Finite Group Databasebeta

← #41013595

Subgroups — #41013595

Order 27008.

316 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 105 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×#20269936C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269908C211yesmaximal, characteristic, complemented
4C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#380×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269936—22no—
11#20269936—22no—
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211—21no—
15C2×C4×Dic211—21no—
16#20269910—22no—
17#20269908—22no—
18C22⋊(C211⋊C8)C414yescomplemented
19C23×C4×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C22⋊C4×C211—21no—
23#95×C211—21no—
24C22×Dic211—21no—
25C22×Dic211—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8C2×C412yes—
31C2×C211⋊C8C2×C412yes—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D412yes—
35C4⋊Dic211D412yes—
36C22⋊Dic211—24no—
37C4×Dic211—24no—
38C24×C211D411yescharacteristic
39C22×C4×C211—21no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211—21no—
45C22×C4×C211C2×C412yes—
46C22×C4×C211D411yescharacteristic
47C22×C4×C211Q812yes—
48C22×C4×C211C2311yescharacteristic
49C4⋊C4×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C22⋊C4×C211—42no—
54C2×Dic211—22no—
55C2×Dic211—22no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C211⋊C8—22no—
61C211⋊C8—22no—
62C23×C211—21no—
63C23×C211—21no—
64C23×C211—21no—
65C23×C211—21no—
66C23×C211—21no—
67C23×C211—21no—
68C23×C211C4⋊C412yes—
69C23×C211C2×D411yescharacteristic
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—22no—
73C2×C4×C211—22no—
74C2×C4×C211—22no—
75C2×C4×C211—22no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211C4⋊C412yes—
83C2×C4×C211C22×C411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×Q811yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C22×C211—22no—
90C22×C211—22no—
91C22×C211—22no—
92C22×C211—41no—
93C22×C211—41no—
94C22×C211—41no—
95C22×C211—41no—
96C22×C211C2×C4⋊C411yescharacteristic
97C22×C211#12011yescharacteristic
98C22×C211#12311yescharacteristic
99C22×C211#13511yescharacteristic
100C22×C211#13511yescharacteristic
101C22×C211AGL(1,ℤ/8)11yescharacteristic
102C22×C211#13711yescharacteristic
103C4×C211—22no—
104C4×C211—81no—
105C4×C211—81no—
106C4×C211C4×D411yescharacteristic
107C4×C211C4×D411yescharacteristic
108C4×C211C22⋊Q811yescharacteristic
109C4×C211C22⋊Q811yescharacteristic
110Dic211—42no—
111Dic211—42no—
112C2×C211—42no—
113C2×C211#38511yescharacteristic
114C2×C211#42711yescharacteristic
115C2×C211#42811yescharacteristic
116C2×C211#44911yescharacteristic
117C2×C211#45211yescharacteristic
118C2×C211#48211yescharacteristic
119C2×C211#48511yescharacteristic
120C211#185411yescharacteristic, complemented, Sylow
121#1854—2111nomaximal, Sylow
122C2×#117—2111no—
123C2×#107—2111no—
124C2×#106—2111no—
125C2×C22⋊C8—2111no—
126#380D21111yescharacteristic
127#340—2111no—
128#340—2111no—
129C23×C4C2×D21111yescharacteristic
130C22×C8—2111no—
131C22×C8—2111no—
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—21no—
139C2×C22⋊C4—4221no—
140C2×C42—4221no—
141#107—4222no—
142#106—4222no—
143C22⋊C8—2114no—
144#95—21no—
145C24C22⋊D21111yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4C4×D21112yes—
155C22×C4C22⋊D21111yescharacteristic
156C22×C4Dic42212yes—
157C22×C4C22×D21111yescharacteristic
158C2×C8—4221no—
159C2×C8—4221no—
160C2×C8—4221no—
161C2×C8—4221no—
162C2×C8—2112no—
163C2×C8—2112no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—4221no—
169C4⋊C4—4221no—
170C4⋊C4—2112no—
171C4⋊C4—2112no—
172C22⋊C4—42no—
173C22⋊C4—4224no—
174C42—4224no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23Dic211⋊C412yes—
182C23C2×C22⋊D21111yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—8441no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—4222no—
200C2×C4—4222no—
201C2×C4Dic211⋊C412yes—
202C2×C4C2×C4×D21111yescharacteristic
203C2×C4D4×D21111yescharacteristic
204C2×C4D4×D21111yescharacteristic
205C2×C4C2×Dic42211yescharacteristic
206C2×C4#1680996111yescharacteristic
207C2×C4#1680996111yescharacteristic
208C8—4222no—
209C8—4222no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×Dic211⋊C411yescharacteristic
218C22#2026994511yescharacteristic
219C22#2026995811yescharacteristic
220C22#2026999711yescharacteristic
221C22#2026999711yescharacteristic
222C22#2027000311yescharacteristic
223C22#2027000811yescharacteristic
224C4—22no—
225C4—81no—
226C4—81no—
227C4—8442no—
228C4—8442no—
229C4(C22⋊D211)⋊C411yescharacteristic
230C4(C22⋊D211)⋊C411yescharacteristic
231C4#2026995411yescharacteristic
232C4#2026995411yescharacteristic
233C2—42no—
234C2#2721301811yescharacteristic
235C2#2721318211yescharacteristic
236C2#2721318611yescharacteristic
237C2#2721328911yescharacteristic
238C2#2721330611yescharacteristic
239C2#2721346711yescharacteristic
240C2#2721348411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenter#380D211
hypocenterC211#1854
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#20269910C2semidirect
3. C2×#20269908C2semidirect
5. #27212908C2semidirect
6. #27212908C2semidirect
18. C22⋊(C211⋊C8)C4semidirect
120. C211#1854semidirect

Maximal subgroups

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

subgroupindex
C2×#202699362
C2×#202699102
C2×#202699082
C2×C22⋊(C211⋊C8)2
#272129082
#380×C2112
#1854211