Finite Group Databasebeta

← #41013704

Subgroups — #41013704

Order 27008.

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

normal subgroups only

This group has 404 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
1C2×#20269957C211yesmaximal, characteristic
2C2×C4⋊C4×D211C211yesmaximal, characteristic
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×C4×D211—21no—
10C22×C4×D211C2211yescharacteristic
11C22×C211⋊C8C2211yescharacteristic
12#20269957—22no—
13#20269957C414yescomplemented
14C2×C4⋊Dic211—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4—21no—
17C4⋊C4×D211—22no—
18C4⋊C4×D211—22no—
19C2×(C2×D211)⋊C4—21no—
20#20269902—22no—
21#20269900—22no—
22(C2×D211)⋊C8—22no—
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×C412yescomplemented
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D411yescharacteristic
36C2×C4×D211—24no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—42no—
42C2×C4×D211—42no—
43C2×C4×D211C2×C412yes—
44C2×C4×D211D412yes—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8—21no—
48C4⋊Dic211—24no—
49C4⋊Dic211—21no—
50C4⋊Dic211D412yes—
51Dic211⋊C4—24no—
52(C2×D211)⋊C4—24no—
53C22×C4×C211—21no—
54C22×C4×C211—21no—
55C22×C4×C211C2311yescharacteristic
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211—21no—
61C4⋊C4×C211C2×C412yescomplemented
62C4⋊C4×C211D412yes—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—21no—
66C22×D211—21no—
67C22×D211C22⋊C412yes—
68C22×D211SD1612yes—
69C22×D211Q1612yes—
70Dic422—22no—
71Dic422—22no—
72Dic422—41no—
73Dic422—41no—
74C2×Dic211—21no—
75C2×Dic211—21no—
76C2×Dic211—22no—
77C2×Dic211—22no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211—41no—
81C2×Dic211—41no—
82C2×Dic211C22⋊C412yes—
83C4×D211—22no—
84C4×D211—22no—
85C4×D211—22no—
86C4×D211—22no—
87C4×D211—48no—
88C211⋊C8—41no—
89C23×C211C2×D411yescharacteristic
90C2×C4×C211—22no—
91C2×C4×C211—22no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211C22×C411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C2×C4×C211#4111yescharacteristic
102C8×C211—41no—
103C22×C211C2×C22⋊C411yescharacteristic
104C22×C211#12011yescharacteristic
105C22×C211#12311yescharacteristic
106C22×C211C2×SD1611yescharacteristic
107C22×C211C2×Q1611yescharacteristic
108C22×C211AGL(1,ℤ/8)11yescharacteristic
109C22×C211#13711yescharacteristic
110C2×D211—22no—
111C2×D211—22no—
112C2×D211—22no—
113C2×D211—22no—
114C2×D211—22no—
115C2×D211—22no—
116C2×D211#10314yescomplemented
117C4×C211—42no—
118C4×C211—42no—
119C4×C211C4×D411yescharacteristic
120C4×C211C4×D411yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12111yescharacteristic
123Dic211—22no—
124Dic211—42no—
125Dic211—42no—
126C2×C211#38611yescharacteristic
127C2×C211C2×#10311yescharacteristic
128C2×C211#41711yescharacteristic
129C2×C211#45011yescharacteristic
130C2×C211#45111yescharacteristic
131C2×C211#48311yescharacteristic
132C2×C211#48411yescharacteristic
133D211—24no—
134C211#187211yescharacteristic, complemented, Sylow
135#1872—2111nomaximal, Sylow
136C2×C22⋊Q8—2111no—
137C22×C4⋊C4—2111no—
138C2×#103—2111no—
139C2×#103—2111no—
140C2×C22⋊C8—2111no—
141#340—2111no—
142#340D21111yescharacteristic, complemented
143C22×Q8—2111no—
144C23×C4—4221no—
145C23×C4—2111no—
146C22×C8—2111no—
147C22×C8C2×D21111yescharacteristic
148C22⋊Q8—4222no—
149C22⋊Q8—2114no—
150C2×C4⋊C4—4222no—
151C2×C4⋊C4—4221no—
152C2×C4⋊C4—4221no—
153C2×C4⋊C4—4222no—
154C2×C4⋊C4—2111no—
155C2×C4⋊C4C2×D21111yescharacteristic
156C2×C4⋊C4C2×D21111yescharacteristic
157C2×C22⋊C4—4221no—
158#103—4222no—
159#103—4222no—
160C22⋊C8—4222no—
161C24—2111no—
162C2×Q8—4221no—
163C2×Q8—4221no—
164C2×Q8—4221no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—2112no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—8442no—
171C22×C4—8442no—
172C22×C4—4224no—
173C22×C4—4221no—
174C22×C4—4221no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4—4221no—
178C22×C4—4221no—
179C22×C4—2111no—
180C22×C4—2112no—
181C22×C4—2112no—
182C22×C4C22×D21111yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—4221no—
187C2×C8—4221no—
188C2×C8—4221no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—4224no—
192C4⋊C4—4224no—
193C4⋊C4—4221no—
194C4⋊C4—2112no—
195C4⋊C4C4×D21112yescomplemented
196C4⋊C4D84412yes—
197C22⋊C4—4224no—
198C23—4221no—
199C23—4221no—
200C23—4221no—
201C23—4221no—
202C23—2112no—
203C23—2112no—
204C23—2112no—
205C23D4×D21111yescharacteristic
206Q8—8441no—
207Q8—8441no—
208Q8—4222no—
209Q8—4222no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—8448no—
217C2×C4—8441no—
218C2×C4—8441no—
219C2×C4—8441no—
220C2×C4—8441no—
221C2×C4—4222no—
222C2×C4—4222no—
223C2×C4—4221no—
224C2×C4—4221no—
225C2×C4—4222no—
226C2×C4—4222no—
227C2×C4—4222no—
228C2×C4—4222no—
229C2×C4—2112no—
230C2×C4C2×C4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4C2×D84411yescharacteristic
234C2×C4#1680996211yescharacteristic
235C2×C4#1680996311yescharacteristic
236C8—41no—
237C8—8441no—
238C22—4222no—
239C22—4222no—
240C22—4222no—
241C22—4222no—
242C22—4222no—
243C22—4222no—
244C22—2114no—
245C22C22⋊C4×D21111yescharacteristic
246C22#2026994711yescharacteristic
247C22#2026996211yescharacteristic
248C22SD16×D21111yescharacteristic
249C22Q16×D21111yescharacteristic
250C22#2027000111yescharacteristic
251C22#2027000611yescharacteristic
252C4—42no—
253C4—42no—
254C4—8442no—
255C4—8442no—
256C4—4222no—
257C4#2026993911yescharacteristic
258C4C4×D84411yescharacteristic
259C4#2026995011yescharacteristic
260C4#2026995211yescharacteristic
261C2—4224no—
262C2#2721302411yescharacteristic
263C2#103×D21111yescharacteristic
264C2#2721314411yescharacteristic
265C2#2721329811yescharacteristic
266C2#2721330311yescharacteristic
267C2#2721347811yescharacteristic
268C2#2721348311yescharacteristic
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#1872
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
13. #20269957C4semidirect
32. C2×Dic422C2×C4semidirect
61. C4⋊C4×C211C2×C4semidirect
116. C2×D211#103semidirect
134. C211#1872semidirect
142. #340D211semidirect
195. C4⋊C4C4×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699572
C2×C4⋊C4×D2112
C2×#202699022
C2×#202699002
C2×(C2×D211)⋊C82
#272129082
#340×C2112
#1872211