Finite Group Databasebeta

← #41014814

Subgroups — #41014814

Order 27008.

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

normal subgroups only

This group has 258 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×#20269898C211yesmaximal, characteristic
2#27213049C211yesmaximal, characteristic
3#27213011C211yesmaximal, characteristic, complemented
4#27212909C211yesmaximal, characteristic, complemented
5#27212908C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C22⋊Dic211—21no—
12C2×C22⋊Dic211—21no—
13C2×C4×Dic211C2211yescharacteristic
14#20269898—22no—
15C211⋊#95—21no—
16C211⋊#95—21no—
17C22×D4×C211C2211yescharacteristic
18C22×C8×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20#102×C211—22no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C22×Dic211D411yescharacteristic
24C22×Dic211D411yescharacteristic
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C22⋊Dic211—42no—
33C22⋊Dic211—42no—
34C4×Dic211—21no—
35C4×Dic211—21no—
36C24×C211—21no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211D412yes—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45C2×C8×C211—21no—
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C2×Dic211—21no—
51C2×Dic211—21no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C211⋊C8—41no—
63C23×C211—22no—
64C23×C211—41no—
65C23×C211—41no—
66C23×C211—41no—
67C23×C211—41no—
68C23×C211—41no—
69C23×C211—41no—
70C23×C211C2×D411yescharacteristic
71D4×C211—22no—
72D4×C211—22no—
73D4×C211—41no—
74D4×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—42no—
86C22×C211—42no—
87C22×C211—42no—
88C22×C211—81no—
89C22×C211—81no—
90C22×C211—81no—
91C22×C211—81no—
92C22×C211#12111yescharacteristic
93C22×C211#12311yescharacteristic
94C22×C211#12411yescharacteristic
95C22×C211#13511yescharacteristic
96C22×C211#13511yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C22×C211AGL(1,ℤ/8)11yescharacteristic
99C4×C211—81no—
100C4×C211#12311yescharacteristic
101C4×C211#12311yescharacteristic
102C4×C211#12611yescharacteristic
103C4×C211#12611yescharacteristic
104Dic211—41no—
105Dic211—41no—
106Dic211—81no—
107Dic211—81no—
108C2×C211—82no—
109C2×C211#39711yescharacteristic
110C2×C211#46311yescharacteristic
111C2×C211#46311yescharacteristic
112C2×C211#48211yescharacteristic
113C2×C211#48211yescharacteristic
114C2×C211#48811yescharacteristic
115C2×C211#49011yescharacteristic
116C211#207311yescharacteristic, complemented, Sylow
117#2073—2111nomaximal, Sylow
118C2×#102—2111no—
119C2×#102D21111yescharacteristic
120#390—2111no—
121#382—2111no—
122#340—2111no—
123#340—2111no—
124#336—2111no—
125C22×D4C2×D21111yescharacteristic
126C22×C8—2111no—
127C22×C8C2×D21111yescharacteristic
128C2×C4⋊C4—2111no—
129C2×C4⋊C4—2111no—
130C2×C4⋊C4C2×D21111yescharacteristic
131C2×C22⋊C4—4221no—
132C2×C22⋊C4—4221no—
133C2×C42—2111no—
134#102—22no—
135#102—4222no—
136#95—4221no—
137#95—4221no—
138C24—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C22⋊D21112yes—
145C22×C4—21no—
146C22×C4—4221no—
147C22×C4—4221no—
148C22×C4—2111no—
149C22×C4—2111no—
150C22×C4C22×D21111yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—4221no—
155C2×C8—4221no—
156C2×C8—4221no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—4221no—
160C4⋊C4—4221no—
161C4⋊C4—4221no—
162C4⋊C4—4221no—
163C22⋊C4—8442no—
164C22⋊C4—8442no—
165C42—4221no—
166C42—4221no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23D4×D21111yescharacteristic
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—8441no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—4221no—
189C2×C4—4221no—
190C2×C4—4221no—
191C2×C4—4221no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4C2×C22⋊D21111yescharacteristic
195C2×C4#1680996111yescharacteristic
196C2×C4#1680996111yescharacteristic
197C2×C4#1680996111yescharacteristic
198C2×C4#1680996211yescharacteristic
199C2×C4#1680996211yescharacteristic
200C8—41no—
201C8—8441no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22#2026994911yescharacteristic
210C22#2026995911yescharacteristic
211C22#2026996311yescharacteristic
212C22#2026999611yescharacteristic
213C22#2026999811yescharacteristic
214C22#2027000111yescharacteristic
215C22#2027000411yescharacteristic
216C4—81no—
217C4—8441no—
218C4—8441no—
219C4—16881no—
220C4—16881no—
221C4#2026995811yescharacteristic
222C4#2026996011yescharacteristic
223C4#2026997011yescharacteristic
224C4#2026997011yescharacteristic
225C2—82no—
226C2#2721307411yescharacteristic
227C2#2721335911yescharacteristic
228C2#2721336311yescharacteristic
229C2#2721346811yescharacteristic
230C2#2721346911yescharacteristic
231C2#2721349911yescharacteristic
232C2#2721350911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenterC2×#102D211
hypocenterC211#2073
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
3. #27213011C2semidirect
4. #27212909C2semidirect
5. #27212908C2semidirect
6. #27212900C2semidirect
116. C211#2073semidirect

Maximal subgroups

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

subgroupindex
C2×#202698982
#272130492
#272130112
#272129092
#272129082
#272129002
C2×#102×C2112
#2073211