Finite Group Databasebeta

← #41013598

Subgroups — #41013598

Order 27008.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20269935C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269909C211yesmaximal, characteristic, complemented
4C2×(C2×D211)⋊C8C211yesmaximal, characteristic
5#27213003C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269935—22no—
11#20269935—22no—
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×(C2×D211)⋊C4—21no—
15C2×(C2×D211)⋊C4—21no—
16C2×C4×Dic211—21no—
17#20269910—22no—
18#20269909—22no—
19(C2×D211)⋊C8C414yescomplemented
20C211⋊#95—21no—
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C23×D211D411yescharacteristic
25C22×Dic211—21no—
26C22×Dic211—21no—
27C22×Dic211D411yescharacteristic
28C2×C4×D211—21no—
29C2×C4×D211—21no—
30C2×C4×D211—21no—
31C2×C4×D211—21no—
32C2×C4×D211C2×C412yes—
33C2×C4×D211Q812yes—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8C2×C412yes—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211D412yes—
41(C2×D211)⋊C4—24no—
42(C2×D211)⋊C4—42no—
43C4×Dic211—24no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22×C4×C211C2311yescharacteristic
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211C2×C412yes—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D412yes—
54C22×D211—21no—
55C22×D211—21no—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211C4⋊C412yes—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—22no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211C4⋊C412yes—
70C4×D211—22no—
71C4×D211—22no—
72C4×D211—22no—
73C4×D211—22no—
74C211⋊C8—22no—
75C23×C211C2×D411yescharacteristic
76C2×C4×C211—22no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211C22×C411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×Q811yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C8×C211—22no—
89C22×C211C2×C4⋊C411yescharacteristic
90C22×C211#12011yescharacteristic
91C22×C211#12311yescharacteristic
92C22×C211#13511yescharacteristic
93C22×C211#13511yescharacteristic
94C22×C211AGL(1,ℤ/8)11yescharacteristic
95C22×C211#13711yescharacteristic
96C2×D211—22no—
97C2×D211—22no—
98C2×D211—22no—
99C2×D211—41no—
100C2×D211—41no—
101C2×D211—41no—
102C2×D211—41no—
103C4×C211—42no—
104C4×C211—81no—
105C4×C211C4×D411yescharacteristic
106C4×C211C4×D411yescharacteristic
107C4×C211C22⋊Q811yescharacteristic
108C4×C211C22⋊Q811yescharacteristic
109Dic211—22no—
110Dic211—42no—
111Dic211—81no—
112C2×C211#38511yescharacteristic
113C2×C211#42711yescharacteristic
114C2×C211#42811yescharacteristic
115C2×C211#44911yescharacteristic
116C2×C211#45211yescharacteristic
117C2×C211#48211yescharacteristic
118C2×C211#48511yescharacteristic
119D211—42no—
120C211#185411yescharacteristic, complemented, Sylow
121#1854—2111nomaximal, Sylow
122C2×#117—2111no—
123C2×#107—2111no—
124C2×#106—2111no—
125C2×C22⋊C8—2111no—
126#380—2111no—
127#340—2111no—
128#340D21111yescharacteristic, complemented
129C23×C4—2111no—
130C22×C8—2111no—
131C22×C8C2×D21111yescharacteristic
132#117—4222no—
133#117—4222no—
134C2×C4⋊C4—2111no—
135C2×C4⋊C4—2111no—
136C2×C4⋊C4C2×D21111yescharacteristic
137C2×C4⋊C4C2×D21111yescharacteristic
138C2×C22⋊C4—4221no—
139C2×C22⋊C4—4221no—
140C2×C42—4221no—
141#107—4222no—
142#106—4222no—
143C22⋊C8—2114no—
144#95—4221no—
145C24—2111no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—4221no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4—2111no—
155C22×C4—2112no—
156C22×C4—2112no—
157C22×C4C22×D21111yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—4221no—
161C2×C8—4221no—
162C2×C8—2112no—
163C2×C8C4×D21112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—4221no—
168C4⋊C4—4221no—
169C4⋊C4—4221no—
170C4⋊C4—2112no—
171C4⋊C4D84412yes—
172C22⋊C4—8442no—
173C22⋊C4—4224no—
174C42—4224no—
175C23—4221no—
176C23—4221no—
177C23—4221no—
178C23—4221no—
179C23—4221no—
180C23—4221no—
181C23—2112no—
182C23D4×D21111yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4—4222no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—4222no—
198C2×C4—4222no—
199C2×C4—4222no—
200C2×C4—4222no—
201C2×C4—2112no—
202C2×C4C2×C4×D21111yescharacteristic
203C2×C4D4×D21111yescharacteristic
204C2×C4C2×D84411yescharacteristic
205C2×C4Q8×D21111yescharacteristic
206C2×C4#1680996111yescharacteristic
207C2×C4#1680996211yescharacteristic
208C8—22no—
209C8—4222no—
210C22—8441no—
211C22—8441no—
212C22—8441no—
213C22—8441no—
214C22—4222no—
215C22—4222no—
216C22—4222no—
217C22C4⋊C4×D21111yescharacteristic
218C22#2026994711yescharacteristic
219C22#2026995911yescharacteristic
220C22#2026999811yescharacteristic
221C22#2026999811yescharacteristic
222C22#2027000111yescharacteristic
223C22#2027000911yescharacteristic
224C4—42no—
225C4—81no—
226C4—8442no—
227C4—16881no—
228C4—4222no—
229C4(C22⋊D211)⋊C411yescharacteristic
230C4C4×D84411yescharacteristic
231C4#2026995511yescharacteristic
232C4#2026995711yescharacteristic
233C2—8442no—
234C2#2721302011yescharacteristic
235C2#2721318111yescharacteristic
236C2#2721318711yescharacteristic
237C2#2721329211yescharacteristic
238C2#2721331111yescharacteristic
239C2#2721346911yescharacteristic
240C2#2721348511yescharacteristic
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#1854
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
2. C2×#20269910C2semidirect
3. C2×#20269909C2semidirect
6. #27212908C2semidirect
7. #340×C211C2semidirect
19. (C2×D211)⋊C8C4semidirect
120. C211#1854semidirect
128. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699352
C2×#202699102
C2×#202699092
C2×(C2×D211)⋊C82
#272130032
#272129082
#340×C2112
#1854211