Finite Group Databasebeta

← #41014484

Subgroups — #41014484

Order 27008.

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

normal subgroups only

This group has 474 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×#16809949C211yesmaximal, characteristic, complemented
2C2×#20269973C211yesmaximal, characteristic
3C2×#20269899C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5#27213053C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C22×SD16×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C22⋊D211—22no—
10C2×#16809949—22no—
11C2×#16809949—22no—
12C2×#16809949—22no—
13C22×C211⋊C8C2211yescharacteristic
14#20269973—22no—
15#20269973—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×C4×Dic211C2211yescharacteristic
18#20269899—22no—
19#20269898—22no—
20C211⋊#95—22no—
21C22×Q8×C211C2211yescharacteristic
22C22×D4×C211C2211yescharacteristic
23C2×SD16×C211—22no—
24C2×SD16×C211—22no—
25C2×SD16×C211—22no—
26C22×C8×C211C2211yescharacteristic
27C23×D211—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844D412yes—
34C2×C22⋊D211—28no—
35C2×C22⋊D211—44no—
36C2×C22⋊D211—44no—
37C22×Dic211—21no—
38C22×Dic211D412yes—
39#16809949—28no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8D412yescomplemented
43C4⋊Dic211—21no—
44C4⋊Dic211—21no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C24×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211D412yescomplemented
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211D412yes—
60C22×C4×C211—21no—
61C22×C4×C211C2311yescharacteristic
62SD16×C211—28no—
63C2×C8×C211—21no—
64C2×C8×C211—21no—
65C2×C8×C211D412yescomplemented
66C22×D211—22no—
67C22×D211—41no—
68C22×D211—41no—
69C22×D211—42no—
70C22×D211—42no—
71D844—22no—
72D844—22no—
73D844—41no—
74D844—41no—
75C22⋊D211—416no—
76C2×Dic211—22no—
77C2×Dic211—22no—
78C2×Dic211—41no—
79C2×Dic211—42no—
80C2×Dic211SD1614yescomplemented
81C211⋊C8—22no—
82C23×C211—22no—
83C23×C211—41no—
84C23×C211—41no—
85C23×C211—42no—
86C23×C211—42no—
87C23×C211C2×D411yescharacteristic
88Q8×C211—22no—
89Q8×C211—22no—
90Q8×C211—41no—
91Q8×C211—41no—
92D4×C211—22no—
93D4×C211—22no—
94D4×C211—41no—
95D4×C211—41no—
96C2×C4×C211—22no—
97C2×C4×C211—41no—
98C2×C4×C211—41no—
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211#4111yescharacteristic
105C8×C211—22no—
106C22×C211—42no—
107C22×C211—44no—
108C22×C211—82no—
109C22×C211—81no—
110C22×C211—81no—
111C22×C211#12011yescharacteristic
112C22×C211#12111yescharacteristic
113C22×C211#12711yescharacteristic
114C22×C211C2×SD1612yes—
115C22×C211AGL(1,ℤ/8)12yes—
116C2×D211—42no—
117C2×D211—44no—
118C2×D211—82no—
119C2×D211—81no—
120C2×D211—81no—
121C4×C211—42no—
122C4×C211#12011yescharacteristic
123C4×C211#12011yescharacteristic
124C4×C211#12111yescharacteristic
125C4×C211#12111yescharacteristic
126Dic211—24no—
127Dic211—81no—
128C2×C211—82no—
129C2×C211#39211yescharacteristic
130C2×C211#45012yes—
131C2×C211#46012yes—
132C2×C211#49211yescharacteristic
133C2×C211#49611yescharacteristic
134D211—82no—
135C211#201411yescharacteristic, complemented, Sylow
136#2014—2111nomaximal, Sylow
137C22×SD16—2111no—
138C22×SD16D21111yescharacteristic, complemented
139C2×#127—2111no—
140C2×#102—2111no—
141C2×#102—2111no—
142#391—2111no—
143#336—2111no—
144C22×Q8C2×D21111yescharacteristic
145C22×D4—4222no—
146C22×D4—2111no—
147C22×D4C2×D21111yescharacteristic
148C2×SD16—22no—
149C2×SD16—22no—
150C2×SD16—22no—
151C2×SD16—4222no—
152C2×SD16—4222no—
153C2×SD16—4222no—
154C22×C8—2111no—
155C22×C8C2×D21111yescharacteristic
156#127—4222no—
157#127—4222no—
158C2×C4⋊C4—2111no—
159C2×C42—2111no—
160#102—4222no—
161#102—4222no—
162#95—4222no—
163C24—21no—
164C24—4221no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8C22⋊D21112yescomplemented
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—8444no—
177C2×D4—8444no—
178C2×D4—4228no—
179C2×D4—4221no—
180C2×D4—4221no—
181C2×D4—4221no—
182C2×D4—4221no—
183C2×D4—4221no—
184C2×D4—2112no—
185C2×D4C22⋊D21112yes—
186C22×C4—21no—
187C22×C4—4221no—
188C22×C4—2112no—
189C22×C4C22×D21111yescharacteristic
190SD16—28no—
191SD16—4228no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—4221no—
195C2×C8—4221no—
196C2×C8—2112no—
197C2×C8C22⋊D21112yescomplemented
198C4⋊C4—4221no—
199C4⋊C4—4221no—
200C42—4221no—
201C42—4221no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—8441no—
208C23—8441no—
209C23—8442no—
210C23—8442no—
211C23—4222no—
212C23D4×D21111yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—41no—
216Q8—41no—
217D4—22no—
218D4—22no—
219D4—41no—
220D4—41no—
221D4—84416no—
222D4—8441no—
223D4—8441no—
224D4—4222no—
225D4—4222no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—8441no—
230C2×C4—8442no—
231C2×C4—4222no—
232C2×C4—4222no—
233C2×C4—2114no—
234C2×C4D4×D21111yescharacteristic
235C2×C4D4×D21111yescharacteristic
236C2×C4C2×C22⋊D21111yescharacteristic
237C2×C4C2×C22⋊D21111yescharacteristic
238C2×C4C2×C22⋊D21111yescharacteristic
239C2×C4#1680996111yescharacteristic
240C8—22no—
241C8—4222no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22—8444no—
248C22—8442no—
249C22—16882no—
250C22—16881no—
251C22—16881no—
252C22#2026994511yescharacteristic
253C22#2026994911yescharacteristic
254C22#2026997311yescharacteristic
255C22SD16×D21112yes—
256C22#2027000412yes—
257C4—42no—
258C4—16881no—
259C4—4224no—
260C4#2026994511yescharacteristic
261C4#2026994611yescharacteristic
262C4#2026994811yescharacteristic
263C4#2026994911yescharacteristic
264C2—82no—
265C2—16882no—
266C2#2721305411yescharacteristic
267C2#2721329512yes—
268C2#2721334412yes—
269C2#2721351711yescharacteristic
270C2#2721353211yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×SD16×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269973C2
hypercenterC22×SD16D211
hypocenterC211#2014
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
1. C22×#16809949C2semidirect
3. C2×#20269899C2semidirect
4. C2×#20269898C2semidirect
5. #27213053C2semidirect
6. #27212900C2semidirect
7. C22×SD16×C211C2semidirect
42. C2×C211⋊C8D4semidirect
53. C2×Q8×C211D4semidirect
65. C2×C8×C211D4semidirect
80. C2×Dic211SD16semidirect
135. C211#2014semidirect
138. C22×SD16D211semidirect
170. C2×Q8C22⋊D211semidirect
197. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099492
C2×#202699732
C2×#202698992
C2×#202698982
#272130532
#272129002
C22×SD16×C2112
#2014211