Finite Group Databasebeta

← #41014773

Subgroups — #41014773

Order 27008.

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

normal subgroups only

This group has 284 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×Dic211⋊C8C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic
4C2×#20269897C211yesmaximal, characteristic
5#27213047C211yesmaximal, characteristic
6#27213042C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×Dic211⋊C4—21no—
12C2×(C2×D211)⋊C4—21no—
13C2×(C2×D211)⋊C4—21no—
14C2×C4×Dic211C2211yescharacteristic
15#20269908—22no—
16Dic211⋊C8—22no—
17#20269899—22no—
18#20269897—22no—
19C211⋊#95—21no—
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⋊C8D412yes—
36C4⋊Dic211—21no—
37C4⋊Dic211D412yes—
38Dic211⋊C4—42no—
39(C2×D211)⋊C4—42no—
40(C2×D211)⋊C4—42no—
41C4×Dic211—21no—
42C4×Dic211—21no—
43C22×C4×C211—21no—
44C22×C4×C211—21no—
45C22×C4×C211C2311yescharacteristic
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C22×D211—22no—
54C22×D211—41no—
55C22×D211—41no—
56C22×D211—41no—
57C22×D211—41no—
58C22×D211—41no—
59C22×D211—41no—
60D844—22no—
61D844—22no—
62D844—41no—
63D844—41no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—22no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211SD1612yes—
73C211⋊C8—22no—
74C23×C211C2×D411yescharacteristic
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×Q811yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—41no—
88C22×C211#12111yescharacteristic
89C22×C211#12111yescharacteristic
90C22×C211C22⋊Q811yescharacteristic
91C22×C211C2×SD1611yescharacteristic
92C22×C211#13511yescharacteristic
93C22×C211AGL(1,ℤ/8)11yescharacteristic
94C22×C211AGL(1,ℤ/8)11yescharacteristic
95C2×D211—42no—
96C2×D211—42no—
97C2×D211—42no—
98C2×D211—81no—
99C2×D211—81no—
100C2×D211—81no—
101C2×D211—81no—
102C4×C211—81no—
103C4×C211—81no—
104C4×C211#12111yescharacteristic
105C4×C211C22⋊Q811yescharacteristic
106C4×C211C22⋊Q811yescharacteristic
107C4×C211#12611yescharacteristic
108Dic211—22no—
109Dic211—42no—
110Dic211—41no—
111C2×C211#39611yescharacteristic
112C2×C211#46011yescharacteristic
113C2×C211#46311yescharacteristic
114C2×C211#46511yescharacteristic
115C2×C211#46911yescharacteristic
116C2×C211#47611yescharacteristic
117C2×C211#47811yescharacteristic
118D211—82no—
119C211#206511yescharacteristic, complemented, Sylow
120#2065—2111nomaximal, Sylow
121C2×#106—2111no—
122C2×C4⋊C8—2111no—
123C2×#102—2111no—
124C2×#102—2111no—
125#390—2111no—
126#389—2111no—
127#340D21111yescharacteristic, complemented
128C22×D4—2111no—
129C22×C8—2111no—
130C22×C8C2×D21111yescharacteristic
131C2×C4⋊C4—4221no—
132C2×C4⋊C4—2111no—
133C2×C4⋊C4C2×D21111yescharacteristic
134C2×C4⋊C4C2×D21111yescharacteristic
135C2×C22⋊C4—4221no—
136C2×C22⋊C4—4221no—
137C2×C42—2111no—
138#106—4222no—
139C4⋊C8—4222no—
140#102—4222no—
141#102—4222no—
142#95—4221no—
143C24—4221no—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—2112no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—4221no—
153C22×C4—2111no—
154C22×C4—2111no—
155C22×C4C22×D21111yescharacteristic
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—4221no—
160C2×C8—4221no—
161C2×C8—2112no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—8442no—
167C4⋊C4—4221no—
168C4⋊C4—2112no—
169C22⋊C4—8442no—
170C22⋊C4—8442no—
171C42—4221no—
172C42—4221no—
173C23—8441no—
174C23—8441no—
175C23—8441no—
176C23—8441no—
177C23—8441no—
178C23—8441no—
179C23—4222no—
180C23D4×D21111yescharacteristic
181D4—8441no—
182D4—8441no—
183D4—4222no—
184D4—4222no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—4221no—
194C2×C4—4222no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—4221no—
198C2×C4—4221no—
199C2×C4—2112no—
200C2×C4D4×D21111yescharacteristic
201C2×C4D4×D21111yescharacteristic
202C2×C4Q8×D21111yescharacteristic
203C2×C4#1680996211yescharacteristic
204C2×C4#1680996211yescharacteristic
205C2×C4#1680996211yescharacteristic
206C8—41no—
207C8—4222no—
208C22—8442no—
209C22—8442no—
210C22—8442no—
211C22—16881no—
212C22—16881no—
213C22—16881no—
214C22—16881no—
215C22#2026995011yescharacteristic
216C22#2026995011yescharacteristic
217C22#2026995511yescharacteristic
218C22SD16×D21111yescharacteristic
219C22#2027000011yescharacteristic
220C22#2027000111yescharacteristic
221C22#2027000411yescharacteristic
222C4—81no—
223C4—81no—
224C4—8442no—
225C4—8441no—
226C4—4222no—
227C4#2026995011yescharacteristic
228C4#2026995511yescharacteristic
229C4#2026995511yescharacteristic
230C4#2026997111yescharacteristic
231C2—16882no—
232C2#2721307011yescharacteristic
233C2#2721334511yescharacteristic
234C2#2721336411yescharacteristic
235C2#2721337811yescharacteristic
236C2#2721339811yescharacteristic
237C2#2721343511yescharacteristic
238C2#2721344911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
hypercenter#340D211
hypocenterC211#2065
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
2. C2×Dic211⋊C8C2semidirect
6. #27213042C2semidirect
7. #340×C211C2semidirect
119. C211#2065semidirect
127. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
C2×Dic211⋊C82
C2×#202698992
C2×#202698972
#272130472
#272130422
#340×C2112
#2065211