Finite Group Databasebeta

← #41014742

Subgroups — #41014742

Order 27008.

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

normal subgroups only

This group has 246 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#27213774C211yesmaximal, characteristic
2#27213189C211yesmaximal, characteristic
3#27213188C211yesmaximal, characteristic
4#27213016C211yesmaximal, characteristic
5#27212914C211yesmaximal, characteristic
6#27212906C211yesmaximal, characteristic
7#339×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269977—21no—
10#20269976—21no—
11#20269969—21no—
12#20269968—21no—
13#20269954—22no—
14#20269936C2211yescharacteristic
15C2×C4⋊Dic211—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×C4⋊Dic211C2211yescharacteristic
18#20269911—21no—
19#20269910—21no—
20#20269909—21no—
21#20269908—21no—
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24C2×C42×C211C2211yescharacteristic
25C2×Dic422—22no—
26C22×Dic211—21no—
27C22×Dic211D412yes—
28#16809946—21no—
29#16809946Q812yescomplemented
30C2×C211⋊C8—21no—
31C4⋊Dic211—22no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—44no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D411yescharacteristic
39Dic211⋊C4—22no—
40Dic211⋊C4—22no—
41C22⋊Dic211—21no—
42C4×Dic211—21no—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45M16×C211—21no—
46M16×C211Q812yescomplemented
47C2×C8×C211—21no—
48C4⋊C4×C211D411yescharacteristic
49C4⋊C4×C211D411yescharacteristic
50C22⋊C4×C211—21no—
51C42×C211—21no—
52C42×C211—21no—
53C42×C211Q812yescomplemented
54Dic422—82no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—22no—
58C2×Dic211—42no—
59C2×Dic211—42no—
60C2×Dic211—41no—
61C211⋊C8—22no—
62C23×C211C2×D411yescharacteristic
63C2×C4×C211—22no—
64C2×C4×C211—22no—
65C2×C4×C211—22no—
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×Q811yescharacteristic
71C2×C4×C211C2×Q811yescharacteristic
72C2×C4×C211C2×Q811yescharacteristic
73C2×C4×C211#4111yescharacteristic
74C8×C211—22no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211#12011yescharacteristic
78C22×C211C22⋊Q811yescharacteristic
79C22×C211C4⋊Q811yescharacteristic
80C4×C211—24no—
81C4×C211—41no—
82C4×C211—41no—
83C4×C211C22⋊Q811yescharacteristic
84C4×C211C22⋊Q811yescharacteristic
85C4×C211C4⋊Q811yescharacteristic
86C4×C211C4⋊Q811yescharacteristic
87Dic211—42no—
88Dic211—41no—
89Dic211—41no—
90Dic211—81no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211#39511yescharacteristic
94C2×C211#45311yescharacteristic
95C2×C211#45611yescharacteristic
96C211#206011yescharacteristic, complemented, Sylow
97#2060—2111nomaximal, Sylow
98#544—2111no—
99#428—2111no—
100#428—2111no—
101#384—2111no—
102#343—2111no—
103#339—2111no—
104#339D21111yescharacteristic
105C2×M16—2111no—
106C2×M16C2×D21111yescharacteristic
107C4⋊Q8—4221no—
108C4⋊Q8—4221no—
109#125—4221no—
110#125—4221no—
111C22⋊Q8—4222no—
112#117—2111no—
113#117C2×D21111yescharacteristic
114C2×C4⋊C4—4222no—
115C2×C4⋊C4—2111no—
116C2×C4⋊C4—2111no—
117C2×C42C2×D21111yescharacteristic
118#107—4221no—
119#107—4221no—
120#106—4221no—
121#106—4221no—
122C2×Q8—4222no—
123C22×C4—21no—
124C22×C4—4221no—
125C22×C4—2112no—
126C22×C4C22×D21111yescharacteristic
127M16—21no—
128M16—4221no—
129M16—2112no—
130M16Dic42212yescomplemented
131C2×C8—21no—
132C2×C8—4221no—
133C4⋊C4—8444no—
134C4⋊C4—4222no—
135C4⋊C4—4222no—
136C4⋊C4—4222no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—2111no—
142C4⋊C4—2111no—
143C4⋊C4D84411yescharacteristic
144C4⋊C4D84411yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—4221no—
147C42—21no—
148C42—21no—
149C42—4221no—
150C42Dic42212yescomplemented
151C23D4×D21111yescharacteristic
152Q8—16882no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—8442no—
159C2×C4—8442no—
160C2×C4—8441no—
161C2×C4—4221no—
162C2×C4—4221no—
163C2×C4—4222no—
164C2×C4D4×D21111yescharacteristic
165C2×C4C2×D84411yescharacteristic
166C2×C4Q8×D21111yescharacteristic
167C2×C4C2×Dic42211yescharacteristic
168C2×C4C2×Dic42211yescharacteristic
169C2×C4#1680996111yescharacteristic
170C8—22no—
171C8—4222no—
172C22—21no—
173C22—21no—
174C22#2026994711yescharacteristic
175C22#2026995411yescharacteristic
176C22#2026997611yescharacteristic
177C4—24no—
178C4—41no—
179C4—41no—
180C4—8442no—
181C4—8441no—
182C4—8441no—
183C4—16881no—
184C4#2026995411yescharacteristic
185C4#2026995711yescharacteristic
186C4#2026997611yescharacteristic
187C4#2026997711yescharacteristic
188C2—21no—
189C2—21no—
190C2#2721306711yescharacteristic
191C2#2721331511yescharacteristic
192C2#2721333011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#339D211
hypocenterC211#2060
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
29. #16809946Q8semidirect
46. M16×C211Q8semidirect
53. C42×C211Q8semidirect
96. C211#2060semidirect
130. M16Dic422semidirect
150. C42Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137742
#272131892
#272131882
#272130162
#272129142
#272129062
#339×C2112
#2060211