Finite Group Databasebeta

← #41014784

Subgroups — #41014784

Order 27008.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269911C211yesmaximal, characteristic, complemented
2C2×Dic211⋊C8C211yesmaximal, characteristic, complemented
3C2×#20269898C211yesmaximal, characteristic
4#27213049C211yesmaximal, characteristic
5#27213042C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×Dic211⋊C4—21no—
12C2×C22⋊Dic211—21no—
13C2×C22⋊Dic211—21no—
14C2×C4×Dic211C2211yescharacteristic
15#20269911—22no—
16Dic211⋊C8—22no—
17#20269898—22no—
18C211⋊#95—21no—
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—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211D412yes—
34Dic211⋊C4—42no—
35C22⋊Dic211—42no—
36C22⋊Dic211—42no—
37C4×Dic211—21no—
38C4×Dic211—21no—
39C24×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211Q812yes—
46C22×C4×C211—21no—
47C22×C4×C211C2311yescharacteristic
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C2×C8×C211D412yes—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211D812yes—
65C211⋊C8—41no—
66C23×C211—22no—
67C23×C211—41no—
68C23×C211—41no—
69C23×C211—41no—
70C23×C211—41no—
71C23×C211—41no—
72C23×C211—41no—
73C23×C211C2×D411yescharacteristic
74D4×C211—22no—
75D4×C211—22no—
76D4×C211—41no—
77D4×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—22no—
88C22×C211—42no—
89C22×C211—42no—
90C22×C211—42no—
91C22×C211—81no—
92C22×C211—81no—
93C22×C211—81no—
94C22×C211—81no—
95C22×C211#12111yescharacteristic
96C22×C211#12111yescharacteristic
97C22×C211C22⋊Q811yescharacteristic
98C22×C211C2×D811yescharacteristic
99C22×C211#13511yescharacteristic
100C22×C211AGL(1,ℤ/8)11yescharacteristic
101C22×C211#13711yescharacteristic
102C4×C211—81no—
103C4×C211#12111yescharacteristic
104C4×C211C22⋊Q811yescharacteristic
105C4×C211C22⋊Q811yescharacteristic
106C4×C211#12611yescharacteristic
107Dic211—22no—
108Dic211—42no—
109Dic211—41no—
110Dic211—81no—
111C2×C211—82no—
112C2×C211#39611yescharacteristic
113C2×C211#45911yescharacteristic
114C2×C211#46411yescharacteristic
115C2×C211#46611yescharacteristic
116C2×C211#46811yescharacteristic
117C2×C211#47411yescharacteristic
118C2×C211#47811yescharacteristic
119C211#206711yescharacteristic, complemented, Sylow
120#2067—2111nomaximal, Sylow
121C2×#107—2111no—
122C2×C4⋊C8—2111no—
123C2×#102—2111no—
124C2×#102D21111yescharacteristic
125#390—2111no—
126#389—2111no—
127#340—2111no—
128C22×D4C2×D21111yescharacteristic
129C22×C8—2111no—
130C22×C8C2×D21111yescharacteristic
131C2×C4⋊C4—4221no—
132C2×C4⋊C4—2111no—
133C2×C4⋊C4—2111no—
134C2×C4⋊C4C2×D21111yescharacteristic
135C2×C22⋊C4—4221no—
136C2×C22⋊C4—4221no—
137C2×C42—2111no—
138#107—4222no—
139C4⋊C8—4222no—
140#102—22no—
141#102—4222no—
142#95—4221no—
143C24—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4Dic42212yes—
150C22×C4—21no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—2111no—
154C22×C4—2111no—
155C22×C4C22×D21111yescharacteristic
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—4221no—
159C2×C8—4221no—
160C2×C8—4221no—
161C2×C8C22⋊D21112yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—8442no—
165C4⋊C4—4221no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C4⋊C4—2112no—
169C22⋊C4—8442no—
170C22⋊C4—8442no—
171C42—4221no—
172C42—4221no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23D4×D21111yescharacteristic
181D4—22no—
182D4—22no—
183D4—41no—
184D4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—8441no—
189C2×C4—8441no—
190C2×C4—8441no—
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×C4C2×C22⋊D21111yescharacteristic
202C2×C4C2×Dic42211yescharacteristic
203C2×C4#1680996111yescharacteristic
204C2×C4#1680996111yescharacteristic
205C2×C4#1680996211yescharacteristic
206C8—22no—
207C8—8441no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#2026994811yescharacteristic
216C22#2026995011yescharacteristic
217C22#2026995411yescharacteristic
218C22D8×D21111yescharacteristic
219C22#2026999611yescharacteristic
220C22#2027000411yescharacteristic
221C22#2027000611yescharacteristic
222C4—81no—
223C4—8442no—
224C4—8441no—
225C4—16881no—
226C4—4222no—
227C4#2026994911yescharacteristic
228C4#2026995411yescharacteristic
229C4#2026995611yescharacteristic
230C4#2026997011yescharacteristic
231C2—82no—
232C2#2721306911yescharacteristic
233C2#2721334011yescharacteristic
234C2#2721336711yescharacteristic
235C2#2721337911yescharacteristic
236C2#2721338911yescharacteristic
237C2#2721342411yescharacteristic
238C2#2721344411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenterC2×#102D211
hypocenterC211#2067
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×#20269911C2semidirect
2. C2×Dic211⋊C8C2semidirect
5. #27213042C2semidirect
6. #27212908C2semidirect
119. C211#2067semidirect

Maximal subgroups

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

subgroupindex
C2×#202699112
C2×Dic211⋊C82
C2×#202698982
#272130492
#272130422
#272129082
C2×#102×C2112
#2067211