Finite Group Databasebeta

← #41013843

Subgroups — #41013843

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#27213653C211yesmaximal, characteristic
2C2×#20269934C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213109C211yesmaximal, characteristic, complemented
5#27212907C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269958—21no—
11#20269953C412yescomplemented
12#20269948C412yescomplemented
13#20269945—21no—
14C4×C22⋊D211—22no—
15#20269936C2211yescharacteristic
16#20269934—22no—
17#20269934—22no—
18C2×Dic211⋊C4—21no—
19C2×(C2×D211)⋊C4—21no—
20C2×C4×Dic211C2211yescharacteristic
21#20269904—22no—
22(C2×D211)⋊C8—22no—
23C2×#41×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×C42×C211C2211yescharacteristic
26#104×C211—22no—
27C23×D211D411yescharacteristic
28C2×C22⋊D211—22no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C2×C4×D211—21no—
32C2×C4×D211—21no—
33C2×C4×D211—21no—
34C2×C4×D211—21no—
35C2×C4×D211C2×C412yes—
36C2×C4×D211D412yes—
37#16809946—21no—
38#16809946—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211C2×C412yes—
41Dic211⋊C4—24no—
42Dic211⋊C4—22no—
43C22⋊Dic211—22no—
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—24no—
46(C2×D211)⋊C4—22no—
47C4×Dic211—21no—
48C4×Dic211—21no—
49C4×Dic211D412yescomplemented
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53#41×C211—21no—
54C2×Q8×C211C2×C411yescharacteristic
55C2×D4×C211—21no—
56C2×D4×C211C2×C411yescharacteristic
57C22×C4×C211—21no—
58C22×C4×C211—21no—
59C22×C4×C211C2311yescharacteristic
60M16×C211—21no—
61M16×C211—21no—
62C2×C8×C211—21no—
63C42×C211—21no—
64C42×C211D412yescomplemented
65C22×D211—21no—
66C22×D211—21no—
67C22×D211—42no—
68C22×D211C22⋊C412yes—
69C22⋊D211—44no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—22no—
73C2×Dic211—22no—
74C2×Dic211—42no—
75C2×Dic211C22⋊C412yes—
76C4×D211—22no—
77C4×D211—22no—
78C4×D211—22no—
79C4×D211—22no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211C2×D411yescharacteristic
83Q8×C211—21no—
84Q8×C211—21no—
85D4×C211—21no—
86D4×C211—21no—
87D4×C211—41no—
88D4×C211—41no—
89C2×C4×C211—21no—
90C2×C4×C211—22no—
91C2×C4×C211—42no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
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×C211—41no—
103C22×C211—42no—
104C22×C211C2×C22⋊C411yescharacteristic
105C22×C211#12011yescharacteristic
106C22×C211#12311yescharacteristic
107C2×D211—22no—
108C2×D211—22no—
109C2×D211—22no—
110C2×D211—42no—
111C2×D211—42no—
112C4×C211—42no—
113C4×C211—41no—
114C4×C211C4×D411yescharacteristic
115C4×C211C4×D411yescharacteristic
116C4×C211#12111yescharacteristic
117C4×C211#12111yescharacteristic
118Dic211—22no—
119Dic211—42no—
120Dic211—42no—
121C2×C211—21no—
122C2×C211—21no—
123C2×C211—81no—
124C2×C211#38611yescharacteristic
125C2×C211#42112yes—
126D211—42no—
127C211#189711yescharacteristic, complemented, Sylow
128#1897—2111nomaximal, Sylow
129#525—2111no—
130C2×#117—2111no—
131C2×#104—2111no—
132C2×#104D21111yescharacteristic, complemented
133#407—2111no—
134#339—2111no—
135#339—2111no—
136C2×#41C2×D21111yescharacteristic
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#117—2111no—
148C2×C4⋊C4—4221no—
149C2×C22⋊C4—4221no—
150C2×C42—2111no—
151C2×C42C2×D21111yescharacteristic
152#104—22no—
153#104—4222no—
154C22⋊C8—4222no—
155C24—2111no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160C2×Q8C4×D21111yescharacteristic
161C2×D4—21no—
162C2×D4—4222no—
163C2×D4C4×D21111yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
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⋊C4—2112no—
184C22⋊C4—4224no—
185C22⋊C4—4222no—
186C22⋊C4—4221no—
187C22⋊C4—4222no—
188C42—21no—
189C42—4221no—
190C42—4221no—
191C42—2112no—
192C42C22⋊D21112yescomplemented
193C23—21no—
194C23—8442no—
195C23—4221no—
196C23—4221no—
197C23—2112no—
198C23D4×D21111yescharacteristic
199Q8—21no—
200Q8—21no—
201D4—21no—
202D4—21no—
203D4—41no—
204D4—41no—
205D4—8444no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—8442no—
212C2×C4—4222no—
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×C22⋊D21111yescharacteristic
225C2×C4#1680996111yescharacteristic
226C2×C4#1680996211yescharacteristic
227C8—41no—
228C8—8441no—
229C22—22no—
230C22—42no—
231C22—41no—
232C22—8442no—
233C22—8442no—
234C22—4222no—
235C22—4222no—
236C22—4222no—
237C22C22⋊C4×D21111yescharacteristic
238C22#2026994511yescharacteristic
239C22#2026995911yescharacteristic
240C4—42no—
241C4—41no—
242C4—8442no—
243C4—8442no—
244C4—4222no—
245C4(C22⋊D211)⋊C411yescharacteristic
246C4C4×C22⋊D21111yescharacteristic
247C4#2026994811yescharacteristic
248C4#2026995011yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—8442no—
253C2#2721302211yescharacteristic
254C2#2721316112yes—
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213653C2
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1897
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#20269934C2semidirect
3. C2×#20269904C2semidirect
4. #27213109C2semidirect
5. #27212907C2semidirect
6. #27212906C2semidirect
7. C2×#104×C211C2semidirect
11. #20269953C4semidirect
12. #20269948C4semidirect
49. C4×Dic211D4semidirect
64. C42×C211D4semidirect
127. C211#1897semidirect
132. C2×#104D211semidirect
192. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272136532
C2×#202699342
C2×#202699042
#272131092
#272129072
#272129062
C2×#104×C2112
#1897211