Finite Group Databasebeta

← #41013765

Subgroups — #41013765

Order 27008.

334 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20270007C211yesmaximal, characteristic
2#27213629C211yesmaximal, characteristic
3#27213156C211yesmaximal, characteristic, complemented
4#27213052C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007C414yescomplemented
11C2×#16809952—21no—
12C2×#16809950—21no—
13C2×#16809946C2211yescharacteristic
14Q8×Dic211—21no—
15D4×Dic211—22no—
16D4×Dic211—21no—
17#20269936—21no—
18#20269936C2211yescharacteristic
19C2×C4⋊Dic211—21no—
20C2×C4⋊Dic211C2211yescharacteristic
21#20269902—21no—
22#20269902—21no—
23C211⋊#95—22no—
24C2×#41×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26C2×C42×C211C2211yescharacteristic
27#104×C211—22no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422C2×C412yescomplemented
34C22×Dic211—22no—
35C22×Dic211—21no—
36C22×Dic211D412yes—
37#16809952—24no—
38#16809950—24no—
39#16809946—21no—
40#16809946C2×C412yescomplemented
41C2×C211⋊C8—21no—
42C4⋊Dic211—22no—
43C4⋊Dic211—21no—
44C4⋊Dic211—21no—
45C4⋊Dic211D411yescharacteristic
46C4⋊Dic211D411yescharacteristic
47C22⋊Dic211—22no—
48C22⋊Dic211—21no—
49C4×Dic211—22no—
50C4×Dic211—21no—
51#41×C211—21no—
52#41×C211—21no—
53#41×C211C2×C412yes—
54#41×C211D412yes—
55C2×Q8×C211D411yescharacteristic
56C2×D4×C211—21no—
57C2×D4×C211D411yescharacteristic
58C22×C4×C211—21no—
59C22×C4×C211—21no—
60C22×C4×C211C2311yescharacteristic
61M16×C211—21no—
62M16×C211—21no—
63C2×C8×C211—21no—
64C42×C211—21no—
65C42×C211—21no—
66Dic422—22no—
67Dic422—22no—
68Dic422—41no—
69Dic422—41no—
70C2×Dic211—22no—
71C2×Dic211—21no—
72C2×Dic211—22no—
73C2×Dic211—21no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—42no—
77C2×Dic211—82no—
78C211⋊C8—22no—
79C23×C211—21no—
80C23×C211C2×D411yescharacteristic
81Q8×C211—21no—
82Q8×C211C22⋊C412yes—
83D4×C211—22no—
84D4×C211—22no—
85D4×C211—21no—
86D4×C211C22⋊C412yes—
87C2×C4×C211—21no—
88C2×C4×C211—22no—
89C2×C4×C211—22no—
90C2×C4×C211—22no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211C22×C411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C8×C211—41no—
100C22×C211—21no—
101C22×C211—21no—
102C22×C211—22no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211C4×D411yescharacteristic
106C22×C211#12011yescharacteristic
107C22×C211#12111yescharacteristic
108C4×C211—22no—
109C4×C211—42no—
110C4×C211C2×C22⋊C411yescharacteristic
111C4×C211C4×D411yescharacteristic
112C4×C211#12111yescharacteristic
113C4×C211#12311yescharacteristic
114Dic211—42no—
115Dic211—42no—
116Dic211—41no—
117Dic211—41no—
118C2×C211—21no—
119C2×C211—21no—
120C2×C211—42no—
121C2×C211#38611yescharacteristic
122C2×C211#45511yescharacteristic
123C2×C211#45611yescharacteristic
124C211#188211yescharacteristic, complemented, Sylow
125#1882—2111nomaximal, Sylow
126C2×#137—2111no—
127#520—2111no—
128C2×#104D21111yescharacteristic
129#419—2111no—
130#391—2111no—
131#343—2111no—
132#339—2111no—
133C2×#41C2×D21111yescharacteristic
134C22×Q8—2111no—
135#137—4222no—
136#137—2114no—
137C2×Q16—4221no—
138C2×SD16—4221no—
139C2×M16—2111no—
140C2×M16C2×D21111yescharacteristic
141C4×Q8—4221no—
142C4×D4—4221no—
143C4×D4—4222no—
144#117—4221no—
145#117—2111no—
146C2×C4⋊C4—4221no—
147C2×C4⋊C4—2111no—
148C2×C42C2×D21111yescharacteristic
149#104—22no—
150#103—4221no—
151#103—4221no—
152#95—4222no—
153#41—21no—
154#41—21no—
155#41C4×D21112yes—
156#41C22⋊D21112yes—
157C2×Q8—4221no—
158C2×Q8—4221no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—4221no—
162C2×Q8—2112no—
163C2×Q8D84411yescharacteristic
164C2×D4—21no—
165C2×D4D84411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—4221no—
169C22×C4—4222no—
170C22×C4—2112no—
171C22×C4C22×D21111yescharacteristic
172Q16—4224no—
173SD16—4224no—
174M16—21no—
175M16—21no—
176M16—4221no—
177M16—2112no—
178C2×C8—21no—
179C2×C8—4221no—
180C4⋊C4—4222no—
181C4⋊C4—4221no—
182C4⋊C4—4221no—
183C4⋊C4—2111no—
184C4⋊C4—2111no—
185C22⋊C4—4222no—
186C22⋊C4—4221no—
187C42—21no—
188C42—21no—
189C42—4222no—
190C42—4221no—
191C23—21no—
192C23D4×D21111yescharacteristic
193Q8—21no—
194Q8—8441no—
195Q8—8441no—
196Q8—4222no—
197Q8—4222no—
198Q8(C2×D211)⋊C412yes—
199D4—22no—
200D4—22no—
201D4—21no—
202D4(C2×D211)⋊C412yes—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—8441no—
210C2×C4—8441no—
211C2×C4—8442no—
212C2×C4—16882no—
213C2×C4—4222no—
214C2×C4—4221no—
215C2×C4—4222no—
216C2×C4—4221no—
217C2×C4C2×C4×D21111yescharacteristic
218C2×C4D4×D21111yescharacteristic
219C2×C4C2×C22⋊D21111yescharacteristic
220C2×C4C2×D84411yescharacteristic
221C2×C4#1680996111yescharacteristic
222C2×C4#1680996111yescharacteristic
223C8—41no—
224C8—4222no—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22(C22⋊D211)⋊C411yescharacteristic
231C22#2026994711yescharacteristic
232C22#2026994911yescharacteristic
233C4—22no—
234C4—42no—
235C4—8442no—
236C4—8441no—
237C4—8441no—
238C4—8442no—
239C4C2×(C2×D211)⋊C411yescharacteristic
240C4(C22⋊D211)⋊C411yescharacteristic
241C4#2026994911yescharacteristic
242C4#2026996111yescharacteristic
243C2—21no—
244C2—21no—
245C2—42no—
246C2#2721302311yescharacteristic
247C2#2721332511yescharacteristic
248C2#2721333011yescharacteristic
subgroupG/H
centerC22#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1882
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
3. #27213156C2semidirect
4. #27213052C2semidirect
5. #27212914C2semidirect
6. #27212906C2semidirect
10. #20270007C4semidirect
33. C2×Dic422C2×C4semidirect
40. #16809946C2×C4semidirect
124. C211#1882semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
#272136292
#272131562
#272130522
#272129142
#272129062
C2×#104×C2112
#1882211