Finite Group Databasebeta

← #41014532

Subgroups — #41014532

Order 27008.

326 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20269976C211yesmaximal, 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×Dic422—21no—
9C22×C211⋊C8C2211yescharacteristic
10#20269976—22no—
11#20269976—22no—
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—
23C2×Dic422—24no—
24C2×Dic422—42no—
25C2×Dic422—42no—
26C22×Dic211—21no—
27C22×Dic211—21no—
28C22×Dic211D411yescharacteristic
29C22×Dic211D411yescharacteristic
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D412yes—
37Dic211⋊C4—24no—
38C22⋊Dic211—42no—
39C22⋊Dic211—42no—
40C4×Dic211—21no—
41C4×Dic211—21no—
42C24×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211Q812yescomplemented
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54C4⋊C4×C211—21no—
55C4⋊C4×C211D412yes—
56Dic422—48no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—22no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211D812yes—
68C2×Dic211SD1612yes—
69C211⋊C8—41no—
70C23×C211—22no—
71C23×C211—41no—
72C23×C211—41no—
73C23×C211—41no—
74C23×C211—41no—
75C23×C211—41no—
76C23×C211—41no—
77C23×C211C2×D411yescharacteristic
78D4×C211—22no—
79D4×C211—22no—
80D4×C211—41no—
81D4×C211—41no—
82C2×C4×C211—22no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×Q811yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C2×C4×C211#4111yescharacteristic
90C2×C4×C211#4111yescharacteristic
91C8×C211—41no—
92C22×C211—42no—
93C22×C211—42no—
94C22×C211—42no—
95C22×C211—81no—
96C22×C211—81no—
97C22×C211—81no—
98C22×C211—81no—
99C22×C211#12011yescharacteristic
100C22×C211C22⋊Q811yescharacteristic
101C22×C211#12411yescharacteristic
102C22×C211C2×D811yescharacteristic
103C22×C211C2×SD1611yescharacteristic
104C22×C211AGL(1,ℤ/8)11yescharacteristic
105C22×C211#13711yescharacteristic
106C4×C211—42no—
107C4×C211C22⋊Q811yescharacteristic
108C4×C211C22⋊Q811yescharacteristic
109C4×C211#12411yescharacteristic
110C4×C211#12411yescharacteristic
111Dic211—22no—
112Dic211—22no—
113Dic211—42no—
114Dic211—81no—
115C2×C211—82no—
116C2×C211#39311yescharacteristic
117C2×C211#44711yescharacteristic
118C2×C211#44811yescharacteristic
119C2×C211#47411yescharacteristic
120C2×C211#47611yescharacteristic
121C2×C211#48611yescharacteristic
122C2×C211#48911yescharacteristic
123C211#202311yescharacteristic, complemented, Sylow
124#2023—2111nomaximal, Sylow
125C2×C4⋊Q8—2111no—
126C2×#102—2111no—
127C2×#102D21111yescharacteristic
128#390—2111no—
129#340—2111no—
130#340—2111no—
131#336—2111no—
132C22×Q8—4221no—
133C22×D4C2×D21111yescharacteristic
134C22×C8—2111no—
135C22×C8C2×D21111yescharacteristic
136C4⋊Q8—4222no—
137C4⋊Q8—4222no—
138C2×C4⋊C4—4221no—
139C2×C4⋊C4—2111no—
140C2×C4⋊C4—2111no—
141C2×C4⋊C4C2×D21111yescharacteristic
142C2×C22⋊C4—4221no—
143C2×C22⋊C4—4221no—
144C2×C42—2111no—
145#102—22no—
146#102—4222no—
147C24—21no—
148C2×Q8—8442no—
149C2×Q8—8442no—
150C2×Q8—4224no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4Dic42212yescomplemented
157C22×C4—21no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4—2111no—
162C22×C4C22×D21111yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—4221no—
167C2×C8—4221no—
168C2×C8—4221no—
169C4⋊C4—21no—
170C4⋊C4—4224no—
171C4⋊C4—4221no—
172C4⋊C4—4221no—
173C4⋊C4—4221no—
174C4⋊C4—2112no—
175C4⋊C4C22⋊D21112yes—
176C22⋊C4—8442no—
177C22⋊C4—8442no—
178C42—4221no—
179C42—4221no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23D4×D21111yescharacteristic
188Q8—8448no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—8441no—
200C2×C4—8441no—
201C2×C4—4222no—
202C2×C4—4221no—
203C2×C4—4221no—
204C2×C4—4221no—
205C2×C4—4221no—
206C2×C4—2112no—
207C2×C4—2112no—
208C2×C4D4×D21111yescharacteristic
209C2×C4C2×C22⋊D21111yescharacteristic
210C2×C4C2×Dic42211yescharacteristic
211C2×C4#1680996111yescharacteristic
212C2×C4#1680996111yescharacteristic
213C2×C4#1680996211yescharacteristic
214C8—41no—
215C8—8441no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22#2026994511yescharacteristic
224C22#2026995411yescharacteristic
225C22#2026996311yescharacteristic
226C22D8×D21111yescharacteristic
227C22SD16×D21111yescharacteristic
228C22#2027000111yescharacteristic
229C22#2027000611yescharacteristic
230C4—42no—
231C4—8442no—
232C4—16881no—
233C4—4222no—
234C4—4222no—
235C4#2026995411yescharacteristic
236C4#2026995611yescharacteristic
237C4#2026996311yescharacteristic
238C4#2026996411yescharacteristic
239C2—82no—
240C2#2721305811yescharacteristic
241C2#2721327811yescharacteristic
242C2#2721328311yescharacteristic
243C2#2721342411yescharacteristic
244C2#2721343411yescharacteristic
245C2#2721349111yescharacteristic
246C2#2721350411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenterC2×#102D211
hypocenterC211#2023
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×#20269976C2semidirect
4. #27212909C2semidirect
5. #27212908C2semidirect
6. #27212900C2semidirect
48. C2×D4×C211Q8semidirect
123. C211#2023semidirect
156. C2×D4Dic422semidirect

Maximal subgroups

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

subgroupindex
C2×#202699762
C2×#202698982
#272130492
#272129092
#272129082
#272129002
C2×#102×C2112
#2023211