Finite Group Databasebeta

← #41013852

Subgroups — #41013852

Order 27008.

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

normal subgroups only

This group has 598 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
1C22×C22⋊Dic211C211yesmaximal, characteristic, complemented
2C2×#20269892C212yesmaximal, complemented
3#27213002C211yesmaximal, characteristic, complemented
4#27212913C212yesmaximal, complemented
5C2×#120×C211C211yesmaximal, characteristic
6C23×Dic211—21no—
7C2×C22⋊Dic211—24no—
8C2×C22⋊Dic211—21no—
9C2×C22⋊Dic211—22no—
10C2×C22⋊Dic211C2212yes—
11C2×C22⋊Dic211C2212yes—
12#20269892—24no—
13C25×C211C2211yescharacteristic
14C22×D4×C211C412yescomplemented
15C22×D4×C211C2211yescharacteristic
16#120×C211—21no—
17#120×C211—22no—
18#120×C211—21no—
19C2×C22⋊C4×C211C412yescomplemented
20C2×C22⋊C4×C211C2211yescharacteristic
21C22×Dic211—24no—
22C22×Dic211—22no—
23C22×Dic211—22no—
24C22×Dic211—44no—
25C22⋊Dic211—24no—
26C22⋊Dic211—22no—
27C22⋊Dic211—22no—
28C22⋊Dic211—22no—
29C22⋊Dic211—44no—
30C22⋊Dic211D414yescomplemented
31C24×C211—22no—
32C24×C211—22no—
33C24×C211—44no—
34C24×C211C2×C412yes—
35C24×C211C2×C411yescharacteristic, complemented
36C24×C211D411yescharacteristic
37C24×C211D412yes—
38C24×C211D411yescharacteristic
39C24×C211C2311yescharacteristic
40C2×D4×C211—22no—
41C2×D4×C211—22no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—22no—
45C2×D4×C211—22no—
46C2×D4×C211—44no—
47C22×C4×C211C2×C411yescharacteristic, complemented
48C22×C4×C211C2×C412yescomplemented
49C22⋊C4×C211—22no—
50C22⋊C4×C211—22no—
51C22⋊C4×C211—21no—
52C22⋊C4×C211—21no—
53C2×Dic211—24no—
54C2×Dic211—48no—
55C2×Dic211—44no—
56C2×Dic211—44no—
57C2×Dic211—42no—
58C23×C211—24no—
59C23×C211—21no—
60C23×C211—21no—
61C23×C211—24no—
62C23×C211—24no—
63C23×C211—24no—
64C23×C211—21no—
65C23×C211—24no—
66C23×C211—21no—
67C23×C211—21no—
68C23×C211—22no—
69C23×C211—42no—
70C23×C211—42no—
71C23×C211—42no—
72C23×C211—42no—
73C23×C211—48no—
74C23×C211—48no—
75C23×C211—42no—
76C23×C211C22⋊C412yes—
77C23×C211C22⋊C412yes—
78C23×C211C22×C411yescharacteristic
79C23×C211C2×D412yes—
80C23×C211C2×D411yescharacteristic
81C23×C211C2×D411yescharacteristic
82C23×C211#4112yes—
83D4×C211—44no—
84D4×C211—44no—
85D4×C211—41no—
86D4×C211—42no—
87D4×C211—41no—
88C2×C4×C211—21no—
89C2×C4×C211—22no—
90C2×C4×C211—22no—
91C2×C4×C211—42no—
92C22×C211—24no—
93C22×C211—24no—
94C22×C211—22no—
95C22×C211—22no—
96C22×C211—22no—
97C22×C211—22no—
98C22×C211—22no—
99C22×C211—22no—
100C22×C211—22no—
101C22×C211—22no—
102C22×C211—42no—
103C22×C211—42no—
104C22×C211—48no—
105C22×C211—48no—
106C22×C211—42no—
107C22×C211—42no—
108C22×C211—41no—
109C22×C211—42no—
110C22×C211—82no—
111C22×C211#9914yescomplemented
112C22×C211C2×C22⋊C411yescharacteristic
113C22×C211C4×D412yes—
114C22×C211#12011yescharacteristic
115C22×C211#12112yes—
116C22×C211#12311yescharacteristic
117C4×C211—41no—
118C4×C211—42no—
119Dic211—44no—
120Dic211—82no—
121C2×C211—21no—
122C2×C211—22no—
123C2×C211—24no—
124C2×C211—21no—
125C2×C211—22no—
126C2×C211—42no—
127C2×C211—81no—
128C2×C211#38611yescharacteristic
129C2×C211C2×#9912yes—
130C211#189811yescharacteristic, complemented, Sylow
131#1898—2111nomaximal, Sylow
132C2×#120D21111yescharacteristic
133C22×C22⋊C4—2111no—
134C2×#99—2112no—
135#379—2111no—
136#342—2112no—
137C25C2×D21111yescharacteristic
138C22×D4Dic21112yescomplemented
139C22×D4C2×D21111yescharacteristic
140C23×C4—4221no—
141#120—21no—
142#120—22no—
143#120—21no—
144C2×C22⋊C4—4222no—
145C2×C22⋊C4—4221no—
146C2×C22⋊C4—4224no—
147C2×C22⋊C4—2112no—
148C2×C22⋊C4—2112no—
149C2×C22⋊C4Dic21112yescomplemented
150C2×C22⋊C4C2×D21111yescharacteristic
151#99—4224no—
152C24—22no—
153C24—22no—
154C24—44no—
155C24C2×Dic21112yes—
156C24C2×Dic21111yescharacteristic, complemented
157C24C22⋊D21111yescharacteristic
158C24C22⋊D21112yes—
159C24C22⋊D21111yescharacteristic
160C24C22×D21111yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—44no—
168C22×C4—8444no—
169C22×C4—4222no—
170C22×C4—4224no—
171C22×C4—4222no—
172C22×C4C2×Dic21111yescharacteristic, complemented
173C22×C4C2×Dic21112yescomplemented
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—8444no—
179C22⋊C4—4222no—
180C22⋊C4—4222no—
181C22⋊C4—4224no—
182C22⋊C4—4222no—
183C22⋊C4—2114no—
184C23—24no—
185C23—21no—
186C23—21no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—21no—
191C23—24no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—48no—
200C23—48no—
201C23—42no—
202C23C22⋊Dic21112yes—
203C23C22⋊Dic21112yes—
204C23C22×Dic21111yescharacteristic
205C23D4×D21112yes—
206C23C2×C22⋊D21111yescharacteristic
207C23C2×C22⋊D21111yescharacteristic
208C23#1680996112yes—
209D4—44no—
210D4—44no—
211D4—41no—
212D4—42no—
213D4—41no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—8448no—
219C2×C4—8444no—
220C2×C4—8444no—
221C2×C4—8442no—
222C2×C4—4224no—
223C22—24no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—48no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—82no—
242C22#2026989214yescomplemented
243C22C2×C22⋊Dic21111yescharacteristic
244C22D4×Dic21112yes—
245C22#2026994511yescharacteristic
246C22#2026994912yes—
247C22#2026995811yescharacteristic
248C4—42no—
249C4—41no—
250C4—8444no—
251C4—16882no—
252C2—21no—
253C2—22no—
254C2—24no—
255C2—21no—
256C2—22no—
257C2—42no—
258C2—81no—
259C2#2721302611yescharacteristic
260C2C2×#2026989212yes—
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting, prime radicalC2×#120×C211C2
socleC22×C211C2×C22⋊C4
cosocleC24×C211C23
hypercenterC2×#120D211
hypocenterC211#1898
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C25 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊Dic211C2semidirect
2. C2×#20269892C2semidirect
3. #27213002C2semidirect
4. #27212913C2semidirect
14. C22×D4×C211C4semidirect
19. C2×C22⋊C4×C211C4semidirect
30. C22⋊Dic211D4semidirect
35. C24×C211C2×C4semidirect
47. C22×C4×C211C2×C4semidirect
48. C22×C4×C211C2×C4semidirect
111. C22×C211#99semidirect
130. C211#1898semidirect
138. C22×D4Dic211semidirect
149. C2×C22⋊C4Dic211semidirect
156. C24C2×Dic211semidirect
172. C22×C4C2×Dic211semidirect
173. C22×C4C2×Dic211semidirect
242. C22#20269892semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic2112
C2×#202698922
#272130022
#272129132
C2×#120×C2112
#1898211