Finite Group Databasebeta

← #41013845

Subgroups — #41013845

Order 27008.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#27213658C211yesmaximal, characteristic
2C2×#20269934C211yesmaximal, characteristic, complemented
3C2×#20269905C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27213109C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×C4×D211C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269961—21no—
12#20269957C412yescomplemented
13#20269952C412yescomplemented
14#20269947—21no—
15C4×D844—22no—
16#20269934—22no—
17#20269934—22no—
18C2×Dic211⋊C4—21no—
19C2×(C2×D211)⋊C4—21no—
20C2×C4×Dic211C2211yescharacteristic
21#20269905—22no—
22#20269903—22no—
23(C2×D211)⋊C8—22no—
24C2×M16×C211C2211yescharacteristic
25#117×C211C2211yescharacteristic
26C2×C42×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32C2×Dic422C2×C411yescharacteristic
33C2×D844—22no—
34C2×D844C2×C411yescharacteristic
35C2×C22⋊D211—21no—
36C22×Dic211—21no—
37C22×Dic211D411yescharacteristic
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—21no—
42C2×C4×D211—21no—
43C2×C4×D211C2×C412yes—
44C2×C4×D211D412yes—
45#16809946—21no—
46#16809946—21no—
47C2×C211⋊C8—21no—
48C4⋊Dic211—22no—
49Dic211⋊C4—24no—
50(C2×D211)⋊C4—24no—
51(C2×D211)⋊C4—22no—
52(C2×D211)⋊C4—22no—
53C4×Dic211—21no—
54C4×Dic211D412yescomplemented
55C22×C4×C211—21no—
56C22×C4×C211C2311yescharacteristic
57M16×C211—21no—
58M16×C211—21no—
59C2×C8×C211—21no—
60C4⋊C4×C211C2×C412yes—
61C22⋊C4×C211—21no—
62C42×C211—21no—
63C42×C211—21no—
64C42×C211D412yescomplemented
65C22×D211—21no—
66C22×D211—21no—
67C22×D211—21no—
68C22×D211—42no—
69C22×D211C22⋊C412yes—
70Dic422—21no—
71Dic422—21no—
72D844—21no—
73D844—21no—
74D844—44no—
75C22⋊D211—41no—
76C22⋊D211—41no—
77C2×Dic211—21no—
78C2×Dic211—21no—
79C2×Dic211—21no—
80C2×Dic211—22no—
81C2×Dic211—42no—
82C2×Dic211C22⋊C412yes—
83C4×D211—22no—
84C4×D211—22no—
85C4×D211—22no—
86C4×D211—22no—
87C4×D211—41no—
88C4×D211—41no—
89C211⋊C8—41no—
90C23×C211C2×D411yescharacteristic
91C2×C4×C211—22no—
92C2×C4×C211—22no—
93C2×C4×C211—42no—
94C2×C4×C211C22×C411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C8×C211—41no—
101C22×C211—22no—
102C22×C211C2×C22⋊C411yescharacteristic
103C22×C211#12011yescharacteristic
104C22×C211#12311yescharacteristic
105C2×D211—22no—
106C2×D211—22no—
107C2×D211—22no—
108C2×D211—42no—
109C2×D211—42no—
110C2×D211—41no—
111C2×D211—42no—
112C4×C211—42no—
113C4×C211—42no—
114C4×C211C4×D411yescharacteristic
115C4×C211C4×D411yescharacteristic
116C4×C211#12111yescharacteristic
117C4×C211#12111yescharacteristic
118Dic211—22no—
119Dic211—42no—
120Dic211—41no—
121C2×C211—21no—
122C2×C211—21no—
123C2×C211#38611yescharacteristic
124C2×C211#42112yes—
125D211—42no—
126D211—81no—
127C211#189711yescharacteristic, complemented, Sylow
128#1897—2111nomaximal, Sylow
129#525—2111no—
130C2×#117—2111no—
131C2×#104—2111no—
132C2×#104—2111no—
133#407—2111no—
134#339—2111no—
135#339D21111yescharacteristic, complemented
136C2×#41—2111no—
137C23×C4—2111no—
138C2×M16—2111no—
139C2×M16C2×D21111yescharacteristic
140#123—4221no—
141C22⋊Q8—2112no—
142#121—2112no—
143#120—4221no—
144C4×D4—4222no—
145#117—4222no—
146#117—4222no—
147#117C2×D21111yescharacteristic
148C2×C4⋊C4—4221no—
149C2×C22⋊C4—4221no—
150C2×C42—2111no—
151C2×C42C2×D21111yescharacteristic
152#104—4222no—
153#104—4222no—
154C22⋊C8—4222no—
155C24—2111no—
156#41—4221no—
157#41—4221no—
158#41—4221no—
159#41—4221no—
160C2×Q8—2111no—
161C2×D4—4222no—
162C2×D4—4221no—
163C2×D4—2111no—
164C22×C4—21no—
165C22×C4—4221no—
166C22×C4—4221no—
167C22×C4—4221no—
168C22×C4—4221no—
169C22×C4—4221no—
170C22×C4—4221no—
171C22×C4—2111no—
172C22×C4—2112no—
173C22×C4—2112no—
174C22×C4C22×D21111yescharacteristic
175M16—21no—
176M16—21no—
177M16—4221no—
178M16—4221no—
179C2×C8—21no—
180C2×C8—4221no—
181C4⋊C4—4224no—
182C4⋊C4—4222no—
183C4⋊C4C4×D21112yes—
184C22⋊C4—21no—
185C22⋊C4—4224no—
186C22⋊C4—4222no—
187C22⋊C4—4222no—
188C42—21no—
189C42—21no—
190C42—4221no—
191C42—2112no—
192C42D84412yescomplemented
193C23—8442no—
194C23—4221no—
195C23—4221no—
196C23—4221no—
197C23—2112no—
198C23D4×D21111yescharacteristic
199Q8—4221no—
200Q8—4221no—
201D4—8444no—
202D4—8441no—
203D4—8441no—
204D4—4221no—
205D4—4221no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—8442no—
210C2×C4—8441no—
211C2×C4—8441no—
212C2×C4—4221no—
213C2×C4—4222no—
214C2×C4—4221no—
215C2×C4—4221no—
216C2×C4—4222no—
217C2×C4—4222no—
218C2×C4—4222no—
219C2×C4—4222no—
220C2×C4—2112no—
221C2×C4C2×C4×D21111yescharacteristic
222C2×C4D4×D21111yescharacteristic
223C2×C4D4×D21111yescharacteristic
224C2×C4C2×D84411yescharacteristic
225C2×C4#1680996211yescharacteristic
226C2×C4#1680996311yescharacteristic
227C8—41no—
228C8—8441no—
229C22—22no—
230C22—8442no—
231C22—8442no—
232C22—8442no—
233C22—8441no—
234C22—4222no—
235C22—4222no—
236C22—4222no—
237C22C22⋊C4×D21111yescharacteristic
238C22#2026994711yescharacteristic
239C22#2026996211yescharacteristic
240C4—42no—
241C4—42no—
242C4—8442no—
243C4—8441no—
244C4—4222no—
245C4#2026993911yescharacteristic
246C4C4×D84411yescharacteristic
247C4#2026995011yescharacteristic
248C4#2026995211yescharacteristic
249C2—21no—
250C2—21no—
251C2—8442no—
252C2—16881no—
253C2#2721302411yescharacteristic
254C2#2721316112yes—
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213658C2
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1897
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#20269934C2semidirect
3. C2×#20269905C2semidirect
4. C2×#20269903C2semidirect
5. #27213109C2semidirect
6. #27212905C2semidirect
7. #339×C211C2semidirect
12. #20269957C4semidirect
13. #20269952C4semidirect
54. C4×Dic211D4semidirect
64. C42×C211D4semidirect
127. C211#1897semidirect
135. #339D211semidirect
192. C42D844semidirect

Maximal subgroups

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

subgroupindex
#272136582
C2×#202699342
C2×#202699052
C2×#202699032
#272131092
#272129052
#339×C2112
#1897211