Finite Group Databasebeta

← #41013865

Subgroups — #41013865

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×#20269957C211yesmaximal, characteristic
3C2×#20269895C211yesmaximal, characteristic, complemented
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27213109C211yesmaximal, characteristic
6#27212915C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×C4×D211C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11C2×#16809946C2211yescharacteristic
12C2×#16809945—21no—
13M16×D211—22no—
14M16×D211—22no—
15C2×C8×D211—21no—
16#20269957—21no—
17#20269957—21no—
18#20269957—21no—
19#20269957—21no—
20C2×C4⋊Dic211C412yes—
21C2×(C2×D211)⋊C4C412yes—
22#20269895—22no—
23#20269895—22no—
24(C2×D211)⋊C8—21no—
25(C2×D211)⋊C8—21no—
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28C2×C4⋊C4×C211C2211yescharacteristic
29C23×D211C2×C411yescharacteristic
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422—21no—
36C22×Dic211C2×C412yes—
37C22×Dic211C2×C411yescharacteristic
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—21no—
42C2×C4×D211D411yescharacteristic
43C2×C4×D211D411yescharacteristic
44C2×C4×D211D411yescharacteristic
45C2×C4×D211D411yescharacteristic
46#16809946—21no—
47#16809946—21no—
48#16809946—21no—
49#16809946D412yes—
50#16809945—24no—
51C2×C211⋊C8—21no—
52C2×C211⋊C8—21no—
53C8×D211—24no—
54C4⋊Dic211—22no—
55C4⋊Dic211—22no—
56(C2×D211)⋊C4—22no—
57(C2×D211)⋊C4—22no—
58C22×C4×C211C2×C412yes—
59C22×C4×C211C2311yescharacteristic
60M16×C211—21no—
61M16×C211—21no—
62M16×C211—21no—
63M16×C211D412yes—
64C2×C8×C211—21no—
65C2×C8×C211—21no—
66C4⋊C4×C211—21no—
67C4⋊C4×C211—21no—
68C22×D211—21no—
69C22×D211—21no—
70C22×D211—22no—
71C22×D211—41no—
72C22×D211C22⋊C411yescharacteristic
73C22×D211C22⋊C411yescharacteristic
74Dic422—41no—
75Dic422—41no—
76Dic422—41no—
77Dic422—41no—
78C2×Dic211—21no—
79C2×Dic211—21no—
80C2×Dic211—22no—
81C2×Dic211—22no—
82C2×Dic211—22no—
83C2×Dic211C22⋊C411yescharacteristic
84C2×Dic211C22⋊C411yescharacteristic
85C4×D211—21no—
86C4×D211—21no—
87C4×D211—21no—
88C4×D211—21no—
89C4×D211—21no—
90C4×D211—21no—
91C4×D211—21no—
92C4×D211—21no—
93C211⋊C8—22no—
94C211⋊C8—41no—
95C23×C211C22×C411yescharacteristic
96C2×C4×C211—22no—
97C2×C4×C211—42no—
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×#101—2111no—
140#407—2111no—
141#343—2111no—
142#343D21111yescharacteristic, complemented
143C22×Q8—2111no—
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⋊C4C2×D21111yescharacteristic
159C2×C22⋊C4—2112no—
160#101—4222no—
161#101—4222no—
162C22⋊C8—4221no—
163C22⋊C8—4221no—
164C24—2111no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8—4221no—
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—
191M16D84412yes—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—4224no—
195C2×C8—4221no—
196C2×C8—4221no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—4222no—
200C4⋊C4—4222no—
201C22⋊C4—4222no—
202C22⋊C4—4222no—
203C23—8441no—
204C23—4222no—
205C23—4221no—
206C23—4221no—
207C23—2111no—
208C23—2111no—
209C23C2×C4×D21111yescharacteristic
210Q8—8441no—
211Q8—8441no—
212Q8—8441no—
213Q8—8441no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—4222no—
217C2×C4—4222no—
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×D84411yescharacteristic
235C2×C4#1680996211yescharacteristic
236C2×C4#1680996311yescharacteristic
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#2026993911yescharacteristic
253C22C4×D84411yescharacteristic
254C4—42no—
255C4—8442no—
256C4—4221no—
257C4—4221no—
258C4#2026994711yescharacteristic
259C4#2026995011yescharacteristic
260C4#2026995211yescharacteristic
261C4#2026996211yescharacteristic
262C2—21no—
263C2—21no—
264C2—8441no—
265C2—4222no—
266C2#2721302411yescharacteristic
267C2#101×D21111yescharacteristic
268C2#2721312711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×D211C2×C4
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1901
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. C2×#20269895C2semidirect
4. C2×#20269895C2semidirect
6. #27212915C2semidirect
7. #343×C211C2semidirect
118. C2×D211#101semidirect
134. C211#1901semidirect
142. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2112
C2×#202699572
C2×#202698952
#272131092
#272129152
#343×C2112
#1901211