221no—
115C22⋊C8—4224no—
116C24C2×Dic21111yescharacteristic
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D21111yescharacteristic
120C2×D4—22no—
121C2×D4C2×Dic21112yes—
122C2×D4C22⋊D21111yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4C2×Dic21111yescharacteristic
128C22×C4C22⋊D21112yes—
129C22×C4C22×D21111yescharacteristic
130M16—4222no—
131M16—4222no—
132M16—4224no—
133M16—4222no—
134M16—4222no—
135C2×C8—4222no—
136C2×C8—4224no—
137C2×C8—2114no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4C2×Dic21112yes—
143C42—21no—
144C23—21no—
145C23—22no—
146C23—42no—
147C23C22⋊Dic21112yes—
148C23C22×Dic21111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—44no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4C22⋊Dic21112yes—
162C2×C4D4×D21112yes—
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4C2×C22⋊D21111yescharacteristic
165C2×C4#1680996112yes—
166C8—8442no—
167C8—4224no—
168C22—22no—
169C22—24no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—41no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic21111yescharacteristic
177C22D4×Dic21112yes—
178C4—22no—
179C4—42no—
180C4—41no—
181C4#2026994511yescharacteristic
182C4#2026994912yes—
183C4#2026995811yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—42no—
188C2#2721302611yescharacteristic
subgroupG/H
centerC4#20269945
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#525×C211C2
socleC2×C211#386
cosocleC22×C4×C211C23
hypercenter#525D211
hypocenterC211#1891
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #27213913C2semidirect
2. #27213130C2semidirect
3. #27213110C2semidirect
4. #27212916C2semidirect
28. C2×C211⋊C8D4semidirect
94. C211#1891semidirect

Maximal subgroups

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

subgroupindex
#272139132
#272131302
#272131102
#272129162
#525×C2112
#1891211