Finite Group Databasebeta

← #41013589

Subgroups — #41013589

Order 27008.

360 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4⋊C4×D211C211yesmaximal, characteristic
2C2×#20269908C211yesmaximal, characteristic, complemented
3C2×(C2×D211)⋊C8C211yesmaximal, characteristic
4#27213003C211yesmaximal, characteristic
5#27212909C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#106×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211—21no—
9C22×C4×D211C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×C4⋊Dic211C2211yescharacteristic
13C2×Dic211⋊C4—21no—
14C4⋊C4×D211—22no—
15C4⋊C4×D211—22no—
16C2×(C2×D211)⋊C4—21no—
17#20269908—22no—
18(C2×D211)⋊C8C414yescomplemented
19C211⋊#95—21no—
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#106×C211—22no—
24C23×D211D411yescharacteristic
25C22×Dic211—21no—
26C22×Dic211—21no—
27C22×Dic211D411yescharacteristic
28C2×C4×D211—24no—
29C2×C4×D211—21no—
30C2×C4×D211—21no—
31C2×C4×D211—21no—
32C2×C4×D211—21no—
33C2×C4×D211—42no—
34C2×C4×D211—42no—
35C2×C4×D211C2×C412yes—
36C2×C4×D211Q812yes—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8C2×C412yescomplemented
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42C4⋊Dic211—21no—
43C4⋊Dic211D412yes—
44Dic211⋊C4—24no—
45(C2×D211)⋊C4—42no—
46C22×C4×C211—21no—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C2×C8×C211C2×C412yescomplemented
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211—21no—
55C4⋊C4×C211D412yes—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211C4⋊C412yes—
61C22×D211SD1612yes—
62C22×D211SD1612yes—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—22no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211C4⋊C412yes—
72C4×D211—22no—
73C4×D211—22no—
74C4×D211—22no—
75C4×D211—22no—
76C4×D211—48no—
77C211⋊C8—22no—
78C23×C211C2×D411yescharacteristic
79C2×C4×C211—22no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211C22×C411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×Q811yescharacteristic
89C2×C4×C211#4111yescharacteristic
90C2×C4×C211#4111yescharacteristic
91C8×C211—22no—
92C22×C211C2×C4⋊C411yescharacteristic
93C22×C211#12011yescharacteristic
94C22×C211#12311yescharacteristic
95C22×C211C2×SD1611yescharacteristic
96C22×C211C2×SD1611yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C22×C211#13711yescharacteristic
99C2×D211—22no—
100C2×D211—22no—
101C2×D211—22no—
102C2×D211—22no—
103C2×D211—22no—
104C2×D211—22no—
105C2×D211#10614yescomplemented
106C4×C211—42no—
107C4×C211—81no—
108C4×C211C4×D411yescharacteristic
109C4×C211C4×D411yescharacteristic
110C4×C211C22⋊Q811yescharacteristic
111C4×C211C22⋊Q811yescharacteristic
112Dic211—22no—
113Dic211—42no—
114Dic211—81no—
115C2×C211#38511yescharacteristic
116C2×C211C2×#10611yescharacteristic
117C2×C211#42811yescharacteristic
118C2×C211#44811yescharacteristic
119C2×C211#45011yescharacteristic
120C2×C211#48111yescharacteristic
121C2×C211#48311yescharacteristic
122D211—24no—
123C211#185311yescharacteristic, complemented, Sylow
124#1853—2111nomaximal, Sylow
125C22×C4⋊C4—2111no—
126C2×#106—2111no—
127C2×#106D21111yescharacteristic, complemented
128C2×C22⋊C8—2111no—
129#380—2111no—
130#340—2111no—
131#340—2111no—
132C23×C4—4221no—
133C23×C4—2111no—
134C22×C8—2111no—
135C22×C8C2×D21111yescharacteristic
136C2×C4⋊C4—4222no—
137C2×C4⋊C4—4221no—
138C2×C4⋊C4—4222no—
139C2×C4⋊C4—2111no—
140C2×C4⋊C4—2111no—
141C2×C4⋊C4C2×D21111yescharacteristic
142C2×C4⋊C4C2×D21111yescharacteristic
143C2×C22⋊C4—4221no—
144#106—22no—
145#106—4222no—
146C22⋊C8—2114no—
147#95—4221no—
148C24—2111no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—8442no—
152C22×C4—8442no—
153C22×C4—4224no—
154C22×C4—4221no—
155C22×C4—4221no—
156C22×C4—4221no—
157C22×C4—4221no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4—2112no—
162C22×C4—2112no—
163C22×C4C22×D21111yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—4221no—
167C2×C8—4221no—
168C2×C8—2112no—
169C2×C8C4×D21112yescomplemented
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—4224no—
174C4⋊C4—4221no—
175C4⋊C4—4221no—
176C4⋊C4—4221no—
177C4⋊C4—2112no—
178C4⋊C4C22⋊D21112yes—
179C22⋊C4—8442no—
180C23—4221no—
181C23—4221no—
182C23—4221no—
183C23—4221no—
184C23—2112no—
185C23—2112no—
186C23—2112no—
187C23D4×D21111yescharacteristic
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—8448no—
195C2×C4—8441no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—8441no—
200C2×C4—4222no—
201C2×C4—4221no—
202C2×C4—4221no—
203C2×C4—4222no—
204C2×C4—4222no—
205C2×C4—4222no—
206C2×C4—4222no—
207C2×C4—2112no—
208C2×C4C2×C4×D21111yescharacteristic
209C2×C4D4×D21111yescharacteristic
210C2×C4C2×C22⋊D21111yescharacteristic
211C2×C4Q8×D21111yescharacteristic
212C2×C4#1680996211yescharacteristic
213C2×C4#1680996311yescharacteristic
214C8—22no—
215C8—4222no—
216C22—4222no—
217C22—4222no—
218C22—4222no—
219C22—4222no—
220C22—4222no—
221C22—4222no—
222C22—2114no—
223C22C4⋊C4×D21111yescharacteristic
224C22#2026994511yescharacteristic
225C22#2026996211yescharacteristic
226C22SD16×D21111yescharacteristic
227C22SD16×D21111yescharacteristic
228C22#2027000411yescharacteristic
229C22#2027000611yescharacteristic
230C4—42no—
231C4—81no—
232C4—8442no—
233C4—16881no—
234C4—4222no—
235C4C4×C22⋊D21111yescharacteristic
236C4#2026993911yescharacteristic
237C4#2026995311yescharacteristic
238C4#2026995511yescharacteristic
239C2—4224no—
240C2#2721301711yescharacteristic
241C2#106×D21111yescharacteristic
242C2#2721318511yescharacteristic
243C2#2721328311yescharacteristic
244C2#2721329511yescharacteristic
245C2#2721346611yescharacteristic
246C2#2721347811yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#106×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
cosocleC22×C4×C211C23
hypercenterC2×#106D211
hypocenterC211#1853
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#20269908C2semidirect
5. #27212909C2semidirect
6. #27212908C2semidirect
7. C2×#106×C211C2semidirect
18. (C2×D211)⋊C8C4semidirect
39. C2×C211⋊C8C2×C4semidirect
51. C2×C8×C211C2×C4semidirect
105. C2×D211#106semidirect
123. C211#1853semidirect
127. C2×#106D211semidirect
169. C2×C8C4×D211semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2112
C2×#202699082
C2×(C2×D211)⋊C82
#272130032
#272129092
#272129082
C2×#106×C2112
#1853211