Finite Group Databasebeta

← #41012339

Subgroups — #41012339

Order 27008.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213775C211yesmaximal, characteristic, complemented
2#27213299C211yesmaximal, characteristic
3#27213296C211yesmaximal, characteristic
4#27213064C211yesmaximal, characteristic
5#27212882C211yesmaximal, characteristic, complemented
6#27212871C211yesmaximal, characteristic, complemented
7#328×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809951—21no—
10C2×#16809949—21no—
11#20269976—21no—
12#20269975—21no—
13#20269967—21no—
14#20269967—21no—
15#20269956C2211yescharacteristic
16#20269954—21no—
17#20269933—21no—
18C2×Dic211⋊C4C2211yescharacteristic
19C2×(C2×D211)⋊C4—21no—
20C2×(C2×D211)⋊C4—21no—
21#20269899—21no—
22#20269897—21no—
23C22⋊(C211⋊C8)C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211—21no—
28C2×Dic422—21no—
29C2×Dic422D411yescharacteristic
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—42no—
33C22×Dic211D411yescharacteristic
34C22×Dic211D411yescharacteristic
35#16809951—42no—
36#16809949—42no—
37C2×C211⋊C8—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45Dic211⋊C4—21no—
46Dic211⋊C4—21no—
47C22⋊Dic211—21no—
48(C2×D211)⋊C4—42no—
49(C2×D211)⋊C4—42no—
50C4×Dic211—21no—
51C4×Dic211—21no—
52C2×Q8×C211D411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55C2×C8×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211D411yescharacteristic
58C22⋊C4×C211—21no—
59C22×D211—22no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211—41no—
65C22×D211—41no—
66Dic422—41no—
67Dic422—81no—
68D844—22no—
69D844—22no—
70D844—41no—
71D844—41no—
72C2×Dic211—21no—
73C2×Dic211—21no—
74C2×Dic211—21no—
75C2×Dic211—21no—
76C2×Dic211—21no—
77C2×Dic211—41no—
78C2×Dic211SD1612yescomplemented
79C211⋊C8—41no—
80C23×C211C2×D411yescharacteristic
81Q8×C211—41no—
82C2×C4×C211—21no—
83C2×C4×C211—21no—
84C2×C4×C211—41no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C8×C211—41no—
91C22×C211—21no—
92C22×C211—21no—
93C22×C211#12011yescharacteristic
94C22×C211C2×SD1611yescharacteristic
95C22×C211#13711yescharacteristic
96C2×D211—42no—
97C2×D211—42no—
98C2×D211—42no—
99C2×D211—81no—
100C2×D211—81no—
101C2×D211—81no—
102C2×D211—81no—
103C4×C211—21no—
104C4×C211—21no—
105C4×C211—41no—
106C4×C211—41no—
107C4×C211—81no—
108Dic211—22no—
109Dic211—41no—
110Dic211—41no—
111Dic211—41no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211#44811yescharacteristic
115C2×C211#45411yescharacteristic
116C2×C211B(4,𝔽2)11yescharacteristic
117D211—82no—
118C211#159911yescharacteristic, complemented, Sylow
119#1599—2111nomaximal, Sylow
120#544—2111no—
121#450—2111no—
122#450—2111no—
123#394—2111no—
124#328—2111no—
125#328D21111yescharacteristic, complemented
126#324—2111no—
127C22×D4—2111no—
128C2×SD16—4221no—
129C2×SD16—4221no—
130C4⋊Q8—4221no—
131C4⋊Q8—4221no—
132#125—4221no—
133#125—4221no—
134C22⋊Q8—4221no—
135C22⋊Q8—2111no—
136C22⋊Q8C2×D21111yescharacteristic
137#117—4221no—
138C2×C4⋊C4—2111no—
139C2×C22⋊C4—4221no—
140C2×C22⋊C4—4221no—
141#102—4221no—
142#102—4221no—
143C22⋊C8—2111no—
144C22⋊C8C2×D21111yescharacteristic
145#95C2×D21111yescharacteristic
146C24—4221no—
147C2×Q8—4221no—
148C2×Q8—2111no—
149C2×Q8D84411yescharacteristic
150C2×D4—8442no—
151C2×D4—4221no—
152C2×D4—4221no—
153C22×C4—21no—
154C22×C4—2111no—
155C22×C4—2111no—
156C22×C4C22×D21111yescharacteristic
157SD16—8442no—
158SD16—8442no—
159C2×C8—21no—
160C2×C8—4221no—
161C4⋊C4—21no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—4221no—
165C4⋊C4—4221no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C4⋊C4—4221no—
169C4⋊C4—4221no—
170C4⋊C4—2111no—
171C4⋊C4D84411yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—8442no—
174C22⋊C4—8442no—
175C22⋊C4—4221no—
176C42—4221no—
177C42—4221no—
178C23—8441no—
179C23—8441no—
180C23—8441no—
181C23—8441no—
182C23—8441no—
183C23—8441no—
184C23—4222no—
185C23D4×D21111yescharacteristic
186Q8—41no—
187Q8—8441no—
188Q8—16881no—
189D4—8441no—
190D4—8441no—
191D4—4222no—
192D4—4222no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—8441no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4—4221no—
203C2×C4—4221no—
204C2×C4—4221no—
205C2×C4—2112no—
206C2×C4D4×D21111yescharacteristic
207C2×C4C2×D84411yescharacteristic
208C8—41no—
209C8—8441no—
210C22—21no—
211C22—21no—
212C22—8442no—
213C22—8442no—
214C22—8442no—
215C22—16881no—
216C22—16881no—
217C22—16881no—
218C22—16881no—
219C22#2026994711yescharacteristic
220C22SD16×D21111yescharacteristic
221C22#2027000911yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—8441no—
228C4—8441no—
229C4—8441no—
230C4—4222no—
231C2—21no—
232C2—21no—
233C2—16882no—
234C2#2721328611yescharacteristic
235C2#2721332011yescharacteristic
236C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211#120
prime radicalC22×D844C22
hypercenter#328D211
hypocenterC211#1599
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27213775C2semidirect
5. #27212882C2semidirect
6. #27212871C2semidirect
7. #328×C211C2semidirect
78. C2×Dic211SD16semidirect
118. C211#1599semidirect
125. #328D211semidirect

Maximal subgroups

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

subgroupindex
#272137752
#272132992
#272132962
#272130642
#272128822
#272128712
#328×C2112
#1599211