Finite Group Databasebeta

← #41014542

Subgroups — #41014542

Order 27008.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20269968C211yesmaximal, characteristic, complemented
2C2×#20269898C211yesmaximal, characteristic
3#27213049C211yesmaximal, characteristic
4#27212909C211yesmaximal, characteristic, complemented
5#27212908C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9#20269968—22no—
10#20269968—22no—
11C2×C4⋊Dic211—21no—
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×C22⋊Dic211—21no—
16C2×C22⋊Dic211—21no—
17C2×C4×Dic211C2211yescharacteristic
18#20269898—22no—
19C22×D4×C211C2211yescharacteristic
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22#102×C211—22no—
23C22×Dic211—21no—
24C22×Dic211—21no—
25C22×Dic211D411yescharacteristic
26C22×Dic211D411yescharacteristic
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—24no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D412yes—
35Dic211⋊C4—24no—
36C22⋊Dic211—42no—
37C22⋊Dic211—42no—
38C4×Dic211—21no—
39C4×Dic211—21no—
40C24×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211Q812yes—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D412yes—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C211⋊C8—41no—
67C23×C211—22no—
68C23×C211—41no—
69C23×C211—41no—
70C23×C211—41no—
71C23×C211—41no—
72C23×C211—41no—
73C23×C211—41no—
74C23×C211C2×D411yescharacteristic
75D4×C211—22no—
76D4×C211—22no—
77D4×C211—41no—
78D4×C211—41no—
79C2×C4×C211—22no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×Q811yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C8×C211—41no—
89C22×C211—42no—
90C22×C211—42no—
91C22×C211—42no—
92C22×C211—81no—
93C22×C211—81no—
94C22×C211—81no—
95C22×C211—81no—
96C22×C211#12011yescharacteristic
97C22×C211C22⋊Q811yescharacteristic
98C22×C211#12411yescharacteristic
99C22×C211#13511yescharacteristic
100C22×C211#13511yescharacteristic
101C22×C211AGL(1,ℤ/8)11yescharacteristic
102C22×C211AGL(1,ℤ/8)11yescharacteristic
103C4×C211—42no—
104C4×C211C22⋊Q811yescharacteristic
105C4×C211C22⋊Q811yescharacteristic
106C4×C211#12411yescharacteristic
107C4×C211#12411yescharacteristic
108Dic211—41no—
109Dic211—41no—
110Dic211—42no—
111Dic211—81no—
112C2×C211—82no—
113C2×C211#39311yescharacteristic
114C2×C211#44911yescharacteristic
115C2×C211#44911yescharacteristic
116C2×C211#47811yescharacteristic
117C2×C211#47811yescharacteristic
118C2×C211#48811yescharacteristic
119C2×C211#49011yescharacteristic
120C211#202511yescharacteristic, complemented, Sylow
121#2025—2111nomaximal, Sylow
122C2×#125—2111no—
123C2×#102—2111no—
124C2×#102D21111yescharacteristic
125#390—2111no—
126#340—2111no—
127#340—2111no—
128#336—2111no—
129C22×D4C2×D21111yescharacteristic
130C22×C8—2111no—
131C22×C8C2×D21111yescharacteristic
132#125—4222no—
133#125—4222no—
134C2×C4⋊C4—4221no—
135C2×C4⋊C4—4221no—
136C2×C4⋊C4—2111no—
137C2×C4⋊C4—2111no—
138C2×C4⋊C4C2×D21111yescharacteristic
139C2×C22⋊C4—4221no—
140C2×C22⋊C4—4221no—
141C2×C42—2111no—
142#102—22no—
143#102—4222no—
144C24—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4Dic42212yes—
151C22×C4—21no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4—2111no—
155C22×C4—2111no—
156C22×C4C22×D21111yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—4221no—
161C2×C8—4221no—
162C2×C8—4221no—
163C4⋊C4—21no—
164C4⋊C4—4224no—
165C4⋊C4—4224no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C4⋊C4—4221no—
169C4⋊C4—2112no—
170C4⋊C4C22⋊D21112yes—
171C22⋊C4—8442no—
172C22⋊C4—8442no—
173C42—4221no—
174C42—4221no—
175C23—22no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23D4×D21111yescharacteristic
183D4—22no—
184D4—22no—
185D4—41no—
186D4—41no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—4222no—
198C2×C4—4221no—
199C2×C4—4221no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4D4×D21111yescharacteristic
203C2×C4C2×C22⋊D21111yescharacteristic
204C2×C4C2×Dic42211yescharacteristic
205C2×C4#1680996111yescharacteristic
206C2×C4#1680996111yescharacteristic
207C2×C4#1680996211yescharacteristic
208C8—41no—
209C8—8441no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22#2026994511yescharacteristic
218C22#2026995411yescharacteristic
219C22#2026996311yescharacteristic
220C22#2026999611yescharacteristic
221C22#2026999811yescharacteristic
222C22#2027000111yescharacteristic
223C22#2027000411yescharacteristic
224C4—42no—
225C4—8441no—
226C4—8442no—
227C4—8441no—
228C4—16881no—
229C4#2026995411yescharacteristic
230C4#2026995611yescharacteristic
231C4#2026996311yescharacteristic
232C4#2026996411yescharacteristic
233C2—82no—
234C2#2721305811yescharacteristic
235C2#2721328811yescharacteristic
236C2#2721329011yescharacteristic
237C2#2721344411yescharacteristic
238C2#2721344811yescharacteristic
239C2#2721349911yescharacteristic
240C2#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#2025
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
1. C2×#20269968C2semidirect
4. #27212909C2semidirect
5. #27212908C2semidirect
6. #27212900C2semidirect
120. C211#2025semidirect

Maximal subgroups

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

subgroupindex
C2×#202699682
C2×#202698982
#272130492
#272129092
#272129082
#272129002
C2×#102×C2112
#2025211