Finite Group Databasebeta

← #41011970

Subgroups — #41011970

Order 27008.

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

normal subgroups only

This group has 288 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
1C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
2#27213006C211yesmaximal, characteristic
3#27212883C214yesmaximal, complemented
4C2×C22⋊Q8×C211C211yesmaximal, characteristic
5C22×C211⋊C8—21no—
6C2×C4⋊Dic211—21no—
7C2×C22⋊Dic211—21no—
8C22⋊(C211⋊C8)—21no—
9C22⋊(C211⋊C8)C2212yes—
10C211⋊#95C2212yes—
11C22×Q8×C211C411yescharacteristic, complemented
12C23×C4×C211C2211yescharacteristic
13C22⋊Q8×C211—21no—
14C22⋊Q8×C211—21no—
15C22⋊Q8×C211C412yescomplemented
16C22⋊Q8×C211C2212yes—
17C2×C4⋊C4×C211—21no—
18C2×C4⋊C4×C211C411yescharacteristic, complemented
19C2×C22⋊C4×C211—21no—
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—42no—
23C4⋊Dic211—42no—
24C22⋊Dic211—42no—
25C24×C211D411yescharacteristic
26C2×Q8×C211—21no—
27C2×Q8×C211—42no—
28C2×Q8×C211C2×C412yes—
29C22×C4×C211—21no—
30C22×C4×C211—21no—
31C22×C4×C211—42no—
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211C2×C411yescharacteristic
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211C2311yescharacteristic
38C4⋊C4×C211—22no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211—22no—
41C4⋊C4×C211C2×C412yes—
42C22⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C2×Dic211—42no—
45C2×Dic211—42no—
46C2×Dic211—42no—
47C211⋊C8—42no—
48C23×C211—41no—
49C23×C211—41no—
50C23×C211C22⋊C411yescharacteristic
51C23×C211C22⋊C411yescharacteristic
52C23×C211SD1612yes—
53C23×C211Q1612yes—
54C23×C211C2×D411yescharacteristic
55Q8×C211—42no—
56Q8×C211—42no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—22no—
61C2×C4×C211—22no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—42no—
66C2×C4×C211—42no—
67C2×C4×C211C22⋊C411yescharacteristic
68C2×C4×C211C22⋊C411yescharacteristic
69C2×C4×C211C22×C411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211—42no—
78C22×C211—42no—
79C22×C211#10312yes—
80C22×C211#10312yes—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211C2×SD1611yescharacteristic
83C22×C211C2×Q1611yescharacteristic
84C4×C211—21no—
85C4×C211—21no—
86C4×C211—21no—
87C4×C211—42no—
88C4×C211—42no—
89C4×C211#9912yes—
90Dic211—82no—
91C2×C211—21no—
92C2×C211—22no—
93C2×C211—22no—
94C2×C211—21no—
95C2×C211C2×#9911yescharacteristic
96C2×C211C2×#10311yescharacteristic
97C2×C211#42111yescharacteristic
98C211#152011yescharacteristic, complemented, Sylow
99#1520—2111nomaximal, Sylow
100C2×C22⋊Q8D21111yescharacteristic
101C2×C22⋊C8—2111no—
102#380—2111no—
103#328—2114no—
104C22×Q8Dic21111yescharacteristic, complemented
105C23×C4C2×D21111yescharacteristic
106C22×C8—4221no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8Dic21112yescomplemented
110C22⋊Q8C2×D21112yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—4221no—
113C2×C4⋊C4Dic21111yescharacteristic, complemented
114C2×C22⋊C4—21no—
115C2×C22⋊C4—4221no—
116C22⋊C8—4221no—
117C22⋊C8—2112no—
118#95—2112no—
119C24C22⋊D21111yescharacteristic
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×Dic21112yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—4222no—
127C22×C4C2×Dic21111yescharacteristic
128C22×C4C2×Dic21111yescharacteristic
129C22×C4C22⋊D21111yescharacteristic
130C22×C4C22⋊D21111yescharacteristic
131C22×C4C22⋊D21111yescharacteristic
132C22×C4C22×D21111yescharacteristic
133C2×C8—8442no—
134C2×C8—4222no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—8442no—
139C4⋊C4C2×Dic21112yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—8442no—
143C23—41no—
144C23—41no—
145C23C22⋊Dic21111yescharacteristic
146C23C22⋊Dic21111yescharacteristic
147C23#1680994912yes—
148C23#1680995212yes—
149C23C2×C22⋊D21111yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—8442no—
163C2×C4—8442no—
164C2×C4—8442no—
165C2×C4C22⋊Dic21111yescharacteristic
166C2×C4C22⋊Dic21111yescharacteristic
167C2×C4C22×Dic21111yescharacteristic
168C2×C4C2×C22⋊D21111yescharacteristic
169C8—8442no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#2026990112yes—
179C22#2026990112yes—
180C22C2×C22⋊Dic21111yescharacteristic
181C22C2×#1680994911yescharacteristic
182C22C2×#1680995211yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—16882no—
189C4#2026989212yes—
190C2—21no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2C2×#2026989211yescharacteristic
195C2C2×#2026990111yescharacteristic
196C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C22⋊Q8×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC2×C22⋊Q8D211
hypocenterC211#1520
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #27212883C2semidirect
11. C22×Q8×C211C4semidirect
15. C22⋊Q8×C211C4semidirect
18. C2×C4⋊C4×C211C4semidirect
98. C211#1520semidirect
104. C22×Q8Dic211semidirect
109. C22⋊Q8Dic211semidirect
113. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C211⋊C8)2
#272130062
#272128832
C2×C22⋊Q8×C2112
#1520211