Finite Group Databasebeta

← #41014240

Subgroups — #41014240

Order 27008.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269975C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic
3C2×#20269896C211yesmaximal, characteristic
4C2×#20269895C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic
6M16×Dic211C211yesmaximal, characteristic
7C2×#101×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269975—21no—
12#20269975—21no—
13#20269975—22no—
14C2×Dic211⋊C4C414yes—
15C2×C4×Dic211C2211yescharacteristic
16Dic211⋊C8—22no—
17#20269896—22no—
18#20269895—22no—
19#20269895—22no—
20#20269887—21no—
21C8×Dic211—21no—
22C22×Q8×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25#101×C211—22no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C22×Dic211C2×C412yes—
33C22×Dic211C2×C412yes—
34#16809946—21no—
35#16809946—21no—
36#16809946—21no—
37#16809946D412yes—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40Dic211⋊C4—24no—
41Dic211⋊C4—24no—
42C4×Dic211D411yescharacteristic
43C4×Dic211D411yescharacteristic
44C4×Dic211D412yes—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C22×C4×C211C2×C412yes—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211—21no—
55M16×C211—21no—
56M16×C211D412yes—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59Dic422—41no—
60Dic422—41no—
61Dic422—41no—
62Dic422—41no—
63C2×Dic211—24no—
64C2×Dic211—22no—
65C2×Dic211—22no—
66C2×Dic211—22no—
67C2×Dic211C22⋊C412yes—
68C2×Dic211C22⋊C412yes—
69C211⋊C8—22no—
70C211⋊C8—41no—
71C23×C211C22×C411yescharacteristic
72Q8×C211—41no—
73Q8×C211—41no—
74Q8×C211—41no—
75Q8×C211—41no—
76C2×C4×C211—22no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C8×C211—41no—
87C22×C211—22no—
88C22×C211C2×C22⋊C411yescharacteristic
89C22×C211C4×D411yescharacteristic
90C22×C211C4×D411yescharacteristic
91C4×C211—42no—
92C4×C211#12111yescharacteristic
93C4×C211#12111yescharacteristic
94C4×C211#12411yescharacteristic
95C4×C211#12711yescharacteristic
96Dic211—22no—
97Dic211—42no—
98Dic211#10114yescomplemented
99C2×C211—21no—
100C2×C211—21no—
101C2×C211#39011yescharacteristic
102C2×C211C2×#10112yes—
103C211#196611yescharacteristic, complemented, Sylow
104#1966—2111nomaximal, Sylow
105C2×C4⋊Q8—2111no—
106#423—2111no—
107C2×#101—2111no—
108C2×#101—2111no—
109C2×#101—2111no—
110C2×#101D21111yescharacteristic
111C4×M16—2111no—
112C22×Q8—2111no—
113C22×Q8C2×D21111yescharacteristic
114C2×M16—2111no—
115C2×M16—2111no—
116C2×M16C2×D21111yescharacteristic
117C2×M16C2×D21111yescharacteristic
118C4⋊Q8—4221no—
119C4⋊Q8—4221no—
120C4⋊Q8—4222no—
121C2×C4⋊C4—2114no—
122C2×C42—2111no—
123C4⋊C8—4222no—
124#101—22no—
125#101—4222no—
126#101—4222no—
127#101—4222no—
128#97—4221no—
129C4×C8—4221no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—4221no—
140C2×Q8—4221no—
141C2×Q8—4221no—
142C22×C4—2112no—
143C22×C4—2112no—
144C22×C4C4×D21112yes—
145C22×C4C22×D21111yescharacteristic
146M16—21no—
147M16—21no—
148M16—21no—
149M16—4221no—
150M16—4221no—
151M16—4221no—
152M16—2112no—
153M16C22⋊D21112yes—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—4221no—
157C2×C8—4221no—
158C4⋊C4—4224no—
159C4⋊C4—4224no—
160C42—2111no—
161C42—2111no—
162C42—2112no—
163C23C2×C4×D21111yescharacteristic
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—41no—
168Q8—8441no—
169Q8—8441no—
170Q8—8441no—
171Q8—8441no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—4222no—
176C2×C4—4222no—
177C2×C4—4222no—
178C2×C4—4224no—
179C2×C4—2112no—
180C2×C4—2112no—
181C2×C4D4×D21111yescharacteristic
182C2×C4D4×D21111yescharacteristic
183C2×C4D4×D21111yescharacteristic
184C2×C4C2×C22⋊D21111yescharacteristic
185C2×C4#1680996111yescharacteristic
186C2×C4#1680996211yescharacteristic
187C8—22no—
188C8—41no—
189C8—8441no—
190C8—4222no—
191C22—22no—
192C22C22⋊C4×D21111yescharacteristic
193C22(C22⋊D211)⋊C411yescharacteristic
194C22C4×C22⋊D21111yescharacteristic
195C4—42no—
196C4—8442no—
197C4—4222no—
198C4—2114no—
199C4#2026994911yescharacteristic
200C4#2026995011yescharacteristic
201C4#2026996311yescharacteristic
202C4#2026997311yescharacteristic
203C2—21no—
204C2—21no—
205C2#2721304611yescharacteristic
206C2#101×D21112yes—
subgroupG/H
centerC22C22⋊C4×D211
commutator, prime radicalC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1966
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
98. Dic211#101semidirect
103. C211#1966semidirect

Maximal subgroups

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

subgroupindex
C2×#202699752
#272131692
C2×#202698962
C2×#202698952
M16×Dic2112
C2×#101×C2112
#1966211