Finite Group Databasebeta

← #41011952

Subgroups — #41011952

Order 27008.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#27213657C211yesmaximal, characteristic
2C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
3#27212882C212yesmaximal, complemented
4#27212879C212yesmaximal, complemented
5#381×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C411yescharacteristic, complemented
7C22×C211⋊C8—21no—
8#20269960—21no—
9#20269956C411yescharacteristic, complemented
10#20269956C2211yescharacteristic
11#20269951C411yescharacteristic, complemented
12#20269951C2211yescharacteristic
13#20269946—21no—
14C4×C22⋊D211—21no—
15C4×C22⋊D211—21no—
16#20269936C411yescharacteristic, complemented
17C22⋊(C211⋊C8)—21no—
18C22⋊(C211⋊C8)C2212yes—
19C23×C4×C211C2211yescharacteristic
20C2×C22⋊C4×C211—21no—
21#95×C211—21no—
22#95×C211C2212yes—
23#16809962—42no—
24C2×Dic422C2×C411yescharacteristic
25C2×D844C2×C411yescharacteristic
26C2×C22⋊D211—21no—
27C2×C22⋊D211—21no—
28C2×C22⋊D211—21no—
29C2×C4×D211—21no—
30C2×C211⋊C8—22no—
31C2×C211⋊C8—42no—
32C4⋊Dic211C2×C411yescharacteristic
33C4⋊Dic211C2×C411yescharacteristic
34Dic211⋊C4—21no—
35Dic211⋊C4—21no—
36C22⋊Dic211—21no—
37C22⋊Dic211—21no—
38C22⋊Dic211—21no—
39(C2×D211)⋊C4—21no—
40(C2×D211)⋊C4—21no—
41C4×Dic211—21no—
42C24×C211D411yescharacteristic
43C22×C4×C211—22no—
44C22×C4×C211—42no—
45C22×C4×C211C2×C411yescharacteristic
46C22×C4×C211C2×C411yescharacteristic
47C22×C4×C211D411yescharacteristic
48C22×C4×C211D411yescharacteristic
49C22×C4×C211D411yescharacteristic
50C22×C4×C211C2311yescharacteristic
51C22⋊C4×C211—42no—
52C22×D211—21no—
53Dic422—41no—
54D844—41no—
55C22⋊D211—42no—
56C22⋊D211—42no—
57C22⋊D211—42no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C4×D211—42no—
62C211⋊C8—42no—
63C23×C211—22no—
64C23×C211—22no—
65C23×C211—41no—
66C23×C211C22⋊C411yescharacteristic
67C23×C211C22⋊C411yescharacteristic
68C23×C211C2×D411yescharacteristic
69C2×C4×C211—22no—
70C2×C4×C211—22no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—42no—
76C2×C4×C211—42no—
77C2×C4×C211—42no—
78C2×C4×C211C22⋊C411yescharacteristic
79C2×C4×C211C22⋊C411yescharacteristic
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C22×C211—21no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—44no—
90C22×C211#9912yes—
91C22×C211C2×C22⋊C411yescharacteristic
92C22×C211#13512yes—
93C2×D211—41no—
94C2×D211—41no—
95C2×D211—41no—
96C4×C211—21no—
97C4×C211—21no—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—82no—
101Dic211—41no—
102Dic211—41no—
103Dic211—41no—
104C2×C211—22no—
105C2×C211—22no—
106C2×C211—41no—
107C2×C211C2×#9911yescharacteristic
108C2×C211#41611yescharacteristic
109C2×C211#42111yescharacteristic
110D211—81no—
111C211#151611yescharacteristic, complemented, Sylow
112#1516—2111nomaximal, Sylow
113#525—2111no—
114C2×C22⋊C8—2111no—
115#381D21111yescharacteristic, complemented
116#328—2112no—
117#327—2112no—
118C2×#41—2111no—
119C23×C4C2×D21111yescharacteristic
120C22×C8—4221no—
121#123—4221no—
122C22⋊Q8—2111no—
123C22⋊Q8—2111no—
124#121—2111no—
125#121—2111no—
126#120—4221no—
127C4×D4—4221no—
128C4×D4—4221no—
129#117—2111no—
130C2×C22⋊C4—21no—
131C22⋊C8—4221no—
132C22⋊C8—2112no—
133#95—21no—
134#95C2×D21112yes—
135C24C22⋊D21111yescharacteristic
136#41—8442no—
137C2×Q8—2111no—
138C2×D4—4221no—
139C2×D4—4221no—
140C2×D4—4221no—
141C2×D4—2111no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—4221no—
145C22×C4C4×D21111yescharacteristic
146C22×C4C4×D21111yescharacteristic
147C22×C4C22⋊D21111yescharacteristic
148C22×C4D84411yescharacteristic
149C22×C4D84411yescharacteristic
150C22×C4C22×D21111yescharacteristic
151C2×C8—8442no—
152C2×C8—4222no—
153C4⋊C4—4221no—
154C4⋊C4—4221no—
155C4⋊C4—2111no—
156C4⋊C4—2111no—
157C22⋊C4—42no—
158C22⋊C4—4221no—
159C22⋊C4—4221no—
160C22⋊C4—4221no—
161C22⋊C4—4221no—
162C22⋊C4—4221no—
163C42—4221no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—4221no—
168C23(C2×D211)⋊C411yescharacteristic
169C23(C2×D211)⋊C411yescharacteristic
170C23C2×C22⋊D21111yescharacteristic
171Q8—8441no—
172D4—8441no—
173D4—8442no—
174D4—8442no—
175D4—8442no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—8442no—
186C2×C4—4221no—
187C2×C4—4221no—
188C2×C4—4221no—
189C2×C4(C2×D211)⋊C411yescharacteristic
190C2×C4(C2×D211)⋊C411yescharacteristic
191C2×C4C2×C4×D21111yescharacteristic
192C2×C4C2×D84411yescharacteristic
193C8—8442no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—8441no—
203C22—8441no—
204C22—8441no—
205C22#2026989112yes—
206C22C2×(C2×D211)⋊C411yescharacteristic
207C22#2026999712yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—82no—
213C4—8441no—
214C4—8441no—
215C4—8441no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—16881no—
220C2C2×#2026989111yescharacteristic
221C2#2721314011yescharacteristic
222C2#2721316411yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#381×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenter#381D211
hypocenterC211#1516
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C211⋊C8)C2semidirect
3. #27212882C2semidirect
4. #27212879C2semidirect
5. #381×C211C2semidirect
6. C2×#16809962C4semidirect
9. #20269956C4semidirect
11. #20269951C4semidirect
16. #20269936C4semidirect
111. C211#1516semidirect
115. #381D211semidirect

Maximal subgroups

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

subgroupindex
#272136572
C2×C22⋊(C211⋊C8)2
#272128822
#272128792
#381×C2112
#1516211