Finite Group Databasebeta

← #41014406

Subgroups — #41014406

Order 27008.

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

normal subgroups only

This group has 392 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×#16809952C211yesmaximal, characteristic, complemented
2C2×#20269956C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7C2×C22⋊Q8×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809952—22no—
10C2×#16809952—22no—
11C2×#16809952—22no—
12C22×C211⋊C8C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14#20269956—22no—
15#20269956—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×Dic211⋊C4—21no—
18C2×C22⋊Dic211—21no—
19#20269901—22no—
20#20269900—22no—
21C22⋊(C211⋊C8)—22no—
22C22×Q8×C211C2211yescharacteristic
23C23×C4×C211C2211yescharacteristic
24C22⋊Q8×C211—22no—
25C22⋊Q8×C211—22no—
26C2×C4⋊C4×C211—21no—
27C2×C4⋊C4×C211C2211yescharacteristic
28C2×C22⋊C4×C211—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422D412yes—
35C22×Dic211—21no—
36C22×Dic211—21no—
37#16809952—28no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8D412yes—
44C4⋊Dic211—21no—
45C4⋊Dic211D412yes—
46Dic211⋊C4—24no—
47C22⋊Dic211—24no—
48C24×C211D411yescharacteristic
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211D412yes—
55C22×C4×C211—21no—
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211—21no—
59C22×C4×C211—21no—
60C22×C4×C211—21no—
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211D411yescharacteristic
64C22×C4×C211C2311yescharacteristic
65C4⋊C4×C211—24no—
66C4⋊C4×C211—21no—
67C4⋊C4×C211D412yes—
68C22⋊C4×C211—24no—
69Dic422—22no—
70Dic422—22no—
71Dic422—41no—
72Dic422—41no—
73C2×Dic211—22no—
74C2×Dic211—22no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C211⋊C8—22no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211—21no—
83C23×C211—21no—
84C23×C211—21no—
85C23×C211—21no—
86C23×C211—21no—
87C23×C211Q1612yescomplemented
88C23×C211C2×D411yescharacteristic
89Q8×C211—22no—
90Q8×C211—22no—
91Q8×C211—41no—
92Q8×C211—41no—
93C2×C4×C211—22no—
94C2×C4×C211—22no—
95C2×C4×C211—21no—
96C2×C4×C211—21no—
97C2×C4×C211—21no—
98C2×C4×C211—41no—
99C2×C4×C211—41no—
100C2×C4×C211—41no—
101C2×C4×C211—41no—
102C2×C4×C211—41no—
103C2×C4×C211—41no—
104C2×C4×C211—41no—
105C2×C4×C211—41no—
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211C2×D411yescharacteristic
111C2×C4×C211#4111yescharacteristic
112C22×C211—22no—
113C22×C211—22no—
114C22×C211—22no—
115C22×C211—41no—
116C22×C211—41no—
117C22×C211—41no—
118C22×C211—41no—
119C22×C211#12011yescharacteristic
120C22×C211#12011yescharacteristic
121C22×C211#12111yescharacteristic
122C22×C211C2×Q1611yescharacteristic
123C22×C211#13511yescharacteristic
124C22×C211#13711yescharacteristic
125C22×C211#13711yescharacteristic
126C4×C211—41no—
127C4×C211—42no—
128C4×C211—42no—
129C4×C211#12011yescharacteristic
130C4×C211#12111yescharacteristic
131C4×C211#12111yescharacteristic
132C4×C211#12711yescharacteristic
133Dic211—42no—
134Dic211—42no—
135C2×C211—42no—
136C2×C211#39211yescharacteristic
137C2×C211#45111yescharacteristic
138C2×C211#45111yescharacteristic
139C2×C211#45211yescharacteristic
140C2×C211#45211yescharacteristic
141C2×C211#46711yescharacteristic
142C2×C211#47011yescharacteristic
143C211#200111yescharacteristic, complemented, Sylow
144#2001—2111nomaximal, Sylow
145C22×Q16—2111no—
146C2×C22⋊Q8—2111no—
147C2×C22⋊Q8D21111yescharacteristic
148C2×#103—2111no—
149C2×#103—2111no—
150C2×C22⋊C8—2111no—
151#340—2111no—
152C22×Q8—2111no—
153C22×Q8C2×D21111yescharacteristic
154C23×C4C2×D21111yescharacteristic
155C2×Q16—4222no—
156C2×Q16—4222no—
157C2×Q16—4222no—
158C22×C8—2111no—
159C22×C8—2111no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—4222no—
163C22⋊Q8—4222no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—4221no—
166C2×C4⋊C4—2111no—
167C2×C4⋊C4C2×D21111yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—4221no—
170#103—4222no—
171#103—4222no—
172C22⋊C8—4222no—
173C24C22⋊D21111yescharacteristic
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—4221no—
180C2×Q8—4221no—
181C2×Q8—4221no—
182C2×Q8—4221no—
183C2×Q8—4221no—
184C2×Q8—2112no—
185C2×Q8C22⋊D21112yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—4221no—
195C22×C4—4221no—
196C22×C4C22⋊D21111yescharacteristic
197C22×C4C22×D21111yescharacteristic
198Q16—4228no—
199C2×C8—4221no—
200C2×C8—4221no—
201C2×C8—4221no—
202C2×C8—4221no—
203C2×C8—4221no—
204C2×C8—2112no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4—4224no—
208C4⋊C4—4221no—
209C4⋊C4—2112no—
210C4⋊C4C22⋊D21112yes—
211C22⋊C4—24no—
212C22⋊C4—4224no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#1680995212yescomplemented
220C23C2×C22⋊D21111yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—8441no—
226Q8—8441no—
227Q8—4222no—
228Q8—4222no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—8441no—
243C2×C4—8441no—
244C2×C4—8441no—
245C2×C4—8441no—
246C2×C4—4222no—
247C2×C4—4222no—
248C2×C4D4×D21111yescharacteristic
249C2×C4D4×D21111yescharacteristic
250C2×C4D4×D21111yescharacteristic
251C2×C4C2×C22⋊D21111yescharacteristic
252C2×C4C2×C22⋊D21111yescharacteristic
253C2×C4#1680996111yescharacteristic
254C8—8441no—
255C8—4222no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#2026994511yescharacteristic
264C22#2026994611yescharacteristic
265C22#2026994911yescharacteristic
266C22C2×#1680995211yescharacteristic
267C22#2026999711yescharacteristic
268C22#2027000711yescharacteristic
269C22#2027000811yescharacteristic
270C4—42no—
271C4—41no—
272C4—42no—
273C4—8442no—
274C4—8442no—
275C4#2026994511yescharacteristic
276C4#2026994811yescharacteristic
277C4#2026994911yescharacteristic
278C4#2026997311yescharacteristic
279C2—42no—
280C2#2721305411yescharacteristic
281C2#2721330111yescharacteristic
282C2#2721330211yescharacteristic
283C2#2721330611yescharacteristic
284C2#2721330811yescharacteristic
285C2#2721338511yescharacteristic
286C2#2721340211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×C22⋊Q8×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
hypercenterC2×C22⋊Q8D211
hypocenterC211#2001
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#16809952C2semidirect
3. C2×#20269901C2semidirect
4. C2×#20269900C2semidirect
6. #27212908C2semidirect
87. C23×C211Q16semidirect
143. C211#2001semidirect
219. C23#16809952semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
C2×#202699562
C2×#202699012
C2×#202699002
C2×C22⋊(C211⋊C8)2
#272129082
C2×C22⋊Q8×C2112
#2001211