Finite Group Databasebeta

← #41013497

Subgroups — #41013497

Order 27008.

306 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 97 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#27213110C211yesmaximal, characteristic
2C4×C22⋊Dic211C211yesmaximal, characteristic
3#27212905C214yesmaximal, complemented
4C2×#117×C211C211yesmaximal, characteristic
5C2×#16809946C2212yes—
6C2×C22⋊Dic211—21no—
7C2×C4×Dic211C2212yes—
8C22⋊(C211⋊C8)—21no—
9C22⋊(C211⋊C8)—21no—
10C211⋊#95—21no—
11C23×C4×C211C2211yescharacteristic
12#117×C211—21no—
13#117×C211—21no—
14#117×C211C412yescomplemented
15#117×C211C2212yes—
16C2×C4⋊C4×C211C411yescharacteristic, complemented
17C2×C4⋊C4×C211C411yescharacteristic, complemented
18C2×C22⋊C4×C211—21no—
19C2×C42×C211—21no—
20C22×Dic211—22no—
21#16809946—24no—
22C2×C211⋊C8—22no—
23C22⋊Dic211—24no—
24C4×Dic211—24no—
25C24×C211D411yescharacteristic
26C22×C4×C211—24no—
27C22×C4×C211—21no—
28C22×C4×C211—21no—
29C22×C4×C211C2×C411yescharacteristic
30C22×C4×C211C2×C411yescharacteristic
31C22×C4×C211D411yescharacteristic
32C22×C4×C211D411yescharacteristic
33C22×C4×C211D411yescharacteristic
34C22×C4×C211C2311yescharacteristic
35C4⋊C4×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211C2×C412yes—
38C4⋊C4×C211C2×C412yes—
39C22⋊C4×C211—22no—
40C22⋊C4×C211—22no—
41C42×C211—22no—
42C42×C211—22no—
43C2×Dic211—24no—
44C2×Dic211—42no—
45C2×Dic211—42no—
46C211⋊C8—42no—
47C23×C211—22no—
48C23×C211—22no—
49C23×C211—41no—
50C23×C211C22⋊C411yescharacteristic
51C23×C211C22⋊C411yescharacteristic
52C23×C211C2×D411yescharacteristic
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—42no—
61C2×C4×C211—42no—
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C22×C411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211#4114yes—
67C22×C211—21no—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—44no—
75C22×C211#10412yes—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#12312yes—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—42no—
81C4×C211—42no—
82C4×C211#11712yes—
83C4×C211#12312yes—
84Dic211—44no—
85C2×C211—22no—
86C2×C211—22no—
87C2×C211—41no—
88C2×C211#38111yescharacteristic
89C2×C211C2×#10411yescharacteristic
90C2×C211#42111yescharacteristic
91C211#182611yescharacteristic, complemented, Sylow
92#1826—2111nomaximal, Sylow
93C2×#117D21111yescharacteristic
94#407—2111no—
95C4×C22⋊C4—2111no—
96#339—2114no—
97C23×C4C2×D21111yescharacteristic
98C2×M16—2112no—
99#117—21no—
100#117—21no—
101#117Dic21112yescomplemented
102#117C2×D21112yes—
103C2×C4⋊C4Dic21111yescharacteristic, complemented
104C2×C4⋊C4Dic21111yescharacteristic, complemented
105C2×C22⋊C4—21no—
106C2×C22⋊C4—4221no—
107C2×C42—21no—
108C2×C42—2112no—
109C22⋊C8—4221no—
110C22⋊C8—4221no—
111#95—4221no—
112C24C22⋊D21111yescharacteristic
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—4222no—
117C22×C4C2×Dic21111yescharacteristic
118C22×C4C2×Dic21111yescharacteristic
119C22×C4C22⋊D21111yescharacteristic
120C22×C4C22⋊D21111yescharacteristic
121C22×C4C22⋊D21111yescharacteristic
122C22×C4C22×D21111yescharacteristic
123M16—4224no—
124C2×C8—4222no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4C2×Dic21112yes—
128C4⋊C4C2×Dic21112yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—4224no—
132C42—22no—
133C42—22no—
134C42—4224no—
135C23—22no—
136C23—22no—
137C23—41no—
138C23C22⋊Dic21111yescharacteristic
139C23C22⋊Dic21111yescharacteristic
140C23C2×C22⋊D21111yescharacteristic
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—8442no—
151C2×C4—8442no—
152C2×C4—4224no—
153C2×C4C22⋊Dic21111yescharacteristic
154C2×C4C22⋊Dic21111yescharacteristic
155C2×C4C22×Dic21111yescharacteristic
156C2×C4C2×C22⋊D21111yescharacteristic
157C2×C4#1680996214yes—
158C8—8442no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—44no—
167C22#2026990412yes—
168C22C2×C22⋊Dic21111yescharacteristic
169C22#2026996012yes—
170C4—21no—
171C4—21no—
172C4—42no—
173C4—42no—
174C4—8444no—
175C4#2026993612yes—
176C4#2026996012yes—
177C2—22no—
178C2—22no—
179C2—41no—
180C2#2721300911yescharacteristic
181C2C2×#2026990411yescharacteristic
182C2#2721316311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#117×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC2×#117D211
hypocenterC211#1826
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #27212905C2semidirect
14. #117×C211C4semidirect
16. C2×C4⋊C4×C211C4semidirect
17. C2×C4⋊C4×C211C4semidirect
91. C211#1826semidirect
101. #117Dic211semidirect
103. C2×C4⋊C4Dic211semidirect
104. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131102
C4×C22⋊Dic2112
#272129052
C2×#117×C2112
#1826211