Finite Group Databasebeta

← #41013477

Subgroups — #41013477

Order 27008.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C4⋊C4×D211C211yesmaximal, characteristic
2#27213109C212yesmaximal
3#27212915C211yesmaximal, characteristic, complemented
4#27212914C212yesmaximal, complemented
5#343×C211C211yesmaximal, characteristic, complemented
6C22×C4×D211C412yes—
7C22×C4×D211C2211yescharacteristic
8C2×#16809946C2212yes—
9C2×C4⋊Dic211C2211yescharacteristic
10C2×Dic211⋊C4C412yes—
11C4⋊C4×D211—22no—
12C4⋊C4×D211—22no—
13(C2×D211)⋊C8—22no—
14(C2×D211)⋊C8—22no—
15C2×M16×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C23×D211C2×C411yescharacteristic
18C22×Dic211C2×C411yescharacteristic
19C22×Dic211C2×C412yes—
20C2×C4×D211—22no—
21C2×C4×D211—22no—
22C2×C4×D211—21no—
23C2×C4×D211—22no—
24C2×C4×D211—21no—
25C2×C4×D211—44no—
26C2×C4×D211D412yes—
27C2×C4×D211D412yes—
28#16809946—22no—
29#16809946—22no—
30C2×C211⋊C8—22no—
31C4⋊Dic211—22no—
32Dic211⋊C4—24no—
33C22×C4×C211C2×C412yes—
34C22×C4×C211C2311yescharacteristic
35M16×C211—22no—
36M16×C211—22no—
37C2×C8×C211—22no—
38C4⋊C4×C211—22no—
39C22×D211—21no—
40C22×D211—21no—
41C22×D211—22no—
42C22×D211—22no—
43C22×D211C22⋊C411yescharacteristic
44C22×D211C22⋊C411yescharacteristic
45C2×Dic211—21no—
46C2×Dic211—21no—
47C2×Dic211—22no—
48C2×Dic211—42no—
49C2×Dic211C22⋊C412yes—
50C4×D211—22no—
51C4×D211—22no—
52C4×D211—22no—
53C4×D211—22no—
54C4×D211—44no—
55C4×D211—44no—
56C211⋊C8—42no—
57C23×C211C22×C411yescharacteristic
58C2×C4×C211—22no—
59C2×C4×C211—42no—
60C2×C4×C211C2×D412yes—
61C2×C4×C211#4111yescharacteristic
62C2×C4×C211#4112yes—
63C2×C4×C211#4111yescharacteristic
64C8×C211—42no—
65C22×C211—21no—
66C22×C211—21no—
67C22×C211C2×C22⋊C411yescharacteristic
68C22×C211#11711yescharacteristic
69C22×C211#11711yescharacteristic
70C2×D211—24no—
71C2×D211—24no—
72C2×D211—21no—
73C2×D211—21no—
74C2×D211—21no—
75C2×D211—21no—
76C2×D211#10012yes—
77C2×D211#10112yes—
78C4×C211—42no—
79C4×C211#12312yes—
80C4×C211#12312yes—
81Dic211—22no—
82Dic211—42no—
83C2×C211—21no—
84C2×C211—21no—
85C2×C211#38111yescharacteristic
86C2×C211C2×#10011yescharacteristic
87C2×C211C2×#10111yescharacteristic
88D211—22no—
89D211—22no—
90C211#182111yescharacteristic, complemented, Sylow
91#1821—2111nomaximal, Sylow
92C22×C4⋊C4—2111no—
93#407—2112no—
94#343—2112no—
95#343—2111no—
96#343D21111yescharacteristic, complemented
97C23×C4—2112no—
98C23×C4—2111no—
99C2×M16—2112no—
100C2×M16C2×D21112yes—
101C2×C4⋊C4—4222no—
102C2×C4⋊C4—4222no—
103C2×C4⋊C4—2112no—
104C2×C4⋊C4—2111no—
105C2×C4⋊C4C2×D21111yescharacteristic
106C22⋊C8—4222no—
107C22⋊C8—4222no—
108C24—2111no—
109C22×C4—8444no—
110C22×C4—4222no—
111C22×C4—4222no—
112C22×C4—4221no—
113C22×C4—4222no—
114C22×C4—4221no—
115C22×C4—2112no—
116C22×C4—2111no—
117C22×C4—2112no—
118C22×C4—2112no—
119C22×C4C4×D21112yes—
120C22×C4C22×D21111yescharacteristic
121M16—22no—
122M16—22no—
123M16—4222no—
124M16—4222no—
125C2×C8—22no—
126C2×C8—4222no—
127C4⋊C4—22no—
128C4⋊C4—4224no—
129C4⋊C4—4222no—
130C23—4222no—
131C23—4222no—
132C23—4221no—
133C23—4221no—
134C23—2111no—
135C23—2111no—
136C23C2×C4×D21111yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—8444no—
140C2×C4—8444no—
141C2×C4—8442no—
142C2×C4—4222no—
143C2×C4—4221no—
144C2×C4—4221no—
145C2×C4—4222no—
146C2×C4—4222no—
147C2×C4—4222no—
148C2×C4—4222no—
149C2×C4—2112no—
150C2×C4D4×D21112yes—
151C2×C4#1680996111yescharacteristic
152C2×C4#1680996212yes—
153C2×C4#1680996311yescharacteristic
154C8—42no—
155C8—8442no—
156C22—21no—
157C22—21no—
158C22—4224no—
159C22—4224no—
160C22—4221no—
161C22—4221no—
162C22—4221no—
163C22—4221no—
164C22—2112no—
165C22—2112no—
166C22C22⋊C4×D21111yescharacteristic
167C22#2026993411yescharacteristic
168C22#2026993511yescharacteristic
169C4—42no—
170C4—8442no—
171C4—4222no—
172C4#2026995912yes—
173C4#2026996212yes—
174C2—21no—
175C2—21no—
176C2—4222no—
177C2—4222no—
178C2#2721300811yescharacteristic
179C2#100×D21111yescharacteristic
180C2#101×D21111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×D211C2×C4
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1821
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #27212915C2semidirect
4. #27212914C2semidirect
5. #343×C211C2semidirect
90. C211#1821semidirect
96. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2112
#272131092
#272129152
#272129142
#343×C2112
#1821211