Finite Group Databasebeta

← #41013711

Subgroups — #41013711

Order 27008.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20269936C211yesmaximal, characteristic
2C2×#20269901C212yesmaximal, complemented
3C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
4#27212908C212yesmaximal, complemented
5C2×C22⋊Q8×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7#20269936—24no—
8C2×C4⋊Dic211C2212yes—
9C2×C22⋊Dic211—21no—
10C2×C4×Dic211—21no—
11#20269901—24no—
12C22⋊(C211⋊C8)—22no—
13C22×Q8×C211C2211yescharacteristic
14C23×C4×C211C2211yescharacteristic
15C22⋊Q8×C211—22no—
16C22⋊Q8×C211C414yescomplemented
17C2×C4⋊C4×C211—21no—
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C22⋊C4×C211—21no—
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C2×C211⋊C8—22no—
24C4⋊Dic211—22no—
25C4⋊Dic211D414yes—
26C22⋊Dic211—24no—
27C4×Dic211—24no—
28C24×C211D411yescharacteristic
29C2×Q8×C211—21no—
30C2×Q8×C211—22no—
31C2×Q8×C211—22no—
32C2×Q8×C211C2×C412yes—
33C22×C4×C211—22no—
34C22×C4×C211—22no—
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211C2×C412yes—
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D412yes—
40C22×C4×C211C2311yescharacteristic
41C4⋊C4×C211—24no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211C2×C412yes—
44C22⋊C4×C211—24no—
45C2×Dic211—24no—
46C2×Dic211—42no—
47C2×Dic211—42no—
48C211⋊C8—42no—
49C23×C211—21no—
50C23×C211—21no—
51C23×C211—22no—
52C23×C211—22no—
53C23×C211C22⋊C412yes—
54C23×C211C2×D411yescharacteristic
55Q8×C211—24no—
56Q8×C211—42no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211—24no—
60C2×C4×C211—24no—
61C2×C4×C211—22no—
62C2×C4×C211—22no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—42no—
66C2×C4×C211C22⋊C412yes—
67C2×C4×C211C22×C411yescharacteristic
68C2×C4×C211C2×D412yes—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211#4112yes—
71C22×C211—22no—
72C22×C211—22no—
73C22×C211—22no—
74C22×C211—42no—
75C22×C211—42no—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#12011yescharacteristic
78C22×C211#12311yescharacteristic
79C22×C211#13512yes—
80C22×C211#13712yes—
81C4×C211—22no—
82C4×C211—42no—
83C4×C211—42no—
84C4×C211C4×D412yes—
85C4×C211#12112yes—
86Dic211—44no—
87C2×C211—42no—
88C2×C211#38611yescharacteristic
89C2×C211#41611yescharacteristic
90C2×C211#41911yescharacteristic
91C2×C211#45212yes—
92C2×C211#48512yes—
93C211#187311yescharacteristic, complemented, Sylow
94#1873—2111nomaximal, Sylow
95C2×C22⋊Q8D21111yescharacteristic
96C2×#117—2111no—
97C2×#103—2112no—
98C2×C22⋊C8—2111no—
99#340—2112no—
100C22×Q8C2×D21111yescharacteristic
101C23×C4C2×D21111yescharacteristic
102C22×C8—2112no—
103C22⋊Q8—22no—
104C22⋊Q8Dic21114yescomplemented
105#117—4224no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—2112no—
108C2×C4⋊C4C2×D21111yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—4221no—
111C2×C42—4221no—
112#103—4224no—
113C22⋊C8—4222no—
114C24C22⋊D21111yescharacteristic
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8C2×Dic21112yes—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—4222no—
124C22×C4C2×Dic21112yes—
125C22×C4C22⋊D21111yescharacteristic
126C22×C4C22⋊D21112yes—
127C22×C4C22×D21111yescharacteristic
128C2×C8—4222no—
129C2×C8—4222no—
130C2×C8—4222no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—4222no—
134C4⋊C4—2114no—
135C4⋊C4C2×Dic21112yes—
136C22⋊C4—24no—
137C22⋊C4—4224no—
138C42—4224no—
139C23—21no—
140C23—21no—
141C23—22no—
142C23—22no—
143C23C22⋊Dic21112yes—
144C23C2×C22⋊D21111yescharacteristic
145Q8—24no—
146Q8—42no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—8442no—
157C2×C4—8442no—
158C2×C4—4224no—
159C2×C4C22⋊Dic21112yes—
160C2×C4C22×Dic21111yescharacteristic
161C2×C4D4×D21112yes—
162C2×C4C2×C22⋊D21111yescharacteristic
163C2×C4#1680996112yes—
164C8—8442no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22C2×C22⋊Dic21111yescharacteristic
171C22#2026994511yescharacteristic
172C22#2026995811yescharacteristic
173C22#2026999712yes—
174C22#2027000812yes—
175C4—22no—
176C4—42no—
177C4—42no—
178C4—8444no—
179C4D4×Dic21112yes—
180C4#2026994912yes—
181C2—42no—
182C2#2721302611yescharacteristic
183C2#2721314211yescharacteristic
184C2#2721315511yescharacteristic
185C2#2721330612yes—
186C2#2721348412yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×C22⋊Q8×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC2×C22⋊Q8D211
hypocenterC211#1873
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#20269901C2semidirect
4. #27212908C2semidirect
16. C22⋊Q8×C211C4semidirect
93. C211#1873semidirect
104. C22⋊Q8Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699362
C2×#202699012
C2×C22⋊(C211⋊C8)2
#272129082
C2×C22⋊Q8×C2112
#1873211