Finite Group Databasebeta

← #41012662

Subgroups — #41012662

Order 27008.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213811C211yesmaximal, characteristic
2#27213542C211yesmaximal, characteristic, complemented
3#27213432C211yesmaximal, characteristic, complemented
4#27213352C211yesmaximal, characteristic
5#27212876C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic
7#332×C211C211yesmaximal, characteristic, complemented
8C2×Q8×D211—21no—
9C2×#16809951—21no—
10#20269977C2211yescharacteristic
11#20269976—21no—
12#20269955—22no—
13#20269953—21no—
14C4×Dic422C2211yescharacteristic
15C4×D844C2211yescharacteristic
16#20269939—21no—
17#20269909—21no—
18#20269908—22no—
19#20269902—21no—
20#20269902—21no—
21C4×C211⋊C8C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25C2×Dic422—22no—
26C2×Dic422—21no—
27C2×Dic422D411yescharacteristic
28Q8×D211—42no—
29Q8×D211—42no—
30C2×D844D411yescharacteristic
31C2×C4×D211—22no—
32C2×C4×D211—21no—
33#16809951—42no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D411yescharacteristic
39Dic211⋊C4—22no—
40Dic211⋊C4—21no—
41(C2×D211)⋊C4—22no—
42(C2×D211)⋊C4—21no—
43C4×Dic211—21no—
44C2×Q8×C211—21no—
45C2×C8×C211—21no—
46C2×C8×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211D412yes—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—21no—
52Dic422—44no—
53Dic422—41no—
54Dic422—41no—
55Dic422SD1612yescomplemented
56D844—21no—
57D844SD1612yes—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C4×D211—44no—
63C4×D211—41no—
64C4×D211—41no—
65C211⋊C8—22no—
66Q8×C211—41no—
67Q8×C211—41no—
68C2×C4×C211—22no—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C8×C211—41no—
73C8×C211—41no—
74C22×C211#12011yescharacteristic
75C2×D211—22no—
76C2×D211—41no—
77C2×D211—41no—
78C4×C211—21no—
79C4×C211—42no—
80C4×C211C2×SD1611yescharacteristic
81C4×C211C2×SD1611yescharacteristic
82C4×C211AGL(1,ℤ/8)11yescharacteristic
83C4×C211#13711yescharacteristic
84Dic211—22no—
85Dic211—41no—
86Dic211—41no—
87Dic211—81no—
88C2×C211#44811yescharacteristic
89C2×C211#45011yescharacteristic
90C2×C211#45611yescharacteristic
91D211—42no—
92C211#165111yescharacteristic, complemented, Sylow
93#1651—2111nomaximal, Sylow
94D4×Q8—2111no—
95#498—2111no—
96#475—2111no—
97#461—2111no—
98#332D21111yescharacteristic, complemented
99Q8⋊C8—2111no—
100D4⋊C8—2111no—
101C22×Q8—4221no—
102C2×SD16—4221no—
103C4⋊Q8—4221no—
104C4⋊Q8—2111no—
105C4⋊Q8C2×D21111yescharacteristic
106C22⋊Q8—4221no—
107C22⋊Q8—4222no—
108C4×Q8—2111no—
109C4×D4—4221no—
110C4×D4—2111no—
111#106—4221no—
112#106—4222no—
113C4⋊C8C2×D21111yescharacteristic
114C4⋊C8C2×D21111yescharacteristic
115#103—4221no—
116#103—4221no—
117C4×C8—2111no—
118C2×Q8—21no—
119C2×Q8—8442no—
120C2×Q8—8442no—
121C2×Q8—4222no—
122C2×Q8—4221no—
123C2×Q8—2111no—
124C2×D4—2111no—
125C22×C4—4222no—
126C22×C4—4221no—
127SD16—8442no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—4221no—
131C4⋊C4—21no—
132C4⋊C4—4222no—
133C4⋊C4—4221no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C4⋊C4—2111no—
137C4⋊C4—2111no—
138C4⋊C4D84412yes—
139C22⋊C4—4222no—
140C22⋊C4—4221no—
141C42—4221no—
142C42C22×D21111yescharacteristic
143C23—4221no—
144Q8—41no—
145Q8—41no—
146Q8—8444no—
147Q8—8441no—
148Q8—8441no—
149Q8—4221no—
150Q8—2112no—
151D4—4221no—
152D4—2112no—
153C2×C4—22no—
154C2×C4—8444no—
155C2×C4—8441no—
156C2×C4—8441no—
157C2×C4—8441no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×D84411yescharacteristic
164C8—41no—
165C8—41no—
166C8—4222no—
167C22—8441no—
168C22—8441no—
169C22—4222no—
170C22#2026994711yescharacteristic
171C4—21no—
172C4—42no—
173C4—8441no—
174C4—8441no—
175C4—16881no—
176C4—4222no—
177C4SD16×D21111yescharacteristic
178C4SD16×D21111yescharacteristic
179C4#2027000111yescharacteristic
180C4#2027000911yescharacteristic
181C2—8442no—
182C2#2721328611yescharacteristic
183C2#2721329811yescharacteristic
184C2#2721333111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#332D211
hypocenterC211#1651
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27213542C2semidirect
3. #27213432C2semidirect
5. #27212876C2semidirect
7. #332×C211C2semidirect
55. Dic422SD16semidirect
92. C211#1651semidirect
98. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272138112
#272135422
#272134322
#272133522
#272128762
#272128732
#332×C2112
#1651211