Finite Group Databasebeta

← #41012698

Subgroups — #41012698

Order 27008.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#27213900C211yesmaximal, characteristic, complemented
2#27213535C211yesmaximal, characteristic
3#27213512C211yesmaximal, characteristic, complemented
4#27213511C211yesmaximal, characteristic, complemented
5#27212887C211yesmaximal, characteristic, complemented
6#27212885C211yesmaximal, characteristic, complemented
7#330×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—21no—
9C2×D1688—21no—
10#20269976—21no—
11#20269974C2211yescharacteristic
12#20269972—21no—
13#20269972—21no—
14#20269969C2211yescharacteristic
15#20269965C2211yescharacteristic
16#20269962—21no—
17#20269957—21no—
18#20269955—21no—
19#20269899—21no—
20#20269899—21no—
21#20269897—21no—
22#20269897—21no—
23#20269888C2211yescharacteristic
24#125×C211C2211yescharacteristic
25#97×C211C2211yescharacteristic
26#97×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422D411yescharacteristic
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844D411yescharacteristic
33C2×C4×D211—21no—
34#16809951—42no—
35D1688—42no—
36C2×C211⋊C8—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211D411yescharacteristic
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48C4×Dic211—21no—
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C4⋊C4×C211D411yescharacteristic
55C42×C211C2311yescharacteristic
56C22×D211—21no—
57C22×D211—41no—
58Dic422—41no—
59Dic422—81no—
60D844—41no—
61D844—41no—
62D844—41no—
63D844—42no—
64D844—42no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C4×D211—81no—
69C211⋊C8—41no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C8×C211—22no—
76C8×C211—41no—
77C22×C211#12011yescharacteristic
78C2×D211—41no—
79C2×D211—41no—
80C2×D211—41no—
81C2×D211—81no—
82C2×D211—81no—
83C2×D211—81no—
84C4×C211—21no—
85C4×C211—21no—
86C4×C211—21no—
87C4×C211—41no—
88C4×C211—41no—
89Dic211—41no—
90Dic211—41no—
91Dic211—41no—
92C2×C211#45311yescharacteristic
93C2×C211#45411yescharacteristic
94C2×C211#45411yescharacteristic
95D211—81no—
96D211—161no—
97C211#165711yescharacteristic, complemented, Sylow
98#1657—2111nomaximal, Sylow
99#563—2111no—
100#496—2111no—
101#490—2111no—
102#490—2111no—
103#330—2111no—
104#330D21111yescharacteristic, complemented
105#329—2111no—
106C2×SD16—4221no—
107C2×D8—4221no—
108C4⋊Q8—4221no—
109#127—2111no—
110#126—4221no—
111#126—4221no—
112#125—2111no—
113#125C2×D21111yescharacteristic
114#124—2111no—
115#123—4221no—
116C22⋊Q8—4221no—
117C22⋊Q8—4221no—
118#102—4221no—
119#102—4221no—
120#102—4221no—
121#102—4221no—
122#97—2111no—
123#97C2×D21111yescharacteristic
124#97C2×D21111yescharacteristic
125C2×Q8—4221no—
126C2×Q8—2111no—
127C2×D4—4221no—
128C2×D4—4221no—
129C2×D4—4221no—
130C2×D4—2111no—
131C22×C4—4221no—
132SD16—8442no—
133D8—8442no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—4221no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4—2111no—
144C4⋊C4—2111no—
145C4⋊C4D84411yescharacteristic
146C4⋊C4D84411yescharacteristic
147C22⋊C4—4221no—
148C22⋊C4—4221no—
149C22⋊C4—4221no—
150C22⋊C4—4221no—
151C22⋊C4—4221no—
152C42—4221no—
153C42C22×D21111yescharacteristic
154C23—8441no—
155C23—4221no—
156Q8—8441no—
157Q8—16881no—
158D4—8441no—
159D4—8441no—
160D4—8441no—
161D4—8442no—
162D4—8442no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—16881no—
166C2×C4—4221no—
167C2×C4—4221no—
168C2×C4—4221no—
169C2×C4D4×D21111yescharacteristic
170C2×C4D4×D21111yescharacteristic
171C2×C4C2×D84411yescharacteristic
172C8—22no—
173C8—41no—
174C8—8441no—
175C22—8441no—
176C22—8441no—
177C22—8441no—
178C22—16881no—
179C22—16881no—
180C22—16881no—
181C22#2026994711yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—8441no—
188C4—8441no—
189C4—8441no—
190C2—33761no—
191C2—16881no—
192C2#2721331511yescharacteristic
193C2#2721332011yescharacteristic
194C2#2721332111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#330×C211C2
socleC22×C211#120
prime radical#27213535C2
hypercenter#330D211
hypocenterC211#1657
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #27213900C2semidirect
3. #27213512C2semidirect
4. #27213511C2semidirect
5. #27212887C2semidirect
6. #27212885C2semidirect
7. #330×C211C2semidirect
97. C211#1657semidirect
104. #330D211semidirect

Maximal subgroups

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

subgroupindex
#272139002
#272135352
#272135122
#272135112
#272128872
#272128852
#330×C2112
#1657211