Finite Group Databasebeta

← #41014746

Subgroups — #41014746

Order 27008.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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#27213186C211yesmaximal, characteristic
3#27213011C211yesmaximal, characteristic
4#27212914C211yesmaximal, characteristic
5#27212907C211yesmaximal, characteristic
6#27212905C211yesmaximal, characteristic
7#428×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269977—21no—
10#20269976—21no—
11#20269969—21no—
12#20269968—21no—
13#20269954—22no—
14#20269936C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16C2×C4×Dic211C2211yescharacteristic
17#20269910—21no—
18#20269908—21no—
19C211⋊#95—22no—
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#107×C211—21no—
24#106×C211—21no—
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⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—22no—
37Dic211⋊C4—22no—
38C22⋊Dic211—21no—
39C4×Dic211—21no—
40C4×Dic211—21no—
41C4×Dic211Q812yescomplemented
42C22×C4×C211—21no—
43C22×C4×C211C2311yescharacteristic
44M16×C211—21no—
45M16×C211Q812yescomplemented
46C2×C8×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211D411yescharacteristic
50C4⋊C4×C211D411yescharacteristic
51C22⋊C4×C211—21no—
52C42×C211—21no—
53Dic422—82no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C2×Dic211—22no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—42no—
61C211⋊C8—22no—
62C23×C211C2×D411yescharacteristic
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—42no—
66C2×C4×C211—41no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×Q811yescharacteristic
70C2×C4×C211C2×Q811yescharacteristic
71C2×C4×C211C2×Q811yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C8×C211—22no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211#12011yescharacteristic
77C22×C211C22⋊Q811yescharacteristic
78C22×C211C4⋊Q811yescharacteristic
79C4×C211—41no—
80C4×C211—41no—
81C4×C211—81no—
82C4×C211C22⋊Q811yescharacteristic
83C4×C211C22⋊Q811yescharacteristic
84C4×C211C4⋊Q811yescharacteristic
85C4×C211C4⋊Q811yescharacteristic
86Dic211—42no—
87Dic211—41no—
88Dic211—42no—
89Dic211—41no—
90C2×C211—21no—
91C2×C211—21no—
92C2×C211#39511yescharacteristic
93C2×C211#45411yescharacteristic
94C2×C211#45511yescharacteristic
95C211#206111yescharacteristic, complemented, Sylow
96#2061—2111nomaximal, Sylow
97#544—2111no—
98#428—2111no—
99#428D21111yescharacteristic
100#382—2111no—
101#343—2111no—
102#339—2111no—
103#339—2111no—
104C2×M16—2111no—
105C2×M16C2×D21111yescharacteristic
106C4⋊Q8—4221no—
107C4⋊Q8—4221no—
108#125—4221no—
109#125—4221no—
110C22⋊Q8—4222no—
111#117—2111no—
112#117C2×D21111yescharacteristic
113C2×C4⋊C4—2111no—
114C2×C4⋊C4C2×D21111yescharacteristic
115C2×C42—2111no—
116#107—21no—
117#107—4221no—
118#106—21no—
119#106—4221no—
120#95—4222no—
121C2×Q8—4222no—
122C22×C4—21no—
123C22×C4—4221no—
124C22×C4—2112no—
125C22×C4C22×D21111yescharacteristic
126M16—21no—
127M16—4221no—
128M16—2112no—
129M16Dic42212yescomplemented
130C2×C8—21no—
131C2×C8—4221no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—4222no—
135C4⋊C4—4222no—
136C4⋊C4—4222no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—2111no—
140C4⋊C4—2111no—
141C4⋊C4C22⋊D21111yescharacteristic
142C4⋊C4C22⋊D21111yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—4221no—
145C42—21no—
146C42—4221no—
147C42—4221no—
148C42—2112no—
149C23D4×D21111yescharacteristic
150Q8—16882no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—8441no—
156C2×C4—8441no—
157C2×C4—8442no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4—4222no—
161C2×C4—4222no—
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4Q8×D21111yescharacteristic
165C2×C4Q8×D21111yescharacteristic
166C2×C4C2×Dic42211yescharacteristic
167C2×C4#1680996211yescharacteristic
168C8—22no—
169C8—4222no—
170C22—21no—
171C22—21no—
172C22#2026994511yescharacteristic
173C22#2026995511yescharacteristic
174C22#2026997611yescharacteristic
175C4—41no—
176C4—41no—
177C4—81no—
178C4—8442no—
179C4—8441no—
180C4—8442no—
181C4—8441no—
182C4#2026995511yescharacteristic
183C4#2026995611yescharacteristic
184C4#2026997511yescharacteristic
185C4#2026997611yescharacteristic
186C2—21no—
187C2—21no—
188C2#2721306611yescharacteristic
189C2#2721331711yescharacteristic
190C2#2721332211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#428×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#428D211
hypocenterC211#2061
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
29. #16809946Q8semidirect
41. C4×Dic211Q8semidirect
45. M16×C211Q8semidirect
95. C211#2061semidirect
129. M16Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137742
#272131862
#272130112
#272129142
#272129072
#272129052
#428×C2112
#2061211