Finite Group Databasebeta

← #41014407

Subgroups — #41014407

Order 27008.

392 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 91 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×#20269957C211yesmaximal, characteristic
3C2×#20269953C211yesmaximal, characteristic
4C2×#20269902C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7C2×#103×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×C4×D211C2211yescharacteristic
10C2×#16809952—22no—
11C2×#16809952—22no—
12C2×#16809952—22no—
13C22×C211⋊C8C2211yescharacteristic
14#20269957—22no—
15#20269957—22no—
16#20269953—22no—
17#20269953—22no—
18C2×C4⋊Dic211—21no—
19C2×C4⋊Dic211C2211yescharacteristic
20C2×Dic211⋊C4—21no—
21C2×(C2×D211)⋊C4—21no—
22C2×(C2×D211)⋊C4—21no—
23#20269902—22no—
24(C2×D211)⋊C8—22no—
25C22×Q8×C211C2211yescharacteristic
26C22×C8×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28#103×C211—22no—
29C23×D211D411yescharacteristic
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422D412yes—
36C22×Dic211—21no—
37C22×Dic211—21no—
38C22×Dic211D411yescharacteristic
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—21no—
42C2×C4×D211—21no—
43C2×C4×D211—21no—
44C2×C4×D211—21no—
45#16809952—28no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8—21no—
48C2×C211⋊C8D412yes—
49C4⋊Dic211—24no—
50C4⋊Dic211—21no—
51C4⋊Dic211D412yes—
52Dic211⋊C4—24no—
53(C2×D211)⋊C4—24no—
54(C2×D211)⋊C4—24no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211—21no—
58C2×Q8×C211—21no—
59C2×Q8×C211—21no—
60C2×Q8×C211D412yes—
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64C2×C8×C211—21no—
65C2×C8×C211—21no—
66C2×C8×C211—21no—
67C4⋊C4×C211—21no—
68C4⋊C4×C211D412yes—
69C22×D211—21no—
70C22×D211—21no—
71C22×D211—21no—
72C22×D211—21no—
73C22×D211—21no—
74C22×D211—21no—
75C22×D211Q1612yescomplemented
76Dic422—22no—
77Dic422—22no—
78Dic422—41no—
79Dic422—41no—
80C2×Dic211—22no—
81C2×Dic211—22no—
82C2×Dic211—21no—
83C2×Dic211—21no—
84C2×Dic211—21no—
85C2×Dic211—41no—
86C2×Dic211—41no—
87C2×Dic211—41no—
88C2×Dic211—41no—
89C4×D211—41no—
90C4×D211—41no—
91C4×D211—41no—
92C4×D211—41no—
93C211⋊C8—22no—
94C23×C211C2×D411yescharacteristic
95Q8×C211—22no—
96Q8×C211—22no—
97Q8×C211—41no—
98Q8×C211—41no—
99C2×C4×C211—22no—
100C2×C4×C211—22no—
101C2×C4×C211—41no—
102C2×C4×C211—41no—
103C2×C4×C211—41no—
104C2×C4×C211—41no—
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211#4111yescharacteristic
111C8×C211—41no—
112C22×C211#12011yescharacteristic
113C22×C211#12011yescharacteristic
114C22×C211#12111yescharacteristic
115C22×C211C2×Q1611yescharacteristic
116C22×C211#13511yescharacteristic
117C22×C211#13711yescharacteristic
118C22×C211#13711yescharacteristic
119C2×D211—22no—
120C2×D211—22no—
121C2×D211—22no—
122C2×D211—41no—
123C2×D211—41no—
124C2×D211—41no—
125C2×D211—41no—
126C4×C211—42no—
127C4×C211—42no—
128C4×C211#12011yescharacteristic
129C4×C211#12111yescharacteristic
130C4×C211#12111yescharacteristic
131C4×C211#12711yescharacteristic
132Dic211—41no—
133Dic211—42no—
134Dic211—42no—
135C2×C211#39211yescharacteristic
136C2×C211#45111yescharacteristic
137C2×C211#45111yescharacteristic
138C2×C211#45211yescharacteristic
139C2×C211#45211yescharacteristic
140C2×C211#46711yescharacteristic
141C2×C211#47011yescharacteristic
142D211—42no—
143C211#200111yescharacteristic, complemented, Sylow
144#2001—2111nomaximal, Sylow
145C22×Q16—2111no—
146C2×C22⋊Q8—2111no—
147C2×C22⋊Q8—2111no—
148C2×#103—2111no—
149C2×#103D21111yescharacteristic, complemented
150C2×C22⋊C8—2111no—
151#340—2111no—
152C22×Q8—2111no—
153C22×Q8C2×D21111yescharacteristic
154C23×C4—2111no—
155C2×Q16—4222no—
156C2×Q16—4222no—
157C2×Q16—4222no—
158C22×C8—2111no—
159C22×C8C2×D21111yescharacteristic
160C22⋊Q8—4222no—
161C22⋊Q8—4222no—
162C22⋊Q8—4222no—
163C22⋊Q8—4222no—
164C2×C4⋊C4—4221no—
165C2×C4⋊C4—4221no—
166C2×C4⋊C4—2111no—
167C2×C4⋊C4C2×D21111yescharacteristic
168C2×C22⋊C4—4221no—
169C2×C22⋊C4—4221no—
170#103—22no—
171#103—4222no—
172C22⋊C8—4222no—
173C24—2111no—
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×Q8D84412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—4221no—
189C22×C4—4221no—
190C22×C4—4221no—
191C22×C4—4221no—
192C22×C4—4221no—
193C22×C4—4221no—
194C22×C4—4221no—
195C22×C4—4221no—
196C22×C4—2111no—
197C22×C4C22×D21111yescharacteristic
198Q16—4228no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—4221no—
203C2×C8—4221no—
204C2×C8—2112no—
205C4⋊C4—21no—
206C4⋊C4—4224no—
207C4⋊C4—4224no—
208C4⋊C4—4221no—
209C4⋊C4—2112no—
210C4⋊C4C22⋊D21112yes—
211C22⋊C4—4224no—
212C22⋊C4—4224no—
213C23—4221no—
214C23—4221no—
215C23—4221no—
216C23—4221no—
217C23—4221no—
218C23—4221no—
219C23—2112no—
220C23D4×D21111yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—8441no—
226Q8—8441no—
227Q8—4222no—
228Q8—4222no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—8441no—
238C2×C4—8441no—
239C2×C4—8441no—
240C2×C4—8441no—
241C2×C4—8441no—
242C2×C4—8441no—
243C2×C4—4222no—
244C2×C4—4221no—
245C2×C4—4222no—
246C2×C4—4221no—
247C2×C4—4221no—
248C2×C4D4×D21111yescharacteristic
249C2×C4D4×D21111yescharacteristic
250C2×C4D4×D21111yescharacteristic
251C2×C4C2×C22⋊D21111yescharacteristic
252C2×C4C2×D84411yescharacteristic
253C2×C4#1680996211yescharacteristic
254C8—41no—
255C8—4222no—
256C22—8441no—
257C22—8441no—
258C22—8441no—
259C22—8441no—
260C22—4222no—
261C22—4222no—
262C22—4222no—
263C22#2026994511yescharacteristic
264C22#2026994711yescharacteristic
265C22#2026995011yescharacteristic
266C22Q16×D21111yescharacteristic
267C22#2026999811yescharacteristic
268C22#2027000611yescharacteristic
269C22#2027000911yescharacteristic
270C4—42no—
271C4—42no—
272C4—8441no—
273C4—8442no—
274C4—8442no—
275C4#2026994711yescharacteristic
276C4#2026995011yescharacteristic
277C4#2026995111yescharacteristic
278C4#2026997311yescharacteristic
279C2—8442no—
280C2#2721305511yescharacteristic
281C2#2721330011yescharacteristic
282C2#2721330311yescharacteristic
283C2#2721330511yescharacteristic
284C2#2721331111yescharacteristic
285C2#2721338611yescharacteristic
286C2#2721340311yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
hypercenterC2×#103D211
hypocenterC211#2001
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#16809952C2semidirect
4. C2×#20269902C2semidirect
6. #27212908C2semidirect
7. C2×#103×C211C2semidirect
75. C22×D211Q16semidirect
143. C211#2001semidirect
149. C2×#103D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
C2×#202699572
C2×#202699532
C2×#202699022
C2×(C2×D211)⋊C82
#272129082
C2×#103×C2112
#2001211