Finite Group Databasebeta

← #41014054

Subgroups — #41014054

Order 27008.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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#27213188C211yesmaximal, characteristic
2#27213156C211yesmaximal, characteristic
3C2×#20269902C211yesmaximal, characteristic
4C2×#20269895C211yesmaximal, characteristic
5#27213101C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic
7#341×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269936C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269911—21no—
13#20269909—21no—
14#20269902—21no—
15#20269902—21no—
16#20269902—21no—
17#20269902—21no—
18#20269895C414yescomplemented
19#20269887—21no—
20C8×Dic211—21no—
21C2×M16×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C22×C8×C211C2211yescharacteristic
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422C2×C412yes—
30C22×Dic211—21no—
31C22×Dic211—21no—
32#16809946—21no—
33#16809946C2×C412yes—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D411yescharacteristic
39C22⋊Dic211—21no—
40C4×Dic211—21no—
41C22×C4×C211C2311yescharacteristic
42M16×C211—21no—
43M16×C211—21no—
44M16×C211—21no—
45M16×C211C2×C412yes—
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C2×C8×C211D411yescharacteristic
51C2×C8×C211D411yescharacteristic
52Dic422—22no—
53Dic422—22no—
54Dic422—41no—
55Dic422—41no—
56C2×Dic211—22no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—81no—
63C211⋊C8—22no—
64C23×C211#4111yescharacteristic
65C2×C4×C211C22×C411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C8×C211—22no—
72C8×C211—21no—
73C8×C211—21no—
74C8×C211—41no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211#11711yescharacteristic
78C22×C211#12111yescharacteristic
79C22×C211#12611yescharacteristic
80C4×C211C4×D411yescharacteristic
81C4×C211C4×D411yescharacteristic
82C4×C211#12311yescharacteristic
83C4×C211#12411yescharacteristic
84Dic211—42no—
85Dic211—41no—
86Dic211—41no—
87Dic211—81no—
88C2×C211—21no—
89C2×C211—21no—
90C2×C211#38811yescharacteristic
91C2×C211#47111yescharacteristic
92C2×C211#47311yescharacteristic
93C211#193011yescharacteristic, complemented, Sylow
94#1930—2111nomaximal, Sylow
95#428—2111no—
96#419—2111no—
97C2×#103—2111no—
98C2×#101—2111no—
99#405—2111no—
100#343—2111no—
101#341D21111yescharacteristic
102C22×Q8—2111no—
103C2×M16—2111no—
104C2×M16C2×D21111yescharacteristic
105C2×M16C2×D21111yescharacteristic
106C22×C8C2×D21111yescharacteristic
107#117—2111no—
108C2×C4⋊C4—2111no—
109#107—4221no—
110#106—4221no—
111#103—4221no—
112#103—4221no—
113#103—4221no—
114#103—4221no—
115#101—2114no—
116#97—4221no—
117C4×C8—4221no—
118C2×Q8—4221no—
119C2×Q8—4221no—
120C2×Q8—4221no—
121C2×Q8—4221no—
122C2×Q8—4221no—
123C2×Q8—2112no—
124C22×C4—4221no—
125C22×C4—4221no—
126C22×C4C22×D21111yescharacteristic
127M16—21no—
128M16—21no—
129M16—21no—
130M16—4221no—
131M16—2112no—
132M16C4×D21112yes—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—4221no—
138C2×C8D84411yescharacteristic
139C2×C8D84411yescharacteristic
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4—2111no—
143C4⋊C4—2111no—
144C22⋊C4—4221no—
145C42—4221no—
146C23#1680996311yescharacteristic
147Q8—8441no—
148Q8—8441no—
149Q8—4222no—
150Q8—4222no—
151C2×C4—8441no—
152C2×C4—8441no—
153C2×C4—8441no—
154C2×C4—16881no—
155C2×C4—4222no—
156C2×C4—4221no—
157C2×C4—4221no—
158C2×C4C2×C4×D21111yescharacteristic
159C2×C4D4×D21111yescharacteristic
160C2×C4C2×D84411yescharacteristic
161C2×C4#1680996111yescharacteristic
162C2×C4#1680996111yescharacteristic
163C2×C4#1680996211yescharacteristic
164C8—22no—
165C8—21no—
166C8—21no—
167C8—41no—
168C8—4222no—
169C22—21no—
170C22—21no—
171C22#2026993511yescharacteristic
172C22#2026995211yescharacteristic
173C22#2026997211yescharacteristic
174C4—8441no—
175C4—8441no—
176C4—8442no—
177C4—16881no—
178C4(C22⋊D211)⋊C411yescharacteristic
179C4C4×D84411yescharacteristic
180C4#2026996111yescharacteristic
181C4#2026996311yescharacteristic
182C2—21no—
183C2—21no—
184C2#2721303811yescharacteristic
185C2#2721341011yescharacteristic
186C2#2721342211yescharacteristic
subgroupG/H
centerC22#20269952
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#341×C211C2
socleC22×C211#121
prime radicalC23×C211#41
cosocleC22×C4×C211C23
hypercenter#341D211
hypocenterC211#1930
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #341

Complements

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

NH ≅ G/Ntype
18. #20269895C4semidirect
93. C211#1930semidirect

Maximal subgroups

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

subgroupindex
#272131882
#272131562
C2×#202699022
C2×#202698952
#272131012
#272129142
#341×C2112
#1930211