Finite Group Databasebeta

← #41013701

Subgroups — #41013701

Order 27008.

376 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16809946C211yesmaximal, characteristic, complemented
2C2×#20269904C212yesmaximal, complemented
3C4×C22⋊Dic211C211yesmaximal, characteristic, complemented
4#27212905C212yesmaximal, complemented
5#525×C211C211yesmaximal, characteristic
6C2×#16809946—24no—
7C2×#16809946—21no—
8C2×#16809946C2212yes—
9C22×C211⋊C8—21no—
10C2×C22⋊Dic211—21no—
11C2×C4×Dic211C2212yes—
12#20269904—24no—
13C211⋊#95—21no—
14C2×#41×C211C2211yescharacteristic
15C23×C4×C211C2211yescharacteristic
16#123×C211—21no—
17C22⋊Q8×C211C411yescharacteristic, complemented
18C22⋊Q8×C211C411yescharacteristic, complemented
19#121×C211C411yescharacteristic, complemented
20#121×C211C411yescharacteristic, complemented
21#120×C211—21no—
22C4×D4×C211—21no—
23C4×D4×C211—21no—
24#117×C211C2211yescharacteristic
25C22×Dic211—22no—
26#16809946—24no—
27#16809946—22no—
28#16809946D414yescomplemented
29C2×C211⋊C8—24no—
30C2×C211⋊C8—22no—
31C22⋊Dic211—24no—
32C4×Dic211—22no—
33C4×Dic211—22no—
34C24×C211D411yescharacteristic
35#41×C211—22no—
36#41×C211—22no—
37C2×Q8×C211C2×C411yescharacteristic
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211C2×C411yescharacteristic
42C22×C4×C211—22no—
43C22×C4×C211—22no—
44C22×C4×C211—21no—
45C22×C4×C211C2×C411yescharacteristic
46C22×C4×C211C2×C411yescharacteristic
47C22×C4×C211D411yescharacteristic
48C22×C4×C211D411yescharacteristic
49C22×C4×C211D411yescharacteristic
50C22×C4×C211C2311yescharacteristic
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211C2×C411yescharacteristic
54C4⋊C4×C211C2×C411yescharacteristic
55C22⋊C4×C211—21no—
56C22⋊C4×C211—21no—
57C22⋊C4×C211—21no—
58C22⋊C4×C211—21no—
59C22⋊C4×C211—21no—
60C42×C211—21no—
61C2×Dic211—24no—
62C2×Dic211—42no—
63C2×Dic211—42no—
64C211⋊C8—24no—
65C23×C211—22no—
66C23×C211—22no—
67C23×C211—21no—
68C23×C211—41no—
69C23×C211C22⋊C411yescharacteristic
70C23×C211C22⋊C411yescharacteristic
71C23×C211C2×D411yescharacteristic
72Q8×C211—22no—
73D4×C211—22no—
74D4×C211—42no—
75D4×C211—42no—
76D4×C211—42no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—22no—
80C2×C4×C211—21no—
81C2×C4×C211—21no—
82C2×C4×C211—21no—
83C2×C4×C211—21no—
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—21no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211C22⋊C411yescharacteristic
90C2×C4×C211C22⋊C411yescharacteristic
91C2×C4×C211C22×C411yescharacteristic
92C2×C4×C211C2×D412yes—
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211#4112yes—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—21no—
100C22×C211—21no—
101C22×C211—21no—
102C22×C211—44no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211—41no—
106C22×C211#10412yescomplemented
107C22×C211C2×C22⋊C411yescharacteristic
108C22×C211#12112yes—
109C4×C211—21no—
110C4×C211—21no—
111C4×C211—41no—
112C4×C211—41no—
113C4×C211—41no—
114C4×C211C4×D412yes—
115C4×C211#12011yescharacteristic
116C4×C211#12311yescharacteristic
117Dic211—44no—
118C2×C211—22no—
119C2×C211—22no—
120C2×C211—41no—
121C2×C211—81no—
122C2×C211#38611yescharacteristic
123C2×C211C2×#10411yescharacteristic
124C2×C211#42111yescharacteristic
125C211#187111yescharacteristic, complemented, Sylow
126#1871—2111nomaximal, Sylow
127C22×M16—2111no—
128#525D21111yescharacteristic
129C2×#104—2112no—
130C4×C22⋊C4—2111no—
131#339—2112no—
132C2×#41C2×D21111yescharacteristic
133C23×C4C2×D21111yescharacteristic
134C2×M16—4221no—
135C2×M16—4224no—
136C2×M16—2112no—
137C22×C8—4221no—
138#123—21no—
139C22⋊Q8Dic21111yescharacteristic, complemented
140C22⋊Q8Dic21111yescharacteristic, complemented
141#121Dic21111yescharacteristic, complemented
142#121Dic21111yescharacteristic, complemented
143#120—21no—
144C4×D4—21no—
145C4×D4—21no—
146#117C2×D21111yescharacteristic
147C2×C22⋊C4—4221no—
148C2×C42—2112no—
149#104—4224no—
150#95—4221no—
151C24C22⋊D21111yescharacteristic
152#41—22no—
153#41—22no—
154C2×Q8C2×Dic21111yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4C2×Dic21111yescharacteristic
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—4222no—
163C22×C4C2×Dic21111yescharacteristic
164C22×C4C2×Dic21111yescharacteristic
165C22×C4C22⋊D21111yescharacteristic
166C22×C4C22⋊D21111yescharacteristic
167C22×C4C22⋊D21111yescharacteristic
168C22×C4C22×D21111yescharacteristic
169M16—4224no—
170M16—4222no—
171M16—2114no—
172C2×C8—4224no—
173C2×C8—4222no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4C2×Dic21111yescharacteristic
177C4⋊C4C2×Dic21111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—4224no—
184C42—21no—
185C42—4222no—
186C42—4222no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—41no—
191C23C22⋊Dic21111yescharacteristic
192C23C22⋊Dic21111yescharacteristic
193C23C2×C22⋊D21111yescharacteristic
194Q8—22no—
195D4—22no—
196D4—42no—
197D4—42no—
198D4—42no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—8442no—
212C2×C4—8442no—
213C2×C4—4224no—
214C2×C4C22⋊Dic21111yescharacteristic
215C2×C4C22⋊Dic21111yescharacteristic
216C2×C4C22×Dic21111yescharacteristic
217C2×C4D4×D21112yes—
218C2×C4C2×C22⋊D21111yescharacteristic
219C2×C4#1680996112yes—
220C8—4224no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22#2026990412yescomplemented
233C22C2×C22⋊Dic21111yescharacteristic
234C22#2026994912yes—
235C4—21no—
236C4—21no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—8444no—
241C4D4×Dic21112yes—
242C4#2026994511yescharacteristic
243C4#2026995811yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—81no—
248C2#2721302611yescharacteristic
249C2C2×#2026990411yescharacteristic
250C2#2721316311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#525×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenter#525D211
hypocenterC211#1871
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#16809946C2semidirect
2. C2×#20269904C2semidirect
3. C4×C22⋊Dic211C2semidirect
4. #27212905C2semidirect
17. C22⋊Q8×C211C4semidirect
18. C22⋊Q8×C211C4semidirect
19. #121×C211C4semidirect
20. #121×C211C4semidirect
28. #16809946D4semidirect
106. C22×C211#104semidirect
125. C211#1871semidirect
139. C22⋊Q8Dic211semidirect
140. C22⋊Q8Dic211semidirect
141. #121Dic211semidirect
142. #121Dic211semidirect
232. C22#20269904semidirect

Maximal subgroups

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

subgroupindex
C22×#168099462
C2×#202699042
C4×C22⋊Dic2112
#272129052
#525×C2112
#1871211