Finite Group Databasebeta

← #41013820

Subgroups — #41013820

Order 27008.

380 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 107 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×C8×D211C211yesmaximal, characteristic
2C2×#20269957C211yesmaximal, characteristic
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×#20269900C211yesmaximal, characteristic, complemented
5#27213003C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×C4×D211C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C8×D211—22no—
12C2×C8×D211—22no—
13C2×C8×D211—22no—
14#20269957—22no—
15#20269957C414yescomplemented
16C2×C4⋊Dic211—21no—
17C2×C4⋊Dic211C2211yescharacteristic
18C2×(C2×D211)⋊C4—21no—
19C2×(C2×D211)⋊C4—21no—
20#20269902—22no—
21#20269900—22no—
22C211⋊#95—21no—
23C22×C8×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26C23×D211D411yescharacteristic
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422C2×C412yes—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D411yescharacteristic
36C2×C4×D211—21no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211C2×C412yes—
41C2×C4×D211D412yes—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8D412yes—
45C8×D211—28no—
46C4⋊Dic211—24no—
47C4⋊Dic211—21no—
48C4⋊Dic211—21no—
49(C2×D211)⋊C4—24no—
50(C2×D211)⋊C4—42no—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C2×C8×C211D412yes—
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211C2×C412yes—
61C22×D211—21no—
62C22×D211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211C22⋊C412yes—
66C22×D211SD1612yes—
67C22×D211Q1612yes—
68Dic422—22no—
69Dic422—22no—
70Dic422—41no—
71Dic422—41no—
72C2×Dic211—21no—
73C2×Dic211—21no—
74C2×Dic211—22no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211C22⋊C412yes—
81C4×D211—22no—
82C4×D211—22no—
83C4×D211—22no—
84C4×D211—22no—
85C211⋊C8—22no—
86C23×C211C2×D411yescharacteristic
87C2×C4×C211—22no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211C22×C411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C8×C211—22no—
100C22×C211C2×C22⋊C411yescharacteristic
101C22×C211#12111yescharacteristic
102C22×C211#12111yescharacteristic
103C22×C211C2×SD1611yescharacteristic
104C22×C211C2×Q1611yescharacteristic
105C22×C211#13511yescharacteristic
106C22×C211#13511yescharacteristic
107C2×D211—22no—
108C2×D211—22no—
109C2×D211—22no—
110C2×D211—22no—
111C2×D211—22no—
112C2×D211—22no—
113C2×D211#10314yescomplemented
114C4×C211—42no—
115C4×C211—81no—
116C4×C211C4×D411yescharacteristic
117C4×C211C4×D411yescharacteristic
118C4×C211#12011yescharacteristic
119C4×C211#12311yescharacteristic
120Dic211—22no—
121Dic211—42no—
122Dic211—81no—
123C2×C211#38611yescharacteristic
124C2×C211C2×#10311yescharacteristic
125C2×C211#41611yescharacteristic
126C2×C211#46511yescharacteristic
127C2×C211#46511yescharacteristic
128C2×C211#46711yescharacteristic
129C2×C211#46711yescharacteristic
130D211—24no—
131C211#189211yescharacteristic, complemented, Sylow
132#1892—2111nomaximal, Sylow
133C23×C8—2111no—
134C2×C22⋊Q8—2111no—
135C2×#103—2111no—
136C2×#103—2111no—
137#380—2111no—
138#340—2111no—
139#340D21111yescharacteristic, complemented
140C22×Q8—2111no—
141C23×C4—2111no—
142C22×C8—4222no—
143C22×C8—4222no—
144C22×C8—4222no—
145C22×C8—2111no—
146C22×C8C2×D21111yescharacteristic
147C22⋊Q8—4222no—
148C22⋊Q8—2114no—
149C2×C4⋊C4—4221no—
150C2×C4⋊C4—2111no—
151C2×C4⋊C4C2×D21111yescharacteristic
152C2×C4⋊C4C2×D21111yescharacteristic
153C2×C22⋊C4—4221no—
154C2×C22⋊C4—4221no—
155#103—4222no—
156#103—4222no—
157#95—4221no—
158C24—2111no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—4221no—
162C2×Q8—4221no—
163C2×Q8—4221no—
164C2×Q8—2112no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—4221no—
168C22×C4—4221no—
169C22×C4—4221no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4—4221no—
173C22×C4—2111no—
174C22×C4—2112no—
175C22×C4—2112no—
176C22×C4C22×D21111yescharacteristic
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—4228no—
180C2×C8—4221no—
181C2×C8—4221no—
182C2×C8—2112no—
183C2×C8D84412yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—4224no—
188C4⋊C4—4221no—
189C4⋊C4—4221no—
190C4⋊C4C4×D21112yes—
191C22⋊C4—8442no—
192C22⋊C4—4224no—
193C23—4221no—
194C23—4221no—
195C23—4221no—
196C23—4221no—
197C23—2112no—
198C23—2112no—
199C23—2112no—
200C23D4×D21111yescharacteristic
201Q8—8441no—
202Q8—8441no—
203Q8—4222no—
204Q8—4222no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—8441no—
216C2×C4—4222no—
217C2×C4—4221no—
218C2×C4—4221no—
219C2×C4—4222no—
220C2×C4—4222no—
221C2×C4—4222no—
222C2×C4—4222no—
223C2×C4—2112no—
224C2×C4C2×C4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4D4×D21111yescharacteristic
227C2×C4C2×D84411yescharacteristic
228C2×C4#1680996211yescharacteristic
229C2×C4#1680996311yescharacteristic
230C8—22no—
231C8—4222no—
232C22—4222no—
233C22—4222no—
234C22—4222no—
235C22—4222no—
236C22—4222no—
237C22—4222no—
238C22—2114no—
239C22C22⋊C4×D21111yescharacteristic
240C22#2026995011yescharacteristic
241C22#2026995211yescharacteristic
242C22SD16×D21111yescharacteristic
243C22Q16×D21111yescharacteristic
244C22#2026999611yescharacteristic
245C22#2026999811yescharacteristic
246C4—42no—
247C4—81no—
248C4—8442no—
249C4—16881no—
250C4—4222no—
251C4#2026993911yescharacteristic
252C4C4×D84411yescharacteristic
253C4#2026994711yescharacteristic
254C4#2026996211yescharacteristic
255C2—4224no—
256C2#2721302411yescharacteristic
257C2#103×D21111yescharacteristic
258C2#2721313911yescharacteristic
259C2#2721337411yescharacteristic
260C2#2721337611yescharacteristic
261C2#2721338611yescharacteristic
262C2#2721338811yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
cosocleC22×C4×C211C23
hypercenter#340D211
hypocenterC211#1892
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
3. C2×#20269902C2semidirect
4. C2×#20269900C2semidirect
6. #27212908C2semidirect
7. #340×C211C2semidirect
15. #20269957C4semidirect
113. C2×D211#103semidirect
131. C211#1892semidirect
139. #340D211semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D2112
C2×#202699572
C2×#202699022
C2×#202699002
#272130032
#272129082
#340×C2112
#1892211