Finite Group Databasebeta

← #41014261

Subgroups — #41014261

Order 27008.

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

normal subgroups only

This group has 384 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×#16809947C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic, complemented
4C2×C8×Dic211C211yesmaximal, characteristic, complemented
5#27213049C211yesmaximal, characteristic, complemented
6#27213047C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809947—22no—
10C2×#16809947—22no—
11C2×#16809947C414yescomplemented
12C22×C211⋊C8C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211—22no—
15C2×(C2×D211)⋊C4—22no—
16C2×C4×Dic211C2211yescharacteristic
17#20269910—22no—
18#20269899—22no—
19C8×Dic211—22no—
20C22×D4×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#102×C211—22no—
24C23×D211—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844C2×C412yes—
31C22×Dic211—21no—
32C22×Dic211D412yes—
33#16809947—28no—
34C2×C211⋊C8C2×C412yes—
35C2×C211⋊C8D412yes—
36C2×C211⋊C8D412yes—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39C22⋊Dic211—44no—
40(C2×D211)⋊C4—44no—
41C4×Dic211—21no—
42C4×Dic211—21no—
43C24×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211C2×C412yes—
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54C2×C8×C211D412yes—
55C4⋊C4×C211—21no—
56C4⋊C4×C211—21no—
57C22×D211—22no—
58C22×D211—41no—
59C22×D211—41no—
60C22×D211—42no—
61C22×D211—42no—
62D844—22no—
63D844—22no—
64D844—41no—
65D844—41no—
66C2×Dic211—22no—
67C2×Dic211—22no—
68C2×Dic211—41no—
69C2×Dic211—42no—
70C2×Dic211D814yescomplemented
71C211⋊C8C22⋊C414yescomplemented
72C23×C211—22no—
73C23×C211—41no—
74C23×C211—41no—
75C23×C211—42no—
76C23×C211—42no—
77C23×C211C2×D411yescharacteristic
78D4×C211—22no—
79D4×C211—22no—
80D4×C211—41no—
81D4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211—42no—
84C2×C4×C211C22×C411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C2×C4×C211#4111yescharacteristic
90C8×C211—22no—
91C22×C211—42no—
92C22×C211—44no—
93C22×C211—82no—
94C22×C211—81no—
95C22×C211—81no—
96C22×C211C4×D411yescharacteristic
97C22×C211#12111yescharacteristic
98C22×C211#12711yescharacteristic
99C22×C211C2×D812yes—
100C22×C211#13512yes—
101C2×D211—42no—
102C2×D211—44no—
103C2×D211—82no—
104C2×D211—81no—
105C2×D211—81no—
106C4×C211—81no—
107C4×C211C2×C22⋊C411yescharacteristic
108C4×C211C4×D411yescharacteristic
109C4×C211#12111yescharacteristic
110C4×C211#12411yescharacteristic
111Dic211—24no—
112Dic211—81no—
113C2×C211—82no—
114C2×C211#39011yescharacteristic
115C2×C211C4×D812yes—
116C2×C211#46612yes—
117C2×C211#49311yescharacteristic
118C2×C211#49511yescharacteristic
119D211—82no—
120C211#197011yescharacteristic, complemented, Sylow
121#1970—2111nomaximal, Sylow
122C22×D8—2111no—
123C2×#107—2111no—
124C2×#102—2111no—
125C2×#102D21111yescharacteristic, complemented
126C2×C4×C8—2111no—
127#390—2111no—
128#390—2111no—
129C22×D4—2111no—
130C22×D4C2×D21111yescharacteristic
131C2×D8—4222no—
132C2×D8—4222no—
133C2×D8—2114no—
134C22×C8—2111no—
135C22×C8C2×D21111yescharacteristic
136C2×C4⋊C4—2111no—
137C2×C4⋊C4C2×D21111yescharacteristic
138C2×C22⋊C4—4222no—
139C2×C22⋊C4—4222no—
140C2×C42—2111no—
141#107—4222no—
142#102—22no—
143#102—4222no—
144C4×C8—4222no—
145C24—21no—
146C24—4221no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—4221no—
153C2×D4—4221no—
154C2×D4—4221no—
155C2×D4—4221no—
156C2×D4—4221no—
157C2×D4—2112no—
158C2×D4C4×D21112yes—
159C22×C4—21no—
160C22×C4—4221no—
161C22×C4—2112no—
162C22×C4C22×D21111yescharacteristic
163D8—4228no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2112no—
167C2×C8—2112no—
168C2×C8—2112no—
169C2×C8C22⋊D21112yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—4221no—
173C4⋊C4—4221no—
174C22⋊C4—8444no—
175C22⋊C4—8444no—
176C42—4221no—
177C42—4221no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—42no—
183C23—8441no—
184C23—8441no—
185C23—8442no—
186C23—8442no—
187C23—4222no—
188C23D4×D21111yescharacteristic
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4—8441no—
194D4—8441no—
195D4—4222no—
196D4—4222no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—8441no—
200C2×C4—8442no—
201C2×C4—4222no—
202C2×C4—4222no—
203C2×C4—2114no—
204C2×C4C2×C4×D21111yescharacteristic
205C2×C4D4×D21111yescharacteristic
206C2×C4D4×D21111yescharacteristic
207C2×C4C2×C22⋊D21111yescharacteristic
208C2×C4#1680996111yescharacteristic
209C2×C4#1680996211yescharacteristic
210C8—22no—
211C8—2114no—
212C22—44no—
213C22—42no—
214C22—82no—
215C22—81no—
216C22—81no—
217C22—8444no—
218C22—8442no—
219C22—16882no—
220C22—16881no—
221C22—16881no—
222C22(C22⋊D211)⋊C411yescharacteristic
223C22#2026995011yescharacteristic
224C22#2026997311yescharacteristic
225C22D8×D21112yes—
226C22#2026999812yes—
227C4—81no—
228C4—16881no—
229C4—4224no—
230C4C22⋊C4×D21111yescharacteristic
231C4C4×C22⋊D21111yescharacteristic
232C4#2026994911yescharacteristic
233C4#2026996311yescharacteristic
234C2—82no—
235C2—16882no—
236C2#2721304611yescharacteristic
237C2#2721322612yes—
238C2#2721338112yes—
239C2#2721352111yescharacteristic
240C2#2721352711yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×#16809947C2
cosocleC22×C4×C211C23
hypercenterC2×#102D211
hypocenterC211#1970
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×#20269910C2semidirect
3. C2×#20269899C2semidirect
4. C2×C8×Dic211C2semidirect
5. #27213049C2semidirect
6. #27213047C2semidirect
7. C2×#102×C211C2semidirect
11. C2×#16809947C4semidirect
70. C2×Dic211D8semidirect
71. C211⋊C8C22⋊C4semidirect
120. C211#1970semidirect
125. C2×#102D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099472
C2×#202699102
C2×#202698992
C2×C8×Dic2112
#272130492
#272130472
C2×#102×C2112
#1970211