Finite Group Databasebeta

← #41014575

Subgroups — #41014575

Order 27008.

278 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#27213772C211yesmaximal, characteristic, complemented
2C2×#20269892C211yesmaximal, characteristic
3C2×#20269892C211yesmaximal, characteristic
4#27212913C211yesmaximal, characteristic, complemented
5#27212912C211yesmaximal, characteristic, complemented
6#27212912C211yesmaximal, characteristic, complemented
7C2×#99×C211C211yesmaximal, characteristic
8#20269956—22no—
9#20269954—22no—
10#20269954—22no—
11C2×C22⋊Dic211C2211yescharacteristic
12C2×C22⋊Dic211C2211yescharacteristic
13C2×C22⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211C2211yescharacteristic
15#20269892—22no—
16#20269892—22no—
17C22×D4×C211C2211yescharacteristic
18C2×C22⋊C4×C211C2211yescharacteristic
19C2×C22⋊C4×C211C2211yescharacteristic
20#99×C211—22no—
21C2×Dic422—22no—
22C22×Dic211—21no—
23C22×Dic211—21no—
24C22×Dic211—21no—
25C22×Dic211—21no—
26C4⋊Dic211—22no—
27Dic211⋊C4—24no—
28C22⋊Dic211—21no—
29C22⋊Dic211—21no—
30C22⋊Dic211—21no—
31C22⋊Dic211—21no—
32C22⋊Dic211—21no—
33C22⋊Dic211—21no—
34C22⋊Dic211D412yes—
35C22⋊Dic211D412yes—
36C24×C211Q811yescharacteristic, complemented
37C24×C211C2311yescharacteristic
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22×C4×C211Q811yescharacteristic, complemented
47C22⋊C4×C211—21no—
48C22⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211D412yes—
51Dic422—82no—
52C2×Dic211—22no—
53C2×Dic211—22no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—81no—
57C2×Dic211—81no—
58C2×Dic211—81no—
59C2×Dic211—81no—
60C23×C211—21no—
61C23×C211—21no—
62C23×C211—21no—
63C23×C211—81no—
64C23×C211—81no—
65C23×C211C2×D411yescharacteristic
66C23×C211C2×D411yescharacteristic
67C23×C211C2×D411yescharacteristic
68C23×C211C2×Q811yescharacteristic
69C23×C211#4111yescharacteristic
70C23×C211#4111yescharacteristic
71C23×C211#4111yescharacteristic
72D4×C211—41no—
73D4×C211—41no—
74D4×C211—41no—
75D4×C211—41no—
76C2×C4×C211—22no—
77C2×C4×C211—21no—
78C2×C4×C211—21no—
79C2×C4×C211—21no—
80C2×C4×C211—41no—
81C2×C4×C211—81no—
82C2×C4×C211—81no—
83C22×C211—41no—
84C22×C211—41no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211—81no—
92C22×C211—81no—
93C22×C211—81no—
94C22×C211#12011yescharacteristic
95C22×C211C22⋊Q811yescharacteristic
96C22×C211C22⋊Q811yescharacteristic
97C22×C211C22⋊Q811yescharacteristic
98C22×C211#12411yescharacteristic
99C22×C211#12411yescharacteristic
100C22×C211#12411yescharacteristic
101C4×C211—42no—
102C4×C211—41no—
103C4×C211—81no—
104Dic211—42no—
105Dic211—42no—
106Dic211—81no—
107Dic211—81no—
108C2×C211—21no—
109C2×C211—21no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—81no—
116C2×C211#39311yescharacteristic
117C211#203211yescharacteristic, complemented, Sylow
118#2032—2111nomaximal, Sylow
119#543—2111no—
120C2×#99—2111no—
121C2×#99—2111no—
122C2×#99D21111yescharacteristic
123#342—2111no—
124#342—2111no—
125#342—2111no—
126C22×D4C2×D21111yescharacteristic
127C22⋊Q8—4222no—
128C22⋊Q8—4222no—
129C22⋊Q8—4222no—
130C2×C22⋊C4—2111no—
131C2×C22⋊C4—2111no—
132C2×C22⋊C4—2111no—
133C2×C22⋊C4—2111no—
134C2×C22⋊C4C2×D21111yescharacteristic
135C2×C22⋊C4C2×D21111yescharacteristic
136#99—22no—
137#99—4222no—
138#99—4222no—
139C24Dic42211yescharacteristic, complemented
140C24C22×D21111yescharacteristic
141C2×Q8—4222no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—4221no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4Dic42211yescharacteristic, complemented
155C4⋊C4—4224no—
156C4⋊C4—4222no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—4221no—
161C22⋊C4—4221no—
162C22⋊C4—4221no—
163C22⋊C4—4221no—
164C22⋊C4—4221no—
165C22⋊C4—4221no—
166C22⋊C4—2112no—
167C22⋊C4—2112no—
168C22⋊C4C22⋊D21112yes—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—81no—
173C23—81no—
174C23D4×D21111yescharacteristic
175C23D4×D21111yescharacteristic
176C23C2×C22⋊D21111yescharacteristic
177C23C2×Dic42211yescharacteristic
178C23#1680996111yescharacteristic
179C23#1680996111yescharacteristic
180C23#1680996211yescharacteristic
181Q8—16882no—
182D4—41no—
183D4—41no—
184D4—41no—
185D4—41no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—16881no—
196C2×C4—16881no—
197C2×C4—16881no—
198C2×C4—16881no—
199C2×C4—4222no—
200C2×C4—4222no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22#2026994511yescharacteristic
213C22#2026995411yescharacteristic
214C22#2026995411yescharacteristic
215C22#2026995611yescharacteristic
216C22#2026996311yescharacteristic
217C22#2026996311yescharacteristic
218C22#2026996411yescharacteristic
219C4—42no—
220C4—41no—
221C4—81no—
222C4—8442no—
223C4—8442no—
224C4—16881no—
225C4—16881no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—81no—
234C2#2721305811yescharacteristic
subgroupG/H
centerC2#27213058
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
FittingC2×#99×C211C2
socleC2×C211#393
prime radicalC22×D4×C211C22
hypercenterC2×#99D211
hypocenterC211#2032
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. #27213772C2semidirect
4. #27212913C2semidirect
5. #27212912C2semidirect
6. #27212912C2semidirect
36. C24×C211Q8semidirect
46. C22×C4×C211Q8semidirect
117. C211#2032semidirect
139. C24Dic422semidirect
154. C22×C4Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137722
C2×#202698922
#272129132
#272129122
C2×#99×C2112
#2032211