Finite Group Databasebeta

← #41014836

Subgroups — #41014836

Order 27008.

256 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 79 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×(C2×D211)⋊C8C211yesmaximal, characteristic
3#27213003C211yesmaximal, characteristic
4#27213003C211yesmaximal, characteristic
5#27212909C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×(C2×D211)⋊C4—21no—
13C2×(C2×D211)⋊C4—21no—
14#20269908—22no—
15(C2×D211)⋊C8—22no—
16C211⋊#95—21no—
17C211⋊#95—21no—
18C22×C8×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C23×D211D411yescharacteristic
22C22×Dic211—21no—
23C22×Dic211—21no—
24C22×Dic211D411yescharacteristic
25C2×C4×D211—21no—
26C2×C4×D211—21no—
27C2×C4×D211—21no—
28C2×C4×D211—21no—
29C2×C4×D211—21no—
30C2×C4×D211—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8D412yes—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38(C2×D211)⋊C4—42no—
39(C2×D211)⋊C4—42no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211C2311yescharacteristic
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C22×D211—21no—
51C22×D211—21no—
52C22×D211—21no—
53C22×D211—21no—
54C22×D211—21no—
55C22×D211—21no—
56C22×D211SD1612yes—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C4×D211—41no—
67C4×D211—41no—
68C4×D211—41no—
69C4×D211—41no—
70C211⋊C8—22no—
71C23×C211C2×D411yescharacteristic
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×C211C2×D411yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C2×C4×C211#4111yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C8×C211—41no—
85C22×C211#12111yescharacteristic
86C22×C211#12311yescharacteristic
87C22×C211#12311yescharacteristic
88C22×C211C2×SD1611yescharacteristic
89C22×C211#13511yescharacteristic
90C22×C211AGL(1,ℤ/8)11yescharacteristic
91C22×C211#13711yescharacteristic
92C2×D211—22no—
93C2×D211—22no—
94C2×D211—22no—
95C2×D211—41no—
96C2×D211—41no—
97C2×D211—41no—
98C2×D211—41no—
99C4×C211—81no—
100C4×C211—81no—
101C4×C211#12311yescharacteristic
102C4×C211#12411yescharacteristic
103C4×C211#12611yescharacteristic
104C4×C211#12611yescharacteristic
105Dic211—41no—
106Dic211—81no—
107Dic211—81no—
108C2×C211#39711yescharacteristic
109C2×C211#46511yescharacteristic
110C2×C211#46811yescharacteristic
111C2×C211#48111yescharacteristic
112C2×C211#48211yescharacteristic
113C2×C211#48311yescharacteristic
114C2×C211#48511yescharacteristic
115D211—42no—
116C211#207711yescharacteristic, complemented, Sylow
117#2077—2111nomaximal, Sylow
118C2×#106—2111no—
119C2×C22⋊C8—2111no—
120#380—2111no—
121#380—2111no—
122#340—2111no—
123#340—2111no—
124#340D21111yescharacteristic, complemented
125C23×C4—2111no—
126C22×C8—2111no—
127C22×C8C2×D21111yescharacteristic
128C2×C4⋊C4—2111no—
129C2×C4⋊C4—2111no—
130C2×C4⋊C4C2×D21111yescharacteristic
131C2×C4⋊C4C2×D21111yescharacteristic
132C2×C22⋊C4—4221no—
133C2×C22⋊C4—4221no—
134#106—4222no—
135C22⋊C8—4222no—
136#95—4221no—
137#95—4221no—
138C24—2111no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—4221no—
142C22×C4—4221no—
143C22×C4—4221no—
144C22×C4—4221no—
145C22×C4—4221no—
146C22×C4—4221no—
147C22×C4—4221no—
148C22×C4—4221no—
149C22×C4—2111no—
150C22×C4C22×D21111yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
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—8442no—
166C22⋊C4—8442no—
167C23—4221no—
168C23—4221no—
169C23—4221no—
170C23—4221no—
171C23—4221no—
172C23—4221no—
173C23—2112no—
174C23D4×D21111yescharacteristic
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—8441no—
182C2×C4—8441no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—8441no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—4221no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4D4×D21111yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996211yescharacteristic
197C2×C4#1680996211yescharacteristic
198C2×C4#1680996211yescharacteristic
199C2×C4#1680996311yescharacteristic
200C8—41no—
201C8—4222no—
202C22—8441no—
203C22—8441no—
204C22—8441no—
205C22—8441no—
206C22—4222no—
207C22—4222no—
208C22—4222no—
209C22#2026995011yescharacteristic
210C22#2026995911yescharacteristic
211C22#2026996211yescharacteristic
212C22SD16×D21111yescharacteristic
213C22#2026999611yescharacteristic
214C22#2027000411yescharacteristic
215C22#2027000911yescharacteristic
216C4—81no—
217C4—81no—
218C4—8441no—
219C4—16881no—
220C4—16881no—
221C4#2026996211yescharacteristic
222C4#2026996311yescharacteristic
223C4#2026997111yescharacteristic
224C4#2026997211yescharacteristic
225C2—8442no—
226C2#2721307511yescharacteristic
227C2#2721337411yescharacteristic
228C2#2721339511yescharacteristic
229C2#2721346611yescharacteristic
230C2#2721346811yescharacteristic
231C2#2721347511yescharacteristic
232C2#2721348911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
hypercenter#340D211
hypocenterC211#2077
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#20269908C2semidirect
5. #27212909C2semidirect
6. #27212908C2semidirect
7. #340×C211C2semidirect
116. C211#2077semidirect
124. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
C2×(C2×D211)⋊C82
#272130032
#272129092
#272129082
#340×C2112
#2077211