Finite Group Databasebeta

← #41014552

Subgroups — #41014552

Order 27008.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213774C211yesmaximal, characteristic, complemented
2#27213151C211yesmaximal, characteristic
3#27213049C211yesmaximal, characteristic
4#27212914C211yesmaximal, characteristic, complemented
5#27212907C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#418×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269977—21no—
10#20269976—21no—
11#20269969—21no—
12#20269968—21no—
13#20269954—22no—
14#20269936C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16C2×C22⋊Dic211—22no—
17C2×C4×Dic211C2211yescharacteristic
18#20269898—21no—
19#20269898—21no—
20C22×D4×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23#102×C211—21no—
24#102×C211—21no—
25C2×Dic422—22no—
26C22×Dic211—21no—
27C22×Dic211D412yes—
28#16809946—21no—
29#16809946—21no—
30C2×C211⋊C8—21no—
31C4⋊Dic211—22no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—22no—
37Dic211⋊C4—22no—
38C22⋊Dic211—21no—
39C22⋊Dic211—44no—
40C4×Dic211—21no—
41C4×Dic211—21no—
42C4×Dic211—21no—
43C24×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211Q812yescomplemented
50C22×C4×C211C2311yescharacteristic
51M16×C211—21no—
52M16×C211—21no—
53C2×C8×C211—21no—
54C4⋊C4×C211D411yescharacteristic
55C4⋊C4×C211D411yescharacteristic
56C22⋊C4×C211—21no—
57C42×C211—21no—
58Dic422—82no—
59C2×Dic211—22no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—22no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—42no—
66C211⋊C8—41no—
67C23×C211—22no—
68C23×C211—42no—
69C23×C211—42no—
70C23×C211—41no—
71C23×C211—41no—
72C23×C211C2×D411yescharacteristic
73D4×C211—22no—
74D4×C211—22no—
75D4×C211—41no—
76D4×C211—41no—
77C2×C4×C211—21no—
78C2×C4×C211—21no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×Q811yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—41no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—44no—
89C22×C211—42no—
90C22×C211—82no—
91C22×C211—81no—
92C22×C211—81no—
93C22×C211#12011yescharacteristic
94C22×C211C22⋊Q811yescharacteristic
95C22×C211#12411yescharacteristic
96C4×C211—41no—
97C4×C211—41no—
98C4×C211C22⋊Q811yescharacteristic
99C4×C211C22⋊Q811yescharacteristic
100C4×C211#12411yescharacteristic
101C4×C211#12411yescharacteristic
102Dic211—42no—
103Dic211—42no—
104Dic211—41no—
105Dic211—41no—
106C2×C211—21no—
107C2×C211—21no—
108C2×C211—82no—
109C2×C211#39311yescharacteristic
110C2×C211#45311yescharacteristic
111C2×C211#45411yescharacteristic
112C211#202711yescharacteristic, complemented, Sylow
113#2027—2111nomaximal, Sylow
114#544—2111no—
115#418—2111no—
116#418D21111yescharacteristic
117#390—2111no—
118#343—2111no—
119#339—2111no—
120#339—2111no—
121C22×D4C2×D21111yescharacteristic
122C2×M16—2111no—
123C2×M16C2×D21111yescharacteristic
124C4⋊Q8—4221no—
125C4⋊Q8—4221no—
126#125—4221no—
127#125—4221no—
128C22⋊Q8—4222no—
129#117—2111no—
130#117C2×D21111yescharacteristic
131C2×C4⋊C4—2111no—
132C2×C22⋊C4—4222no—
133C2×C42—2111no—
134#102—21no—
135#102—21no—
136#102—4221no—
137#102—4221no—
138C24—21no—
139C2×Q8—4222no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4Dic42212yescomplemented
146C22×C4—4221no—
147C22×C4—2112no—
148C22×C4C22×D21111yescharacteristic
149M16—21no—
150M16—21no—
151M16—4221no—
152M16—4221no—
153C2×C8—21no—
154C2×C8—4221no—
155C4⋊C4—4222no—
156C4⋊C4—4222no—
157C4⋊C4—4222no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—2111no—
161C4⋊C4—2111no—
162C4⋊C4C22⋊D21111yescharacteristic
163C4⋊C4C22⋊D21111yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—8444no—
166C22⋊C4—4221no—
167C42—21no—
168C42—4221no—
169C42—4221no—
170C42—4221no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—41no—
175C23—41no—
176C23D4×D21111yescharacteristic
177Q8—16882no—
178D4—22no—
179D4—22no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—8442no—
187C2×C4—4222no—
188C2×C4—4221no—
189C2×C4—4221no—
190C2×C4—4222no—
191C2×C4D4×D21111yescharacteristic
192C2×C4C2×C22⋊D21111yescharacteristic
193C2×C4C2×Dic42211yescharacteristic
194C2×C4#1680996111yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996211yescharacteristic
197C8—41no—
198C8—8441no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22#2026994511yescharacteristic
207C22#2026995411yescharacteristic
208C22#2026996311yescharacteristic
209C4—41no—
210C4—41no—
211C4—8442no—
212C4—8442no—
213C4—8441no—
214C4—8441no—
215C4#2026995411yescharacteristic
216C4#2026995611yescharacteristic
217C4#2026996311yescharacteristic
218C4#2026996411yescharacteristic
219C2—21no—
220C2—21no—
221C2—82no—
222C2#2721305811yescharacteristic
223C2#2721331211yescharacteristic
224C2#2721331711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#418×C211C2
socleC22×C211#120
prime radicalC22×D4×C211C22
hypercenter#418D211
hypocenterC211#2027
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #27213774C2semidirect
4. #27212914C2semidirect
5. #27212907C2semidirect
6. #27212905C2semidirect
49. C2×D4×C211Q8semidirect
112. C211#2027semidirect
145. C2×D4Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137742
#272131512
#272130492
#272129142
#272129072
#272129052
#418×C2112
#2027211