Finite Group Databasebeta

← #41014543

Subgroups — #41014543

Order 27008.

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

normal subgroups only

This group has 294 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×#20269968C211yesmaximal, characteristic, complemented
2C2×#20269899C211yesmaximal, characteristic
3C2×#20269897C211yesmaximal, characteristic
4#27213047C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269968—22no—
11#20269968—22no—
12C2×C4⋊Dic211—21no—
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×(C2×D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17C2×C4×Dic211C2211yescharacteristic
18#20269899—22no—
19#20269897—22no—
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C23×D211—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844Q812yes—
30C22×Dic211—21no—
31C22×Dic211D411yescharacteristic
32C22×Dic211D411yescharacteristic
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C4⋊Dic211—24no—
37C4⋊Dic211—21no—
38C4⋊Dic211D412yes—
39Dic211⋊C4—24no—
40(C2×D211)⋊C4—42no—
41(C2×D211)⋊C4—42no—
42C4×Dic211—21no—
43C4×Dic211—21no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22×C4×C211C2311yescharacteristic
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D412yes—
54C22×D211—22no—
55C22×D211—41no—
56C22×D211—41no—
57C22×D211—41no—
58C22×D211—41no—
59C22×D211—41no—
60C22×D211—41no—
61D844—22no—
62D844—22no—
63D844—41no—
64D844—41no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—22no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C211⋊C8—41no—
75C23×C211C2×D411yescharacteristic
76C2×C4×C211—22no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
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#12011yescharacteristic
90C22×C211C22⋊Q811yescharacteristic
91C22×C211#12411yescharacteristic
92C22×C211#13511yescharacteristic
93C22×C211#13511yescharacteristic
94C22×C211AGL(1,ℤ/8)11yescharacteristic
95C22×C211AGL(1,ℤ/8)11yescharacteristic
96C2×D211—42no—
97C2×D211—42no—
98C2×D211—42no—
99C2×D211—81no—
100C2×D211—81no—
101C2×D211—81no—
102C2×D211—81no—
103C4×C211—42no—
104C4×C211—81no—
105C4×C211C22⋊Q811yescharacteristic
106C4×C211C22⋊Q811yescharacteristic
107C4×C211#12411yescharacteristic
108C4×C211#12411yescharacteristic
109Dic211—41no—
110Dic211—41no—
111Dic211—42no—
112C2×C211#39311yescharacteristic
113C2×C211#44911yescharacteristic
114C2×C211#44911yescharacteristic
115C2×C211#47811yescharacteristic
116C2×C211#47811yescharacteristic
117C2×C211#48811yescharacteristic
118C2×C211#49011yescharacteristic
119D211—82no—
120C211#202511yescharacteristic, complemented, Sylow
121#2025—2111nomaximal, Sylow
122C2×#125—2111no—
123C2×#102—2111no—
124C2×#102—2111no—
125#390—2111no—
126#340—2111no—
127#340D21111yescharacteristic, complemented
128#336—2111no—
129C22×D4—2111no—
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⋊C4C2×D21111yescharacteristic
138C2×C4⋊C4C2×D21111yescharacteristic
139C2×C22⋊C4—4221no—
140C2×C22⋊C4—4221no—
141C2×C42—2111no—
142#102—4222no—
143#102—4222no—
144C24—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—4221no—
150C2×D4—2112no—
151C22×C4—21no—
152C22×C4—21no—
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—21no—
165C4⋊C4—21no—
166C4⋊C4—4224no—
167C4⋊C4—4224no—
168C4⋊C4—4221no—
169C4⋊C4—2112no—
170C4⋊C4D84412yes—
171C22⋊C4—8442no—
172C22⋊C4—8442no—
173C42—4221no—
174C42—4221no—
175C23—8441no—
176C23—8441no—
177C23—8441no—
178C23—8441no—
179C23—8441no—
180C23—8441no—
181C23—4222no—
182C23D4×D21111yescharacteristic
183D4—8441no—
184D4—8441no—
185D4—4222no—
186D4—4222no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
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×D84411yescharacteristic
204C2×C4Q8×D21111yescharacteristic
205C2×C4#1680996111yescharacteristic
206C2×C4#1680996211yescharacteristic
207C2×C4#1680996211yescharacteristic
208C8—41no—
209C8—8441no—
210C22—8442no—
211C22—8442no—
212C22—8442no—
213C22—16881no—
214C22—16881no—
215C22—16881no—
216C22—16881no—
217C22#2026994711yescharacteristic
218C22#2026995511yescharacteristic
219C22#2026996311yescharacteristic
220C22#2026999811yescharacteristic
221C22#2027000011yescharacteristic
222C22#2027000111yescharacteristic
223C22#2027000411yescharacteristic
224C4—42no—
225C4—81no—
226C4—8441no—
227C4—8442no—
228C4—8441no—
229C4#2026995511yescharacteristic
230C4#2026995711yescharacteristic
231C4#2026996311yescharacteristic
232C4#2026996511yescharacteristic
233C2—16882no—
234C2#2721305911yescharacteristic
235C2#2721329211yescharacteristic
236C2#2721329411yescharacteristic
237C2#2721344511yescharacteristic
238C2#2721344911yescharacteristic
239C2#2721350211yescharacteristic
240C2#2721350911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
hypercenter#340D211
hypocenterC211#2025
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×#20269968C2semidirect
5. #27212908C2semidirect
6. #27212900C2semidirect
7. #340×C211C2semidirect
120. C211#2025semidirect
127. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699682
C2×#202698992
C2×#202698972
#272130472
#272129082
#272129002
#340×C2112
#2025211