Finite Group Databasebeta

← #41014665

Subgroups — #41014665

Order 27008.

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

normal subgroups only

This group has 256 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
1C22×#16809952C211yesmaximal, characteristic
2#27213173C211yesmaximal, characteristic
3#27213154C211yesmaximal, characteristic
4C2×#20269896C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic
6#27212910C211yesmaximal, characteristic
7#419×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809952—22no—
10C2×#16809952—22no—
11C2×#16809952—22no—
12C2×#16809946C2211yescharacteristic
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15C4⋊(C211⋊C8)—22no—
16#20269900—22no—
17#20269896—22no—
18#20269895—22no—
19C22×Q8×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#103×C211—22no—
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422D412yes—
29C22×Dic211—21no—
30#16809952—28no—
31#16809946—21no—
32#16809946—21no—
33#16809946—21no—
34#16809946—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D412yes—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211D412yes—
46C22×C4×C211—21no—
47C22×C4×C211C2311yescharacteristic
48M16×C211—21no—
49M16×C211—21no—
50C2×C8×C211—21no—
51C4⋊C4×C211D412yes—
52C22⋊C4×C211—21no—
53C42×C211—21no—
54Dic422—22no—
55Dic422—22no—
56Dic422—41no—
57Dic422—41no—
58C2×Dic211—22no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C211⋊C8—22no—
62C211⋊C8—41no—
63C211⋊C8—41no—
64C23×C211#4111yescharacteristic
65Q8×C211—22no—
66Q8×C211—22no—
67Q8×C211—41no—
68Q8×C211—41no—
69C2×C4×C211—22no—
70C2×C4×C211—22no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C8×C211—41no—
80C22×C211—22no—
81C22×C211#12111yescharacteristic
82C22×C211#12311yescharacteristic
83C22×C211#12311yescharacteristic
84C4×C211—42no—
85C4×C211—42no—
86C4×C211#12011yescharacteristic
87C4×C211#12111yescharacteristic
88C4×C211#12111yescharacteristic
89C4×C211#12411yescharacteristic
90Dic211—42no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211#39411yescharacteristic
94C2×C211#47312yes—
95C211#204711yescharacteristic, complemented, Sylow
96#2047—2111nomaximal, Sylow
97C22×Q16—2111no—
98#424—2111no—
99#419—2111no—
100#419D21111yescharacteristic
101C2×#101—2111no—
102C2×#101—2111no—
103#341—2111no—
104C22×Q8—2111no—
105C22×Q8C2×D21111yescharacteristic
106C2×Q16—4222no—
107C2×Q16—4222no—
108C2×Q16—4222no—
109C2×M16—2111no—
110C2×M16—2111no—
111C2×M16C2×D21111yescharacteristic
112C22×C8—2111no—
113#117C2×D21111yescharacteristic
114C4⋊C8—4222no—
115#103—22no—
116#103—4222no—
117#101—4222no—
118#101—4222no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—4221no—
125C2×Q8—4221no—
126C2×Q8—4221no—
127C2×Q8—4221no—
128C2×Q8—4221no—
129C2×Q8—2112no—
130C2×Q8D84412yes—
131C22×C4—21no—
132C22×C4—4221no—
133C22×C4C22×D21111yescharacteristic
134Q16—4228no—
135M16—21no—
136M16—21no—
137M16—4221no—
138M16—4221no—
139M16—4221no—
140M16—4221no—
141C2×C8—21no—
142C2×C8—4221no—
143C2×C8—4221no—
144C2×C8—4221no—
145C2×C8—4221no—
146C2×C8—2112no—
147C4⋊C4C22⋊D21112yes—
148C22⋊C4—21no—
149C42—21no—
150C23#1680996111yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—41no—
154Q8—41no—
155Q8—8441no—
156Q8—8441no—
157Q8—4222no—
158Q8—4222no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—8441no—
164C2×C4—8441no—
165C2×C4—4222no—
166C2×C4D4×D21111yescharacteristic
167C2×C4D4×D21111yescharacteristic
168C2×C4C2×C22⋊D21111yescharacteristic
169C2×C4C2×D84411yescharacteristic
170C2×C4#1680996111yescharacteristic
171C2×C4#1680996211yescharacteristic
172C8—41no—
173C8—8441no—
174C8—8441no—
175C8—4222no—
176C22—22no—
177C22#2026994911yescharacteristic
178C22#2026995911yescharacteristic
179C22#2026996111yescharacteristic
180C4—42no—
181C4—42no—
182C4—8442no—
183C4#2026994711yescharacteristic
184C4#2026994811yescharacteristic
185C4#2026995111yescharacteristic
186C4#2026996311yescharacteristic
187C2—21no—
188C2—21no—
189C2#2721306211yescharacteristic
190C2#2721341912yes—
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#121
prime radicalC23×C211#41
hypercenter#419D211
hypocenterC211#2047
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
95. C211#2047semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
#272131732
#272131542
C2×#202698962
C2×#202698952
#272129102
#419×C2112
#2047211