Finite Group Databasebeta

← #41013842

Subgroups — #41013842

Order 27008.

442 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C4×Dic211C211yesmaximal, characteristic, complemented
2C2×#20269904C212yesmaximal, complemented
3#27213110C211yesmaximal, characteristic, complemented
4#27212905C212yesmaximal, complemented
5#525×C211C211yesmaximal, characteristic
6C23×Dic211—21no—
7C2×#16809946C2212yes—
8C2×C4×Dic211—24no—
9C2×C4×Dic211—21no—
10C2×C4×Dic211C2212yes—
11#20269904—24no—
12C22⋊(C211⋊C8)—22no—
13C2×#41×C211C2211yescharacteristic
14C23×C4×C211C2211yescharacteristic
15#123×C211—21no—
16C22⋊Q8×C211C412yescomplemented
17#121×C211C412yescomplemented
18#120×C211—21no—
19C4×D4×C211—22no—
20#117×C211C2211yescharacteristic
21C22×Dic211—24no—
22C22×Dic211—28no—
23C22×Dic211—22no—
24#16809946—22no—
25#16809946—22no—
26C2×C211⋊C8—22no—
27C4×Dic211—24no—
28C4×Dic211—22no—
29C4×Dic211D414yescomplemented
30C24×C211D411yescharacteristic
31#41×C211—22no—
32#41×C211—22no—
33C2×Q8×C211C2×C411yescharacteristic, complemented
34C2×D4×C211—22no—
35C2×D4×C211—21no—
36C2×D4×C211C2×C411yescharacteristic, complemented
37C22×C4×C211—22no—
38C22×C4×C211—22no—
39C22×C4×C211—21no—
40C22×C4×C211C2×C412yes—
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D412yes—
43C22×C4×C211C2311yescharacteristic
44C4⋊C4×C211—22no—
45C4⋊C4×C211C2×C412yescomplemented
46C22⋊C4×C211—22no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211—22no—
49C42×C211—21no—
50C2×Dic211—216no—
51C2×Dic211—28no—
52C2×Dic211—24no—
53C211⋊C8—42no—
54C23×C211—24no—
55C23×C211—21no—
56C23×C211—21no—
57C23×C211—21no—
58C23×C211C22⋊C412yes—
59C23×C211C2×D411yescharacteristic
60Q8×C211—22no—
61D4×C211—22no—
62D4×C211—44no—
63D4×C211—41no—
64D4×C211—41no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—24no—
68C2×C4×C211—24no—
69C2×C4×C211—22no—
70C2×C4×C211—21no—
71C2×C4×C211—42no—
72C2×C4×C211C22⋊C412yes—
73C2×C4×C211C22×C411yescharacteristic
74C2×C4×C211C2×D412yes—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211#4112yes—
77C22×C211—24no—
78C22×C211—24no—
79C22×C211—22no—
80C22×C211—22no—
81C22×C211—22no—
82C22×C211—41no—
83C22×C211—42no—
84C22×C211#10414yescomplemented
85C22×C211C2×C22⋊C411yescharacteristic
86C22×C211#12011yescharacteristic
87C22×C211#12311yescharacteristic
88C4×C211—22no—
89C4×C211—41no—
90C4×C211—42no—
91C4×C211C4×D412yes—
92C4×C211#12112yes—
93Dic211—28no—
94C2×C211—21no—
95C2×C211—24no—
96C2×C211—21no—
97C2×C211—81no—
98C2×C211#38611yescharacteristic
99C2×C211C2×#10412yes—
100C211#189611yescharacteristic, complemented, Sylow
101#1896—2111nomaximal, Sylow
102#525D21111yescharacteristic
103C22×C42—2111no—
104C2×#104—2112no—
105#407—2111no—
106#339—2112no—
107C2×#41C2×D21111yescharacteristic
108C23×C4—4221no—
109C23×C4C2×D21111yescharacteristic
110C2×M16—2112no—
111#123—21no—
112C22⋊Q8Dic21112yescomplemented
113#121Dic21112yescomplemented
114#120—21no—
115C4×D4—22no—
116#117C2×D21111yescharacteristic
117C2×C42—4221no—
118C2×C42—4224no—
119C2×C42—2112no—
120#104—4224no—
121C22⋊C8—4222no—
122C24C22⋊D21111yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic21111yescharacteristic, complemented
126C2×D4—22no—
127C2×D4—21no—
128C2×D4C2×Dic21111yescharacteristic, complemented
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—4224no—
133C22×C4—4228no—
134C22×C4—4222no—
135C22×C4C2×Dic21112yes—
136C22×C4C22⋊D21111yescharacteristic
137C22×C4C22⋊D21112yes—
138C22×C4C22×D21111yescharacteristic
139M16—4222no—
140M16—4222no—
141C2×C8—4222no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic21112yescomplemented
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C42—21no—
148C42—4224no—
149C42—4222no—
150C42—2114no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23C22⋊Dic21112yes—
156C23C2×C22⋊D21111yescharacteristic
157Q8—22no—
158D4—22no—
159D4—44no—
160D4—41no—
161D4—41no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—42216no—
170C2×C4—4228no—
171C2×C4—4224no—
172C2×C4C22⋊Dic21112yes—
173C2×C4C22×Dic21111yescharacteristic
174C2×C4D4×D21112yes—
175C2×C4C2×C22⋊D21111yescharacteristic
176C2×C4#1680996112yes—
177C8—8442no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—41no—
185C22#2026990414yescomplemented
186C22C2×C22⋊Dic21111yescharacteristic
187C22#2026994511yescharacteristic
188C22#2026995811yescharacteristic
189C4—22no—
190C4—42no—
191C4—41no—
192C4—4228no—
193C4D4×Dic21112yes—
194C4#2026994912yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—81no—
199C2#2721302611yescharacteristic
200C2C2×#2026990412yes—
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#525×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenter#525D211
hypocenterC211#1896
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic211C2semidirect
2. C2×#20269904C2semidirect
3. #27213110C2semidirect
4. #27212905C2semidirect
16. C22⋊Q8×C211C4semidirect
17. #121×C211C4semidirect
29. C4×Dic211D4semidirect
33. C2×Q8×C211C2×C4semidirect
36. C2×D4×C211C2×C4semidirect
45. C4⋊C4×C211C2×C4semidirect
84. C22×C211#104semidirect
100. C211#1896semidirect
112. C22⋊Q8Dic211semidirect
113. #121Dic211semidirect
125. C2×Q8C2×Dic211semidirect
128. C2×D4C2×Dic211semidirect
143. C4⋊C4C2×Dic211semidirect
185. C22#20269904semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic2112
C2×#202699042
#272131102
#272129052
#525×C2112
#1896211