Finite Group Databasebeta

← #41014835

Subgroups — #41014835

Order 27008.

256 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269908C211yesmaximal, characteristic, complemented
2C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
3#27213006C211yesmaximal, characteristic
4#27212908C211yesmaximal, characteristic, complemented
5#27212908C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#380×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×C22⋊Dic211—21no—
13#20269908—22no—
14C22⋊(C211⋊C8)—22no—
15C211⋊#95—21no—
16C23×C4×C211C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C22⋊C4×C211—21no—
20#95×C211—21no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C2×C211⋊C8—21no—
24C2×C211⋊C8—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8D412yes—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C22⋊Dic211—42no—
34C24×C211D411yescharacteristic
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211—21no—
38C22×C4×C211—21no—
39C22×C4×C211—21no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211—21no—
43C22×C4×C211D411yescharacteristic
44C22×C4×C211C2311yescharacteristic
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C22⋊C4×C211—42no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C211⋊C8—22no—
57C211⋊C8—41no—
58C23×C211—21no—
59C23×C211—21no—
60C23×C211—21no—
61C23×C211—21no—
62C23×C211—21no—
63C23×C211—21no—
64C23×C211SD1612yes—
65C23×C211C2×D411yescharacteristic
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C2×C4×C211#4111yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C22×C211—22no—
86C22×C211—22no—
87C22×C211—22no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211#12111yescharacteristic
93C22×C211#12311yescharacteristic
94C22×C211#12311yescharacteristic
95C22×C211C2×SD1611yescharacteristic
96C22×C211#13511yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C22×C211#13711yescharacteristic
99C4×C211—41no—
100C4×C211—81no—
101C4×C211—81no—
102C4×C211#12311yescharacteristic
103C4×C211#12411yescharacteristic
104C4×C211#12611yescharacteristic
105C4×C211#12611yescharacteristic
106Dic211—81no—
107Dic211—81no—
108C2×C211—42no—
109C2×C211#39711yescharacteristic
110C2×C211#46511yescharacteristic
111C2×C211#46811yescharacteristic
112C2×C211#48111yescharacteristic
113C2×C211#48211yescharacteristic
114C2×C211#48311yescharacteristic
115C2×C211#48511yescharacteristic
116C211#207711yescharacteristic, complemented, Sylow
117#2077—2111nomaximal, Sylow
118C2×#106—2111no—
119C2×C22⋊C8—2111no—
120#380—2111no—
121#380D21111yescharacteristic
122#340—2111no—
123#340—2111no—
124#340—2111no—
125C23×C4C2×D21111yescharacteristic
126C22×C8—2111no—
127C22×C8—2111no—
128C2×C4⋊C4—2111no—
129C2×C4⋊C4—2111no—
130C2×C4⋊C4C2×D21111yescharacteristic
131C2×C4⋊C4C2×D21111yescharacteristic
132C2×C22⋊C4—21no—
133C2×C22⋊C4—4221no—
134#106—4222no—
135C22⋊C8—4222no—
136#95—21no—
137#95—4221no—
138C24C22⋊D21111yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—4221no—
148C22×C4—4221no—
149C22×C4C22⋊D21111yescharacteristic
150C22×C4C22×D21111yescharacteristic
151C2×C8—4221no—
152C2×C8—4221no—
153C2×C8—4221no—
154C2×C8—4221no—
155C2×C8—4221no—
156C2×C8—2112no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—4221no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—4221no—
165C22⋊C4—42no—
166C22⋊C4—8442no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23#1680994912yes—
174C23C2×C22⋊D21111yescharacteristic
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—8441no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4D4×D21111yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996111yescharacteristic
197C2×C4#1680996111yescharacteristic
198C2×C4#1680996211yescharacteristic
199C2×C4#1680996211yescharacteristic
200C8—8441no—
201C8—4222no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22#2026994911yescharacteristic
210C22#2026995811yescharacteristic
211C22#2026996011yescharacteristic
212C22C2×#1680994911yescharacteristic
213C22#2026999711yescharacteristic
214C22#2027000311yescharacteristic
215C22#2027000711yescharacteristic
216C4—41no—
217C4—81no—
218C4—81no—
219C4—16881no—
220C4—16881no—
221C4#2026995911yescharacteristic
222C4#2026996311yescharacteristic
223C4#2026997011yescharacteristic
224C4#2026997011yescharacteristic
225C2—42no—
226C2#2721307411yescharacteristic
227C2#2721337711yescharacteristic
228C2#2721339011yescharacteristic
229C2#2721346411yescharacteristic
230C2#2721346711yescharacteristic
231C2#2721347411yescharacteristic
232C2#2721348711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
hypercenter#380D211
hypocenterC211#2077
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#20269908C2semidirect
4. #27212908C2semidirect
5. #27212908C2semidirect
6. #27212908C2semidirect
116. C211#2077semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
C2×C22⋊(C211⋊C8)2
#272130062
#272129082
#380×C2112
#2077211