Finite Group Databasebeta

← #41014479

Subgroups — #41014479

Order 27008.

406 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16809950C211yesmaximal, characteristic
2C2×#20269975C211yesmaximal, characteristic, complemented
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×#20269901C211yesmaximal, characteristic, complemented
5#27213049C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic, complemented
7C22×SD16×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—22no—
10C2×#16809950—22no—
11C2×#16809950—22no—
12C22×C211⋊C8C2211yescharacteristic
13#20269975—22no—
14#20269975—22no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4—22no—
17C2×C22⋊Dic211—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269902—22no—
20#20269901—22no—
21C22×Q8×C211C2211yescharacteristic
22C22×D4×C211C2211yescharacteristic
23C2×SD16×C211—22no—
24C2×SD16×C211—22no—
25C2×SD16×C211—22no—
26C22×C8×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422D412yes—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D412yes—
36#16809950—28no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D412yes—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42Dic211⋊C4—28no—
43C22⋊Dic211—44no—
44C4×Dic211—21no—
45C4×Dic211—21no—
46C24×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211D412yes—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211D412yes—
59C22×C4×C211—21no—
60C22×C4×C211C2311yescharacteristic
61SD16×C211—28no—
62C2×C8×C211—21no—
63C2×C8×C211—21no—
64C2×C8×C211D412yes—
65Dic422—22no—
66Dic422—22no—
67Dic422—41no—
68Dic422—41no—
69C2×Dic211—22no—
70C2×Dic211—22no—
71C2×Dic211—22no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—42no—
76C2×Dic211SD1614yescomplemented
77C211⋊C8—22no—
78C23×C211—22no—
79C23×C211—41no—
80C23×C211—41no—
81C23×C211—42no—
82C23×C211—42no—
83C23×C211C2×D411yescharacteristic
84Q8×C211—22no—
85Q8×C211—22no—
86Q8×C211—41no—
87Q8×C211—41no—
88D4×C211—22no—
89D4×C211—22no—
90D4×C211—41no—
91D4×C211—41no—
92C2×C4×C211—22no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C8×C211—22no—
102C22×C211—42no—
103C22×C211—44no—
104C22×C211—82no—
105C22×C211—81no—
106C22×C211—81no—
107C22×C211#12011yescharacteristic
108C22×C211#12111yescharacteristic
109C22×C211#12711yescharacteristic
110C22×C211C2×SD1612yes—
111C22×C211#13712yes—
112C4×C211—42no—
113C4×C211#12011yescharacteristic
114C4×C211#12011yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12111yescharacteristic
117Dic211—24no—
118Dic211—42no—
119Dic211—81no—
120C2×C211—82no—
121C2×C211#39211yescharacteristic
122C2×C211#44812yes—
123C2×C211#46112yes—
124C2×C211#49211yescharacteristic
125C2×C211#49711yescharacteristic
126C211#201311yescharacteristic, complemented, Sylow
127#2013—2111nomaximal, Sylow
128C22×SD16—2111no—
129C22×SD16D21111yescharacteristic
130C2×C4⋊Q8—2111no—
131C2×#103—2111no—
132C2×#103—2111no—
133#390—2111no—
134#336—2111no—
135C22×Q8—2111no—
136C22×Q8C2×D21111yescharacteristic
137C22×D4C2×D21111yescharacteristic
138C2×SD16—22no—
139C2×SD16—22no—
140C2×SD16—22no—
141C2×SD16—4222no—
142C2×SD16—4222no—
143C2×SD16—4222no—
144C22×C8—2111no—
145C22×C8C2×D21111yescharacteristic
146C4⋊Q8—4222no—
147C4⋊Q8—4222no—
148C2×C4⋊C4—4222no—
149C2×C4⋊C4—2111no—
150C2×C22⋊C4—4222no—
151C2×C42—2111no—
152#103—4222no—
153#103—4222no—
154C24—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—4221no—
161C2×Q8—4221no—
162C2×Q8—4221no—
163C2×Q8—4221no—
164C2×Q8—4221no—
165C2×Q8—2112no—
166C2×Q8C22⋊D21112yes—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4C22⋊D21112yes—
173C22×C4—21no—
174C22×C4—4221no—
175C22×C4—4221no—
176C22×C4—2112no—
177C22×C4C22×D21111yescharacteristic
178SD16—28no—
179SD16—4228no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—4221no—
183C2×C8—4221no—
184C2×C8—2112no—
185C2×C8C22⋊D21112yes—
186C4⋊C4—4228no—
187C4⋊C4—4221no—
188C4⋊C4—4221no—
189C22⋊C4—8444no—
190C42—4221no—
191C42—4221no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23D4×D21111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—8441no—
203Q8—8441no—
204Q8—4222no—
205Q8—4222no—
206D4—22no—
207D4—22no—
208D4—41no—
209D4—41no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—8441no—
216C2×C4—8442no—
217C2×C4—4222no—
218C2×C4—4222no—
219C2×C4—4222no—
220C2×C4—2114no—
221C2×C4D4×D21111yescharacteristic
222C2×C4D4×D21111yescharacteristic
223C2×C4C2×C22⋊D21111yescharacteristic
224C2×C4C2×C22⋊D21111yescharacteristic
225C2×C4C2×C22⋊D21111yescharacteristic
226C2×C4#1680996111yescharacteristic
227C8—22no—
228C8—4222no—
229C22—44no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—82no—
234C22#2026994511yescharacteristic
235C22#2026994911yescharacteristic
236C22#2026997311yescharacteristic
237C22SD16×D21112yes—
238C22#2027000612yes—
239C4—42no—
240C4—8442no—
241C4—16881no—
242C4—4224no—
243C4#2026994511yescharacteristic
244C4#2026994611yescharacteristic
245C4#2026994811yescharacteristic
246C4#2026994911yescharacteristic
247C2—82no—
248C2#2721305411yescharacteristic
249C2#2721328312yes—
250C2#2721334912yes—
251C2#2721351711yescharacteristic
252C2#2721353711yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×SD16×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenterC22×SD16D211
hypocenterC211#2013
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#20269975C2semidirect
3. C2×#20269902C2semidirect
4. C2×#20269901C2semidirect
6. #27212900C2semidirect
76. C2×Dic211SD16semidirect
126. C211#2013semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C2×#202699752
C2×#202699022
C2×#202699012
#272130492
#272129002
C22×SD16×C2112
#2013211