Finite Group Databasebeta

← #41013808

Subgroups — #41013808

Order 27008.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269960C211yesmaximal, characteristic
2#27213619C211yesmaximal, characteristic
3C2×#20269891C211yesmaximal, characteristic
4#27213009C211yesmaximal, characteristic, complemented
5#27212913C211yesmaximal, characteristic, complemented
6#27212912C211yesmaximal, characteristic, complemented
7#381×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211C2211yescharacteristic
9#20269960—22no—
10#20269960C414yescomplemented
11C4×C22⋊D211—22no—
12(C22⋊D211)⋊C4—22no—
13#20269933—21no—
14C2×Dic211⋊C4—21no—
15C2×C22⋊Dic211C2211yescharacteristic
16C2×C22⋊Dic211C2211yescharacteristic
17C2×C22⋊Dic211C2211yescharacteristic
18C2×(C2×D211)⋊C4—21no—
19C22⋊C4×D211—21no—
20#20269891—22no—
21C211⋊#95—22no—
22C23×C4×C211C2211yescharacteristic
23C2×C22⋊C4×C211C2211yescharacteristic
24C2×C22⋊C4×C211C2211yescharacteristic
25#95×C211—22no—
26C23×D211D411yescharacteristic
27C2×C22⋊D211—21no—
28C2×C22⋊D211—21no—
29C2×C22⋊D211—21no—
30C2×C22⋊D211—21no—
31C2×C22⋊D211C2×C412yes—
32C2×C22⋊D211D412yes—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D411yescharacteristic
36C22×Dic211D412yes—
37C2×C4×D211—22no—
38Dic211⋊C4—22no—
39Dic211⋊C4—24no—
40C22⋊Dic211—21no—
41C22⋊Dic211—21no—
42C22⋊Dic211—21no—
43C22⋊Dic211—21no—
44C22⋊Dic211—21no—
45C22⋊Dic211C2×C412yescomplemented
46(C2×D211)⋊C4—22no—
47(C2×D211)⋊C4—24no—
48C4×Dic211—22no—
49C24×C211C2311yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211—21no—
54C22×C4×C211—21no—
55C22×C4×C211—21no—
56C22×C4×C211—21no—
57C22×C4×C211C2×C412yescomplemented
58C22×C4×C211D412yes—
59C22⋊C4×C211—21no—
60C22⋊C4×C211—21no—
61C22⋊C4×C211—21no—
62C22⋊C4×C211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—42no—
66C22×D211C22⋊C412yes—
67C22⋊D211—22no—
68C22⋊D211—22no—
69C22⋊D211—22no—
70C22⋊D211—22no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—22no—
74C2×Dic211—22no—
75C2×Dic211—42no—
76C2×Dic211—42no—
77C2×Dic211—42no—
78C2×Dic211—41no—
79C2×Dic211C22⋊C412yes—
80C4×D211—82no—
81C23×C211—42no—
82C23×C211C22×C411yescharacteristic
83C23×C211C2×D411yescharacteristic
84C23×C211C2×D411yescharacteristic
85C23×C211C2×D411yescharacteristic
86C23×C211C2×D411yescharacteristic
87C23×C211#4111yescharacteristic
88C23×C211#4111yescharacteristic
89C2×C4×C211—22no—
90C2×C4×C211—22no—
91C2×C4×C211—22no—
92C2×C4×C211—22no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211—42no—
98C2×C4×C211—42no—
99C2×C4×C211—41no—
100C22×C211—22no—
101C22×C211—22no—
102C22×C211—22no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211—41no—
106C22×C211—41no—
107C22×C211C2×C22⋊C411yescharacteristic
108C22×C211C4×D411yescharacteristic
109C22×C211C4×D411yescharacteristic
110C22×C211#12011yescharacteristic
111C22×C211#12111yescharacteristic
112C22×C211#12111yescharacteristic
113C22×C211#12311yescharacteristic
114C2×D211—22no—
115C2×D211—42no—
116C2×D211—42no—
117C2×D211—81no—
118C4×C211—42no—
119C4×C211—42no—
120C4×C211—81no—
121Dic211—22no—
122Dic211—42no—
123Dic211—42no—
124Dic211—81no—
125C2×C211—21no—
126C2×C211—21no—
127C2×C211—21no—
128C2×C211—21no—
129C2×C211—21no—
130C2×C211—21no—
131C2×C211#38611yescharacteristic
132C2×C211#45812yes—
133D211—42no—
134C211#189011yescharacteristic, complemented, Sylow
135#1890—2111nomaximal, Sylow
136C2×#123—2111no—
137#518—2111no—
138C2×#99—2111no—
139#381—2111no—
140#381D21111yescharacteristic, complemented
141#342—2111no—
142#342—2111no—
143C22×D4—2111no—
144C23×C4C2×D21111yescharacteristic
145#123—4222no—
146#123—2114no—
147C4×D4—4222no—
148C4×D4—4222no—
149#117—4221no—
150C2×C4⋊C4—4221no—
151C2×C22⋊C4—4221no—
152C2×C22⋊C4—4221no—
153C2×C22⋊C4—2111no—
154C2×C22⋊C4—2111no—
155C2×C22⋊C4—2111no—
156C2×C22⋊C4C2×D21111yescharacteristic
157C2×C22⋊C4C2×D21111yescharacteristic
158#99—4222no—
159#95—22no—
160#95—4222no—
161C24—2111no—
162C24C22×D21111yescharacteristic
163C2×D4—4221no—
164C2×D4—4221no—
165C2×D4—4221no—
166C2×D4—4221no—
167C2×D4—2112no—
168C2×D4—2112no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—4222no—
177C22×C4—4221no—
178C22×C4—4221no—
179C22×C4—2112no—
180C22×C4—2111no—
181C22×C4C4×D21112yescomplemented
182C22×C4C22⋊D21112yes—
183C4⋊C4—4222no—
184C4⋊C4—4224no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—4222no—
190C22⋊C4—4224no—
191C22⋊C4—4221no—
192C22⋊C4—4221no—
193C22⋊C4—4221no—
194C22⋊C4—4221no—
195C22⋊C4—4221no—
196C22⋊C4—2112no—
197C42—4222no—
198C23—42no—
199C23—8442no—
200C23—4221no—
201C23—4221no—
202C23—2112no—
203C23C2×C4×D21111yescharacteristic
204C23D4×D21111yescharacteristic
205C23D4×D21111yescharacteristic
206C23D4×D21111yescharacteristic
207C23C2×C22⋊D21111yescharacteristic
208C23#1680996111yescharacteristic
209C23#1680996211yescharacteristic
210D4—4222no—
211D4—4222no—
212D4—4222no—
213D4—4222no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—8442no—
226C2×C4—8442no—
227C2×C4—8442no—
228C2×C4—8441no—
229C2×C4—16882no—
230C2×C4—4222no—
231C2×C4—4222no—
232C2×C4—4221no—
233C2×C4—4221no—
234C2×C4—2112no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—8442no—
243C22—8442no—
244C22—16881no—
245C22—4222no—
246C22C22⋊C4×D21111yescharacteristic
247C22(C22⋊D211)⋊C411yescharacteristic
248C22C4×C22⋊D21111yescharacteristic
249C22#2026994511yescharacteristic
250C22#2026994811yescharacteristic
251C22#2026995011yescharacteristic
252C22#2026995911yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—8442no—
257C4—8442no—
258C4—16881no—
259C4—4222no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—8442no—
267C2#2721302211yescharacteristic
268C2#2721333512yes—
subgroupG/H
centerC22#20269945
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#381×C211C2
socleC22×C211#120
prime radicalC22×C22⋊D211C22
cosocleC24×C211C23
hypercenter#381D211
hypocenterC211#1890
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #27213009C2semidirect
5. #27212913C2semidirect
6. #27212912C2semidirect
7. #381×C211C2semidirect
10. #20269960C4semidirect
45. C22⋊Dic211C2×C4semidirect
57. C22×C4×C211C2×C4semidirect
134. C211#1890semidirect
140. #381D211semidirect
181. C22×C4C4×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699602
#272136192
C2×#202698912
#272130092
#272129132
#272129122
#381×C2112
#1890211