Finite Group Databasebeta

← #41014253

Subgroups — #41014253

Order 27008.

348 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×#16809950C211yesmaximal, characteristic
2C2×#20269908C211yesmaximal, characteristic, complemented
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×C8×Dic211C211yesmaximal, characteristic, complemented
5#27213051C211yesmaximal, characteristic, complemented
6#27213049C211yesmaximal, characteristic
7C2×#102×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—22no—
10C2×#16809950—22no—
11C2×#16809950C414yescomplemented
12C22×C211⋊C8C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211—22no—
15C2×C4×Dic211C2211yescharacteristic
16#20269908—22no—
17#20269902—22no—
18C8×Dic211—22no—
19C211⋊#95—22no—
20C22×D4×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#102×C211—22no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422C2×C412yes—
30C22×Dic211—21no—
31C22×Dic211—21no—
32C22×Dic211D412yes—
33#16809950—28no—
34C2×C211⋊C8C2×C412yes—
35C2×C211⋊C8D412yes—
36C2×C211⋊C8D412yes—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39C22⋊Dic211—44no—
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×C211C2×C412yes—
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C2×C8×C211—21no—
53C2×C8×C211D412yes—
54C4⋊C4×C211—21no—
55C4⋊C4×C211—21no—
56Dic422—22no—
57Dic422—22no—
58Dic422—41no—
59Dic422—41no—
60C2×Dic211—22no—
61C2×Dic211—22no—
62C2×Dic211—22no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—42no—
67C2×Dic211SD1614yescomplemented
68C211⋊C8C22⋊C414yescomplemented
69C23×C211—22no—
70C23×C211—41no—
71C23×C211—41no—
72C23×C211—42no—
73C23×C211—42no—
74C23×C211C2×D411yescharacteristic
75D4×C211—22no—
76D4×C211—22no—
77D4×C211—41no—
78D4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—42no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—22no—
88C22×C211—42no—
89C22×C211—44no—
90C22×C211—82no—
91C22×C211—81no—
92C22×C211—81no—
93C22×C211C4×D411yescharacteristic
94C22×C211#12111yescharacteristic
95C22×C211#12711yescharacteristic
96C22×C211C2×SD1612yes—
97C22×C211#13512yes—
98C4×C211—81no—
99C4×C211C2×C22⋊C411yescharacteristic
100C4×C211C4×D411yescharacteristic
101C4×C211#12111yescharacteristic
102C4×C211#12411yescharacteristic
103Dic211—24no—
104Dic211—42no—
105Dic211—81no—
106C2×C211—82no—
107C2×C211#39011yescharacteristic
108C2×C211C4×SD1612yes—
109C2×C211#46512yes—
110C2×C211#49211yescharacteristic
111C2×C211#49511yescharacteristic
112C211#196811yescharacteristic, complemented, Sylow
113#1968—2111nomaximal, Sylow
114C22×SD16—2111no—
115C2×#106—2111no—
116C2×#103—2111no—
117C2×#102D21111yescharacteristic
118C2×C4×C8—2111no—
119#391—2111no—
120#390—2111no—
121C22×Q8—2111no—
122C22×D4C2×D21111yescharacteristic
123C2×SD16—4222no—
124C2×SD16—4222no—
125C2×SD16—2114no—
126C22×C8—2111no—
127C22×C8C2×D21111yescharacteristic
128C2×C4⋊C4—2111no—
129C2×C4⋊C4C2×D21111yescharacteristic
130C2×C22⋊C4—4222no—
131C2×C42—2111no—
132#106—4222no—
133#103—4222no—
134#102—22no—
135C4×C8—4222no—
136#95—4222no—
137C24—21no—
138C2×Q8—4221no—
139C2×Q8—4221no—
140C2×Q8—4221no—
141C2×Q8—4221no—
142C2×Q8—4221no—
143C2×Q8—2112no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C4×D21112yes—
150C22×C4—21no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—2112no—
154C22×C4C22×D21111yescharacteristic
155SD16—4228no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—2112no—
159C2×C8—2112no—
160C2×C8—2112no—
161C2×C8C22⋊D21112yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—4221no—
165C4⋊C4—4221no—
166C22⋊C4—8444no—
167C42—4221no—
168C42—4221no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—42no—
173C23—42no—
174C23D4×D21111yescharacteristic
175Q8—8441no—
176Q8—8441no—
177Q8—4222no—
178Q8—4222no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—8442no—
189C2×C4—4222no—
190C2×C4—4222no—
191C2×C4—4222no—
192C2×C4—2114no—
193C2×C4C2×C4×D21111yescharacteristic
194C2×C4D4×D21111yescharacteristic
195C2×C4D4×D21111yescharacteristic
196C2×C4C2×C22⋊D21111yescharacteristic
197C2×C4#1680996111yescharacteristic
198C2×C4#1680996211yescharacteristic
199C8—22no—
200C8—2114no—
201C22—44no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22(C22⋊D211)⋊C411yescharacteristic
207C22#2026995011yescharacteristic
208C22#2026997311yescharacteristic
209C22SD16×D21112yes—
210C22#2026999612yes—
211C4—81no—
212C4—8442no—
213C4—16881no—
214C4—4224no—
215C4C22⋊C4×D21111yescharacteristic
216C4C4×C22⋊D21111yescharacteristic
217C4#2026994911yescharacteristic
218C4#2026996311yescharacteristic
219C2—82no—
220C2#2721304611yescharacteristic
221C2#2721323712yes—
222C2#2721337412yes—
223C2#2721351711yescharacteristic
224C2#2721352911yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#102D211
hypocenterC211#1968
LCSG ▷ C2×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
2. C2×#20269908C2semidirect
3. C2×#20269902C2semidirect
4. C2×C8×Dic211C2semidirect
5. #27213051C2semidirect
11. C2×#16809950C4semidirect
67. C2×Dic211SD16semidirect
68. C211⋊C8C22⋊C4semidirect
112. C211#1968semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C2×#202699082
C2×#202699022
C2×C8×Dic2112
#272130512
#272130492
C2×#102×C2112
#1968211