Finite Group Databasebeta

← #41014491

Subgroups — #41014491

Order 27008.

390 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16809952C211yesmaximal, characteristic, complemented
2C22×#16809949C211yesmaximal, characteristic
3C2×#20269965C211yesmaximal, characteristic
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×#20269897C211yesmaximal, characteristic
6#27212901C211yesmaximal, characteristic, complemented
7#391×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809952—22no—
11C2×#16809952—22no—
12C2×#16809952—22no—
13C2×#16809949—22no—
14C2×#16809949—22no—
15C2×#16809949—22no—
16C22×C211⋊C8C2211yescharacteristic
17C22×C211⋊C8C2211yescharacteristic
18#20269965—22no—
19#20269965—22no—
20C2×(C2×D211)⋊C4—22no—
21#20269900—22no—
22#20269897—22no—
23C22×Q8×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C42×C211C2211yescharacteristic
26#95×C211—22no—
27C23×D211—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422D412yes—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844D412yes—
40C22×Dic211—21no—
41#16809952—28no—
42#16809949—28no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48C2×C211⋊C8D412yes—
49(C2×D211)⋊C4—28no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211D412yes—
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211D412yes—
59C22×C4×C211C2311yescharacteristic
60C4⋊C4×C211—21no—
61C4⋊C4×C211—21no—
62C42×C211—21no—
63C42×C211—21no—
64C22×D211—22no—
65C22×D211—41no—
66C22×D211—41no—
67C22×D211—42no—
68C22×D211—42no—
69Dic422—22no—
70Dic422—22no—
71Dic422—41no—
72Dic422—41no—
73D844—22no—
74D844—22no—
75D844—41no—
76D844—41no—
77C2×Dic211—22no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C211⋊C8—22no—
81C211⋊C8—22no—
82C23×C211C2×D411yescharacteristic
83Q8×C211—22no—
84Q8×C211—22no—
85Q8×C211—41no—
86Q8×C211—41no—
87C2×C4×C211—22no—
88C2×C4×C211—22no—
89C2×C4×C211—22no—
90C2×C4×C211—22no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211—42no—
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C22×C211#12011yescharacteristic
102C22×C211#12111yescharacteristic
103C22×C211#12711yescharacteristic
104C22×C211#13512yes—
105C22×C211#13712yes—
106C2×D211—42no—
107C2×D211—44no—
108C2×D211—82no—
109C2×D211—81no—
110C2×D211—81no—
111C4×C211—42no—
112C4×C211—42no—
113C4×C211—81no—
114C4×C211#12011yescharacteristic
115C4×C211#12011yescharacteristic
116C4×C211#12111yescharacteristic
117C4×C211#12111yescharacteristic
118Dic211—42no—
119C2×C211#39211yescharacteristic
120C2×C211#45212yes—
121C2×C211#46412yes—
122C2×C211#49511yescharacteristic
123C2×C211#49711yescharacteristic
124D211—82no—
125C211#201511yescharacteristic, complemented, Sylow
126#2015—2111nomaximal, Sylow
127C22×Q16—2111no—
128C22×SD16—2111no—
129C2×#124—2111no—
130C2×#103—2111no—
131C2×#102—2111no—
132#391D21111yescharacteristic, complemented
133#336—2111no—
134C22×Q8—2111no—
135C22×Q8C2×D21111yescharacteristic
136C22×D4—2111no—
137C2×Q16—4222no—
138C2×Q16—4222no—
139C2×Q16—4222no—
140C2×SD16—4222no—
141C2×SD16—4222no—
142C2×SD16—4222no—
143C22×C8—2111no—
144C22×C8—2111no—
145#124—4222no—
146#124—4222no—
147C2×C4⋊C4C2×D21111yescharacteristic
148C2×C22⋊C4—4222no—
149C2×C42C2×D21111yescharacteristic
150#103—4222no—
151#102—4222no—
152#95—22no—
153C24—4221no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—4221no—
162C2×Q8—4221no—
163C2×Q8—4221no—
164C2×Q8—2112no—
165C2×Q8D84412yes—
166C2×D4—4221no—
167C2×D4—4221no—
168C2×D4—4221no—
169C2×D4—4221no—
170C2×D4—4221no—
171C2×D4—2112no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—4221no—
175C22×C4C22⋊D21112yes—
176C22×C4C22×D21111yescharacteristic
177Q16—4228no—
178SD16—4228no—
179C2×C8—4221no—
180C2×C8—4221no—
181C2×C8—4221no—
182C2×C8—4221no—
183C2×C8—2112no—
184C2×C8—2112no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C22⋊C4—4228no—
188C42—21no—
189C42—21no—
190C23—8441no—
191C23—8441no—
192C23—8442no—
193C23—8442no—
194C23—4222no—
195C23C2×C22⋊D21111yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—41no—
199Q8—41no—
200Q8—8441no—
201Q8—8441no—
202Q8—4222no—
203Q8—4222no—
204D4—8441no—
205D4—8441no—
206D4—4222no—
207D4—4222no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—8441no—
217C2×C4—8441no—
218C2×C4—4222no—
219C2×C4D4×D21111yescharacteristic
220C2×C4D4×D21111yescharacteristic
221C2×C4D4×D21111yescharacteristic
222C2×C4D4×D21111yescharacteristic
223C2×C4C2×D84411yescharacteristic
224C2×C4#1680996211yescharacteristic
225C8—4222no—
226C8—4222no—
227C22—8444no—
228C22—8442no—
229C22—16882no—
230C22—16881no—
231C22—16881no—
232C22#2026994511yescharacteristic
233C22#2026995111yescharacteristic
234C22#2026997311yescharacteristic
235C22#2026999712yes—
236C22#2027000812yes—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—8442no—
241C4#2026994711yescharacteristic
242C4#2026994711yescharacteristic
243C4#2026995011yescharacteristic
244C4#2026995011yescharacteristic
245C2—16882no—
246C2#2721305511yescharacteristic
247C2#2721330612yes—
248C2#2721336812yes—
249C2#2721352811yescharacteristic
250C2#2721353811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#391×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
hypercenter#391D211
hypocenterC211#2015
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×#16809952C2semidirect
4. C2×#20269900C2semidirect
6. #27212901C2semidirect
7. #391×C211C2semidirect
125. C211#2015semidirect
132. #391D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
C22×#168099492
C2×#202699652
C2×#202699002
C2×#202698972
#272129012
#391×C2112
#2015211