Finite Group Databasebeta

← #41013496

Subgroups — #41013496

Order 27008.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 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#117×D211C211yesmaximal, characteristic
2#27213109C211yesmaximal, characteristic
3C4×(C2×D211)⋊C4C211yesmaximal, characteristic
4#27212907C211yesmaximal, characteristic, complemented
5#27212906C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269936C2211yescharacteristic
11#20269935C411yescharacteristic, complemented
12#20269935C411yescharacteristic, complemented
13#20269934—21no—
14#20269933—21no—
15C4⋊C4×D211C411yescharacteristic, complemented
16C4⋊C4×D211C411yescharacteristic, complemented
17C2×(C2×D211)⋊C4—21no—
18C22⋊C4×D211—21no—
19C2×C4×Dic211C2211yescharacteristic
20C42×D211—21no—
21(C2×D211)⋊C8—21no—
22(C2×D211)⋊C8—21no—
23C211⋊#95—21no—
24C2×M16×C211C2211yescharacteristic
25#117×C211C2211yescharacteristic
26C2×C42×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C22×Dic211—21no—
29C22×Dic211D411yescharacteristic
30C2×C4×D211—21no—
31C2×C4×D211—21no—
32C2×C4×D211—21no—
33C2×C4×D211—21no—
34C2×C4×D211—21no—
35C2×C4×D211—21no—
36C2×C4×D211C2×C411yescharacteristic
37C2×C4×D211C2×C411yescharacteristic
38C2×C4×D211D411yescharacteristic
39C2×C4×D211D411yescharacteristic
40#16809946—21no—
41#16809946—21no—
42C2×C211⋊C8—21no—
43C4⋊Dic211C2×C411yescharacteristic
44C4⋊Dic211C2×C411yescharacteristic
45Dic211⋊C4—21no—
46Dic211⋊C4—21no—
47C22⋊Dic211—21no—
48(C2×D211)⋊C4—24no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—21no—
51C4×Dic211—21no—
52C4×Dic211—21no—
53C4×Dic211—21no—
54C4×Dic211—21no—
55C4×Dic211—21no—
56C22×C4×C211—21no—
57C22×C4×C211C2311yescharacteristic
58M16×C211—21no—
59M16×C211—21no—
60C2×C8×C211—21no—
61C4⋊C4×C211C2×C411yescharacteristic
62C4⋊C4×C211C2×C411yescharacteristic
63C22⋊C4×C211—21no—
64C42×C211—21no—
65C42×C211—21no—
66C42×C211—21no—
67C22×D211—21no—
68C22×D211—21no—
69C22×D211—22no—
70C22×D211—41no—
71C22×D211C22⋊C411yescharacteristic
72C22×D211C22⋊C411yescharacteristic
73C2×Dic211—21no—
74C2×Dic211—21no—
75C2×Dic211—22no—
76C2×Dic211—21no—
77C2×Dic211—21no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211C22⋊C411yescharacteristic
81C2×Dic211C22⋊C411yescharacteristic
82C4×D211—21no—
83C4×D211—21no—
84C4×D211—21no—
85C4×D211—21no—
86C4×D211—21no—
87C4×D211—21no—
88C4×D211—21no—
89C4×D211—21no—
90C4×D211—42no—
91C4×D211—42no—
92C211⋊C8—41no—
93C23×C211C2×D411yescharacteristic
94C2×C4×C211—22no—
95C2×C4×C211—21no—
96C2×C4×C211—21no—
97C2×C4×C211—41no—
98C2×C4×C211—41no—
99C2×C4×C211C22×C411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211#4111yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C2×C4×C211#4111yescharacteristic
105C8×C211—41no—
106C22×C211—21no—
107C22×C211—21no—
108C22×C211C2×C22⋊C411yescharacteristic
109C22×C211#12311yescharacteristic
110C22×C211#12311yescharacteristic
111C2×D211—21no—
112C2×D211—21no—
113C2×D211—21no—
114C2×D211—21no—
115C2×D211—21no—
116C2×D211—42no—
117C2×D211—42no—
118C2×D211#10412yes—
119C4×C211—42no—
120C4×C211—41no—
121C4×C211—41no—
122C4×C211#11711yescharacteristic
123C4×C211#11711yescharacteristic
124C4×C211#12311yescharacteristic
125C4×C211#12311yescharacteristic
126Dic211—21no—
127Dic211—21no—
128Dic211—42no—
129Dic211—41no—
130Dic211—41no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211#38111yescharacteristic
134C2×C211C2×#10411yescharacteristic
135C2×C211#42111yescharacteristic
136D211—22no—
137D211—41no—
138C211#182611yescharacteristic, complemented, Sylow
139#1826—2111nomaximal, Sylow
140C2×#117—2111no—
141#407—2111no—
142C4×C22⋊C4—2111no—
143#339—2111no—
144#339—2111no—
145#339—2111no—
146#339D21111yescharacteristic, complemented
147C23×C4—2111no—
148C2×M16—2111no—
149C2×M16C2×D21111yescharacteristic
150#117—4221no—
151#117—4221no—
152#117—2111no—
153#117—2111no—
154#117—2111no—
155#117C2×D21111yescharacteristic
156C2×C4⋊C4—2111no—
157C2×C4⋊C4—2111no—
158C2×C22⋊C4—4221no—
159C2×C22⋊C4—4221no—
160C2×C42—4221no—
161C2×C42—2111no—
162C2×C42C2×D21111yescharacteristic
163C22⋊C8—4221no—
164C22⋊C8—4221no—
165#95—4221no—
166C24—2111no—
167C22×C4—21no—
168C22×C4—4221no—
169C22×C4—4221no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4—4221no—
173C22×C4—4221no—
174C22×C4—4221no—
175C22×C4—2111no—
176C22×C4—2111no—
177C22×C4—2111no—
178C22×C4—2111no—
179C22×C4—2111no—
180C22×C4C22×D21111yescharacteristic
181M16—21no—
182M16—21no—
183M16—4221no—
184M16—4221no—
185C2×C8—21no—
186C2×C8—4221no—
187C4⋊C4—4221no—
188C4⋊C4—4221no—
189C4⋊C4—2111no—
190C4⋊C4—2111no—
191C4⋊C4C4×D21111yescharacteristic
192C4⋊C4C4×D21111yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—4224no—
195C22⋊C4—4221no—
196C22⋊C4—4221no—
197C22⋊C4—4221no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—4221no—
202C42—4221no—
203C42—4221no—
204C42—4221no—
205C42—4221no—
206C23—8441no—
207C23—4222no—
208C23—4221no—
209C23—4221no—
210C23—2111no—
211C23—2111no—
212C23D4×D21111yescharacteristic
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—8442no—
219C2×C4—8442no—
220C2×C4—8441no—
221C2×C4—8441no—
222C2×C4—4222no—
223C2×C4—4221no—
224C2×C4—4221no—
225C2×C4—4221no—
226C2×C4—4221no—
227C2×C4—4221no—
228C2×C4—4221no—
229C2×C4—4221no—
230C2×C4—4221no—
231C2×C4—4221no—
232C2×C4—4221no—
233C2×C4—4221no—
234C2×C4—4221no—
235C2×C4—2111no—
236C2×C4—2111no—
237C2×C4C2×C4×D21111yescharacteristic
238C2×C4D4×D21111yescharacteristic
239C2×C4#1680996111yescharacteristic
240C2×C4#1680996211yescharacteristic
241C2×C4#1680996211yescharacteristic
242C2×C4#1680996311yescharacteristic
243C8—41no—
244C8—8441no—
245C22—21no—
246C22—21no—
247C22—8442no—
248C22—8442no—
249C22—4221no—
250C22—4221no—
251C22—4221no—
252C22—4221no—
253C22—4221no—
254C22—2112no—
255C22C22⋊C4×D21111yescharacteristic
256C22#2026995911yescharacteristic
257C22#2026996211yescharacteristic
258C4—42no—
259C4—41no—
260C4—41no—
261C4—8442no—
262C4—8441no—
263C4—8441no—
264C4—4221no—
265C4—4221no—
266C4#2026993411yescharacteristic
267C4#2026993511yescharacteristic
268C4#2026995911yescharacteristic
269C4#2026996211yescharacteristic
270C2—21no—
271C2—21no—
272C2—8441no—
273C2—4222no—
274C2#2721300811yescharacteristic
275C2#104×D21111yescharacteristic
276C2#2721316111yescharacteristic
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×D211D4
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1826
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
4. #27212907C2semidirect
5. #27212906C2semidirect
6. #27212905C2semidirect
7. #339×C211C2semidirect
11. #20269935C4semidirect
12. #20269935C4semidirect
15. C4⋊C4×D211C4semidirect
16. C4⋊C4×D211C4semidirect
138. C211#1826semidirect
146. #339D211semidirect

Maximal subgroups

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

subgroupindex
#117×D2112
#272131092
C4×(C2×D211)⋊C42
#272129072
#272129062
#272129052
#339×C2112
#1826211