Finite Group Databasebeta

← #41014208

Subgroups — #41014208

Order 27008.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20269964C211yesmaximal, characteristic
2C2×#20269902C211yesmaximal, characteristic, complemented
3C2×#20269900C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic
5C2×C8×Dic211C211yesmaximal, characteristic, complemented
6#27213011C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269964—22no—
11#20269964C414yescomplemented
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C22⋊Dic211—22no—
14C2×C4×Dic211C2211yescharacteristic
15#20269902—22no—
16#20269900—22no—
17#20269898—22no—
18C8×Dic211—22no—
19C211⋊#95—22no—
20C22×D4×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#102×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—
38C22⋊Dic211—28no—
39C4×Dic211C2×C412yes—
40C4×Dic211D412yes—
41C24×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211C2×C412yes—
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C2×C8×C211D412yes—
53C4⋊C4×C211—21no—
54C4⋊C4×C211—21no—
55Dic422—22no—
56Dic422—22no—
57Dic422—41no—
58Dic422—41no—
59C2×Dic211—22no—
60C2×Dic211—22no—
61C2×Dic211—22no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—42no—
66C2×Dic211C22⋊C414yes—
67C211⋊C8—22no—
68C23×C211—22no—
69C23×C211—41no—
70C23×C211—41no—
71C23×C211—42no—
72C23×C211—42no—
73C23×C211C2×D411yescharacteristic
74D4×C211—22no—
75D4×C211—22no—
76D4×C211—41no—
77D4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—42no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
87C22×C211—42no—
88C22×C211—44no—
89C22×C211—82no—
90C22×C211—81no—
91C22×C211—81no—
92C22×C211C2×C22⋊C411yescharacteristic
93C22×C211#12411yescharacteristic
94C22×C211#12711yescharacteristic
95C22×C211#13512yes—
96C22×C211#13512yes—
97C4×C211—81no—
98C4×C211C4×D411yescharacteristic
99C4×C211C4×D411yescharacteristic
100C4×C211#12111yescharacteristic
101C4×C211#12111yescharacteristic
102Dic211—24no—
103Dic211—42no—
104Dic211—81no—
105C2×C211—82no—
106C2×C211#39011yescharacteristic
107C2×C211#41612yes—
108C2×C211#48812yes—
109C2×C211#49511yescharacteristic
110C2×C211#49511yescharacteristic
111C211#195811yescharacteristic, complemented, Sylow
112#1958—2111nomaximal, Sylow
113C2×#124—2111no—
114C2×#103—2111no—
115C2×#103—2111no—
116C2×#102—2111no—
117C2×#102D21111yescharacteristic
118C2×C4×C8—2111no—
119#382—2111no—
120C22×Q8—2111no—
121C22×D4C2×D21111yescharacteristic
122C22×C8—2111no—
123C22×C8C2×D21111yescharacteristic
124#124—4222no—
125#124—2114no—
126C2×C4⋊C4—2111no—
127C2×C4⋊C4C2×D21111yescharacteristic
128C2×C22⋊C4—4222no—
129C2×C42—2111no—
130#103—4222no—
131#103—4222no—
132#102—22no—
133#102—4222no—
134C4×C8—4222no—
135#95—4222no—
136C24—21no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—4221no—
140C2×Q8—4221no—
141C2×Q8—4221no—
142C2×Q8—2112no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C4×D21112yes—
149C22×C4—21no—
150C22×C4—4221no—
151C22×C4—4221no—
152C22×C4—2112no—
153C22×C4C22×D21111yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—4221no—
157C2×C8—4221no—
158C2×C8—2112no—
159C2×C8C22⋊D21112yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C22⋊C4—4228no—
165C42—2112no—
166C42—2112no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23D4×D21111yescharacteristic
173Q8—8441no—
174Q8—8441no—
175Q8—4222no—
176Q8—4222no—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—8442no—
187C2×C4—4222no—
188C2×C4—4222no—
189C2×C4—4222no—
190C2×C4—2114no—
191C2×C4C2×C4×D21111yescharacteristic
192C2×C4D4×D21111yescharacteristic
193C2×C4D4×D21111yescharacteristic
194C2×C4C2×C22⋊D21111yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996211yescharacteristic
197C8—22no—
198C8—4222no—
199C22—44no—
200C22—42no—
201C22—82no—
202C22—81no—
203C22—81no—
204C22C22⋊C4×D21111yescharacteristic
205C22#2026996311yescharacteristic
206C22#2026997311yescharacteristic
207C22#2026999612yes—
208C22#2026999812yes—
209C4—81no—
210C4—8442no—
211C4—16881no—
212C4—4224no—
213C4(C22⋊D211)⋊C411yescharacteristic
214C4C4×C22⋊D21111yescharacteristic
215C4#2026994911yescharacteristic
216C4#2026995011yescharacteristic
217C2—82no—
218C2#2721304611yescharacteristic
219C2#2721313912yes—
220C2#2721349912yes—
221C2#2721352711yescharacteristic
222C2#2721352911yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#102D211
hypocenterC211#1958
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#20269902C2semidirect
3. C2×#20269900C2semidirect
5. C2×C8×Dic211C2semidirect
6. #27213011C2semidirect
11. #20269964C4semidirect
111. C211#1958semidirect

Maximal subgroups

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

subgroupindex
C2×#202699642
C2×#202699022
C2×#202699002
C2×#202698982
C2×C8×Dic2112
#272130112
C2×#102×C2112
#1958211