Finite Group Databasebeta

← #41013863

Subgroups — #41013863

Order 27008.

338 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 85 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×M16×D211C211yesmaximal, characteristic
2C2×#20269953C211yesmaximal, characteristic
3C2×#20269896C211yesmaximal, characteristic, complemented
4#27213109C211yesmaximal, characteristic
5#27212914C211yesmaximal, characteristic, complemented
6#27212914C211yesmaximal, characteristic, complemented
7C2×#101×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11C2×#16809945—21no—
12M16×D211—22no—
13M16×D211—22no—
14C2×C8×D211—21no—
15#20269953—21no—
16#20269953—21no—
17#20269953—21no—
18#20269953—21no—
19C2×C4⋊Dic211C2211yescharacteristic
20C2×Dic211⋊C4C412yes—
21C2×(C2×D211)⋊C4C412yes—
22#20269896—22no—
23(C2×D211)⋊C8—21no—
24(C2×D211)⋊C8—21no—
25C22×Q8×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28#101×C211—22no—
29C23×D211C2×C411yescharacteristic
30C22×Dic211C2×C411yescharacteristic
31C22×Dic211C2×C412yes—
32C2×C4×D211—21no—
33C2×C4×D211—21no—
34C2×C4×D211—21no—
35C2×C4×D211—21no—
36C2×C4×D211D411yescharacteristic
37C2×C4×D211D411yescharacteristic
38C2×C4×D211D411yescharacteristic
39C2×C4×D211D411yescharacteristic
40#16809946—21no—
41#16809946—21no—
42#16809946—21no—
43#16809946D412yes—
44#16809945—24no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C8×D211—24no—
48C4⋊Dic211—21no—
49C4⋊Dic211—21no—
50Dic211⋊C4—22no—
51Dic211⋊C4—22no—
52(C2×D211)⋊C4—22no—
53(C2×D211)⋊C4—22no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211—21no—
58C2×Q8×C211—21no—
59C2×Q8×C211—21no—
60C22×C4×C211C2×C412yes—
61C22×C4×C211C2311yescharacteristic
62M16×C211—21no—
63M16×C211—21no—
64M16×C211—21no—
65M16×C211D412yes—
66C2×C8×C211—21no—
67C2×C8×C211—21no—
68C22×D211—21no—
69C22×D211—21no—
70C22×D211—22no—
71C22×D211—41no—
72C22×D211C22⋊C411yescharacteristic
73C22×D211C22⋊C411yescharacteristic
74C2×Dic211—21no—
75C2×Dic211—21no—
76C2×Dic211—22no—
77C2×Dic211—42no—
78C2×Dic211C22⋊C411yescharacteristic
79C2×Dic211C22⋊C411yescharacteristic
80C4×D211—21no—
81C4×D211—21no—
82C4×D211—21no—
83C4×D211—21no—
84C4×D211—21no—
85C4×D211—21no—
86C4×D211—21no—
87C4×D211—21no—
88C211⋊C8—22no—
89C211⋊C8—41no—
90C23×C211C22×C411yescharacteristic
91Q8×C211—41no—
92Q8×C211—41no—
93Q8×C211—41no—
94Q8×C211—41no—
95C2×C4×C211—22no—
96C2×C4×C211—22no—
97C2×C4×C211—22no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C8×C211—22no—
105C8×C211—41no—
106C22×C211—21no—
107C22×C211—21no—
108C22×C211C2×C22⋊C411yescharacteristic
109C22×C211C4×D411yescharacteristic
110C22×C211C4×D411yescharacteristic
111C2×D211—21no—
112C2×D211—21no—
113C2×D211—21no—
114C2×D211—21no—
115C2×D211—21no—
116C2×D211—42no—
117C2×D211—42no—
118C2×D211#10112yescomplemented
119C4×C211—42no—
120C4×C211#12011yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12111yescharacteristic
123C4×C211#12311yescharacteristic
124Dic211—21no—
125Dic211—21no—
126Dic211—42no—
127C2×C211—21no—
128C2×C211—21no—
129C2×C211#38611yescharacteristic
130C2×C211C2×#10111yescharacteristic
131C2×C211#41311yescharacteristic
132D211—22no—
133D211—41no—
134C211#190111yescharacteristic, complemented, Sylow
135#1901—2111nomaximal, Sylow
136C22×M16—2111no—
137C2×C22⋊Q8—2111no—
138C2×#101—2111no—
139C2×#101D21111yescharacteristic, complemented
140#407—2111no—
141#343—2111no—
142#343—2111no—
143C22×Q8C2×D21111yescharacteristic
144C23×C4—2111no—
145C2×M16—4221no—
146C2×M16—4222no—
147C2×M16—4222no—
148C2×M16—2111no—
149C2×M16—2111no—
150C2×M16C2×D21111yescharacteristic
151C2×M16C2×D21111yescharacteristic
152C22×C8—4221no—
153C22⋊Q8—4221no—
154C22⋊Q8—4221no—
155C22⋊Q8—4221no—
156C22⋊Q8—4221no—
157C2×C4⋊C4—2112no—
158C2×C4⋊C4—2111no—
159C2×C22⋊C4—2112no—
160#101—22no—
161#101—4222no—
162C22⋊C8—4221no—
163C22⋊C8—4221no—
164C24—2111no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C22×C4—4221no—
172C22×C4—4221no—
173C22×C4—4221no—
174C22×C4—4221no—
175C22×C4—2112no—
176C22×C4—2111no—
177C22×C4—2111no—
178C22×C4—2111no—
179C22×C4—2111no—
180C22×C4—2111no—
181C22×C4C4×D21112yes—
182C22×C4C22×D21111yescharacteristic
183M16—21no—
184M16—21no—
185M16—21no—
186M16—4224no—
187M16—4221no—
188M16—4221no—
189M16—4221no—
190M16—2112no—
191M16C22⋊D21112yes—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—4224no—
195C2×C8—4221no—
196C2×C8—4221no—
197C4⋊C4—4222no—
198C4⋊C4—4222no—
199C4⋊C4—4221no—
200C4⋊C4—4221no—
201C22⋊C4—4222no—
202C22⋊C4—4222no—
203C23—8441no—
204C23—4222no—
205C23—4221no—
206C23—4221no—
207C23—2111no—
208C23—2111no—
209C23C2×C4×D21111yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—8442no—
218C2×C4—4222no—
219C2×C4—4221no—
220C2×C4—4221no—
221C2×C4—4221no—
222C2×C4—4221no—
223C2×C4—4221no—
224C2×C4—4221no—
225C2×C4—4221no—
226C2×C4—4221no—
227C2×C4—4221no—
228C2×C4—4221no—
229C2×C4—2111no—
230C2×C4—2111no—
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4D4×D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C2×C4#1680996111yescharacteristic
236C2×C4#1680996211yescharacteristic
237C8—22no—
238C8—41no—
239C8—8441no—
240C8—4222no—
241C22—21no—
242C22—21no—
243C22—8442no—
244C22—8442no—
245C22—4221no—
246C22—4221no—
247C22—4221no—
248C22—4221no—
249C22—4221no—
250C22—2112no—
251C22C22⋊C4×D21111yescharacteristic
252C22(C22⋊D211)⋊C411yescharacteristic
253C22C4×C22⋊D21111yescharacteristic
254C4—42no—
255C4—8442no—
256C4—4221no—
257C4—4221no—
258C4#2026994511yescharacteristic
259C4#2026994811yescharacteristic
260C4#2026995011yescharacteristic
261C4#2026995911yescharacteristic
262C2—21no—
263C2—21no—
264C2—8441no—
265C2—4222no—
266C2#2721302211yescharacteristic
267C2#101×D21111yescharacteristic
268C2#2721312911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×D211C2×C4
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1901
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
3. C2×#20269896C2semidirect
5. #27212914C2semidirect
6. #27212914C2semidirect
7. C2×#101×C211C2semidirect
118. C2×D211#101semidirect
134. C211#1901semidirect
139. C2×#101D211semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2112
C2×#202699532
C2×#202698962
#272131092
#272129142
C2×#101×C2112
#1901211