Finite Group Databasebeta

← #41014764

Subgroups — #41014764

Order 27008.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#20269913C211yesmaximal, characteristic
2#27213189C211yesmaximal, characteristic
3#27213154C211yesmaximal, characteristic
4C2×#20269902C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic
7#424×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269913—22no—
13#20269910—21no—
14#20269908—21no—
15#20269902—22no—
16#20269900—21no—
17#20269900—21no—
18#20269895—22no—
19C2×M16×C211C2211yescharacteristic
20C22×C8×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22C4⋊C8×C211—22no—
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422Q812yes—
29C22×Dic211—21no—
30C22×Dic211—21no—
31#16809946—21no—
32#16809946—21no—
33#16809946—21no—
34#16809946D412yes—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39C22×C4×C211C2311yescharacteristic
40M16×C211—21no—
41M16×C211—21no—
42C2×C8×C211—21no—
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211D412yes—
46C4⋊C4×C211D411yescharacteristic
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211—21no—
50Dic422—22no—
51Dic422—22no—
52Dic422—41no—
53Dic422—41no—
54C2×Dic211—22no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—81no—
59C211⋊C8—22no—
60C211⋊C8—41no—
61C23×C211#4111yescharacteristic
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×Q811yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C8×C211—22no—
71C8×C211—41no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211#12111yescharacteristic
75C22×C211C22⋊Q811yescharacteristic
76C22×C211#12611yescharacteristic
77C4×C211—41no—
78C4×C211—41no—
79C4×C211#12111yescharacteristic
80C4×C211#12111yescharacteristic
81C4×C211C22⋊Q811yescharacteristic
82C4×C211C22⋊Q811yescharacteristic
83Dic211—42no—
84Dic211—81no—
85C2×C211—21no—
86C2×C211—21no—
87C2×C211#39611yescharacteristic
88C2×C211#47111yescharacteristic
89C2×C211#47311yescharacteristic
90C211#206411yescharacteristic, complemented, Sylow
91#2064—2111nomaximal, Sylow
92C2×#108—2111no—
93#428—2111no—
94#424D21111yescharacteristic
95#419—2111no—
96C2×#103—2111no—
97C2×#101—2111no—
98#343—2111no—
99C22×Q8—2111no—
100C2×M16—2111no—
101C2×M16—2111no—
102C2×M16C2×D21111yescharacteristic
103C22×C8C2×D21111yescharacteristic
104#117C2×D21111yescharacteristic
105C2×C4⋊C4—2111no—
106#108—4222no—
107#107—4221no—
108#106—4221no—
109C4⋊C8—22no—
110#103—4222no—
111#103—4221no—
112#103—4221no—
113#101—4222no—
114C2×Q8—4221no—
115C2×Q8—4221no—
116C2×Q8—4221no—
117C2×Q8—4221no—
118C2×Q8—4221no—
119C2×Q8—2112no—
120C22×C4—4221no—
121C22×C4—4221no—
122C22×C4C22×D21111yescharacteristic
123M16—21no—
124M16—21no—
125M16—4221no—
126M16—4221no—
127M16—4221no—
128M16—2112no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—4221no—
133C2×C8—4221no—
134C2×C8C22⋊D21112yes—
135C4⋊C4—4221no—
136C4⋊C4—4221no—
137C4⋊C4D84411yescharacteristic
138C4⋊C4D84411yescharacteristic
139C22⋊C4—21no—
140C42—21no—
141C23#1680996211yescharacteristic
142Q8—8441no—
143Q8—8441no—
144Q8—4222no—
145Q8—4222no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—8441no—
149C2×C4—8441no—
150C2×C4—8441no—
151C2×C4—16881no—
152C2×C4—4222no—
153C2×C4D4×D21111yescharacteristic
154C2×C4C2×C22⋊D21111yescharacteristic
155C2×C4C2×D84411yescharacteristic
156C2×C4Q8×D21111yescharacteristic
157C2×C4#1680996111yescharacteristic
158C2×C4#1680996311yescharacteristic
159C8—22no—
160C8—41no—
161C8—8441no—
162C8—4222no—
163C22—21no—
164C22—21no—
165C22#2026995111yescharacteristic
166C22#2026995511yescharacteristic
167C22#2026997211yescharacteristic
168C4—41no—
169C4—41no—
170C4—8442no—
171C4—16881no—
172C4#2026994911yescharacteristic
173C4#2026995211yescharacteristic
174C4#2026995311yescharacteristic
175C4#2026995711yescharacteristic
176C2—21no—
177C2—21no—
178C2#2721306811yescharacteristic
179C2#2721340911yescharacteristic
180C2#2721342111yescharacteristic
subgroupG/H
centerC22#20269951
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#424×C211C2
socleC22×C211#121
prime radicalC23×C211#41
hypercenter#424D211
hypocenterC211#2064
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
90. C211#2064semidirect

Maximal subgroups

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

subgroupindex
C2×#202699132
#272131892
#272131542
C2×#202699022
C2×#202698952
#272129142
#424×C2112
#2064211