Finite Group Databasebeta

← #41014198

Subgroups — #41014198

Order 27008.

332 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 115 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
2C2×#20269902C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic
4C2×#20269900C211yesmaximal, characteristic
5C2×C8×Dic211C211yesmaximal, characteristic
6#27213015C211yesmaximal, characteristic
7C2×#103×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269975—22no—
11#20269975C414yescomplemented
12C2×C4⋊Dic211C2211yescharacteristic
13C2×Dic211⋊C4—22no—
14C2×Dic211⋊C4—22no—
15C2×C4×Dic211C2211yescharacteristic
16#20269902—22no—
17#20269901—22no—
18#20269900—22no—
19C8×Dic211—22no—
20C22×Q8×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#103×C211—22no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422C2×C412yes—
30C22×Dic211—21no—
31C22×Dic211—21no—
32C22×Dic211D412yes—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8D412yes—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38Dic211⋊C4—28no—
39Dic211⋊C4—44no—
40C4×Dic211C2×C412yes—
41C4×Dic211D412yes—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211C2×C412yes—
48C22×C4×C211—21no—
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C2×C8×C211—21no—
53C2×C8×C211D412yes—
54C4⋊C4×C211—21no—
55C4⋊C4×C211—21no—
56Dic422—22no—
57Dic422—22no—
58Dic422—41no—
59Dic422—41no—
60C2×Dic211—22no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—42no—
65C2×Dic211C22⋊C414yes—
66C2×Dic211SD1614yes—
67C2×Dic211Q1614yes—
68C211⋊C8—22no—
69C23×C211C2×D411yescharacteristic
70Q8×C211—22no—
71Q8×C211—22no—
72Q8×C211—41no—
73Q8×C211—41no—
74C2×C4×C211—22no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—42no—
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C22×C211C2×C22⋊C411yescharacteristic
87C22×C211#12411yescharacteristic
88C22×C211#12711yescharacteristic
89C22×C211C2×SD1612yes—
90C22×C211C2×Q1612yes—
91C4×C211—42no—
92C4×C211—81no—
93C4×C211C4×D411yescharacteristic
94C4×C211C4×D411yescharacteristic
95C4×C211#12111yescharacteristic
96C4×C211#12111yescharacteristic
97Dic211—42no—
98Dic211—81no—
99Dic211#10318yescomplemented
100C2×C211#39011yescharacteristic
101C2×C211C2×#10312yes—
102C2×C211#48712yes—
103C2×C211#49211yescharacteristic
104C2×C211#49411yescharacteristic
105C211#195611yescharacteristic, complemented, Sylow
106#1956—2111nomaximal, Sylow
107C2×C4⋊Q8—2111no—
108C2×#103—2111no—
109C2×#103—2111no—
110C2×#103—2111no—
111C2×#103D21111yescharacteristic
112C2×C4×C8—2111no—
113#384—2111no—
114C22×Q8—2111no—
115C22×Q8C2×D21111yescharacteristic
116C22×C8—2111no—
117C22×C8C2×D21111yescharacteristic
118C4⋊Q8—4222no—
119C4⋊Q8—2114no—
120C2×C4⋊C4—4222no—
121C2×C4⋊C4—4222no—
122C2×C4⋊C4—2111no—
123C2×C4⋊C4C2×D21111yescharacteristic
124C2×C42—2111no—
125#103—22no—
126#103—4222no—
127#103—4222no—
128#103—4222no—
129C4×C8—4222no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—4221no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—4221no—
140C2×Q8—2112no—
141C2×Q8C4×D21112yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—4221no—
145C22×C4—4221no—
146C22×C4—2112no—
147C22×C4C22×D21111yescharacteristic
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—4221no—
151C2×C8—4221no—
152C2×C8—2112no—
153C2×C8C22⋊D21112yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—8444no—
157C4⋊C4—4228no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C42—2112no—
161C42—2112no—
162C23D4×D21111yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—8441no—
168Q8—8441no—
169Q8—4222no—
170Q8—4222no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—8441no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8442no—
180C2×C4—4222no—
181C2×C4—2114no—
182C2×C4—2114no—
183C2×C4—2114no—
184C2×C4C2×C4×D21111yescharacteristic
185C2×C4D4×D21111yescharacteristic
186C2×C4D4×D21111yescharacteristic
187C2×C4C2×C22⋊D21111yescharacteristic
188C2×C4#1680996111yescharacteristic
189C2×C4#1680996211yescharacteristic
190C8—22no—
191C8—4222no—
192C22C22⋊C4×D21111yescharacteristic
193C22#2026996311yescharacteristic
194C22#2026997311yescharacteristic
195C22SD16×D21112yes—
196C22Q16×D21112yes—
197C4—42no—
198C4—81no—
199C4—8442no—
200C4—16881no—
201C4—2118no—
202C4(C22⋊D211)⋊C411yescharacteristic
203C4C4×C22⋊D21111yescharacteristic
204C4#2026994911yescharacteristic
205C4#2026995011yescharacteristic
206C2#2721304611yescharacteristic
207C2#103×D21112yes—
208C2#2721349512yes—
209C2#2721351711yescharacteristic
210C2#2721352411yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC2×#103D211
hypocenterC211#1956
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
11. #20269975C4semidirect
99. Dic211#103semidirect
105. C211#1956semidirect

Maximal subgroups

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

subgroupindex
C2×#202699752
C2×#202699022
C2×#202699012
C2×#202699002
C2×C8×Dic2112
#272130152
C2×#103×C2112
#1956211