Finite Group Databasebeta

← #41014649

Subgroups — #41014649

Order 27008.

382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#27213708C211yesmaximal, characteristic
2C2×#20269891C212yesmaximal, complemented
3#27213025C212yesmaximal, complemented
4#27212913C211yesmaximal, characteristic, complemented
5#385×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊D211C2212yes—
7#20269961—22no—
8#20269951—21no—
9#20269951—21no—
10#20269947—22no—
11C2×C22⋊Dic211C2212yes—
12C2×(C2×D211)⋊C4—22no—
13#20269891—24no—
14C211⋊#95—22no—
15C23×C4×C211C2211yescharacteristic
16C2×C4⋊C4×C211—21no—
17C2×C22⋊C4×C211C2211yescharacteristic
18C2×C22⋊C4×C211C2211yescharacteristic
19#95×C211—21no—
20C23×D211D412yes—
21C2×D844—21no—
22C2×D844—21no—
23C2×C22⋊D211—22no—
24C2×C22⋊D211—22no—
25C2×C22⋊D211—22no—
26C2×C22⋊D211—22no—
27C2×C22⋊D211—22no—
28C2×C22⋊D211—22no—
29C22×Dic211—22no—
30C22×Dic211D412yes—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C22⋊Dic211—22no—
34C22⋊Dic211—22no—
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—22no—
37(C2×D211)⋊C4—44no—
38C24×C211C2311yescharacteristic
39C22×C4×C211—21no—
40C22×C4×C211—21no—
41C22×C4×C211—22no—
42C22×C4×C211—42no—
43C22×C4×C211D411yescharacteristic
44C22×C4×C211D411yescharacteristic
45C4⋊C4×C211—42no—
46C22⋊C4×C211—22no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211D412yescomplemented
49C22×D211—22no—
50C22×D211—22no—
51C22×D211—22no—
52C22×D211—42no—
53C22×D211—42no—
54D844—81no—
55D844—81no—
56C22⋊D211—42no—
57C22⋊D211—42no—
58C22⋊D211—42no—
59C22⋊D211—42no—
60C2×Dic211—22no—
61C2×Dic211—22no—
62C2×Dic211—22no—
63C2×Dic211—42no—
64C2×Dic211—42no—
65C2×Dic211—42no—
66C23×C211—41no—
67C23×C211—41no—
68C23×C211C2×D412yes—
69C23×C211C2×D411yescharacteristic
70C23×C211C2×D411yescharacteristic
71C23×C211#4111yescharacteristic
72C23×C211#4112yes—
73C2×C4×C211—22no—
74C2×C4×C211—24no—
75C2×C4×C211—21no—
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—42no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C22×C211—21no—
86C22×C211—22no—
87C22×C211—21no—
88C22×C211—22no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211—42no—
92C22×C211#12011yescharacteristic
93C22×C211#12111yescharacteristic
94C22×C211#12112yes—
95C22×C211#12312yes—
96C22×C211#12411yescharacteristic
97C2×D211—44no—
98C2×D211—42no—
99C2×D211—42no—
100C2×D211—42no—
101C2×D211—82no—
102C4×C211—42no—
103C4×C211—42no—
104C4×C211—41no—
105C4×C211—41no—
106Dic211—42no—
107Dic211—82no—
108C2×C211—22no—
109C2×C211—21no—
110C2×C211—22no—
111C2×C211—21no—
112C2×C211#39411yescharacteristic
113C2×C211B(4,𝔽2)11yescharacteristic
114C2×C211#45811yescharacteristic
115D211—82no—
116C211#204411yescharacteristic, complemented, Sylow
117#2044—2111nomaximal, Sylow
118#534—2111no—
119C2×#99—2112no—
120#386—2112no—
121#385D21111yescharacteristic, complemented
122#342—2111no—
123C22×D4—2112no—
124C23×C4C2×D21111yescharacteristic
125#123—4222no—
126#121—4221no—
127#121—4221no—
128#120—4222no—
129C2×C4⋊C4—21no—
130C2×C22⋊C4—4222no—
131C2×C22⋊C4—2112no—
132C2×C22⋊C4C2×D21111yescharacteristic
133C2×C22⋊C4C2×D21111yescharacteristic
134#99—4224no—
135#95—21no—
136#95—4222no—
137C24—2112no—
138C24C22×D21111yescharacteristic
139C2×D4—4221no—
140C2×D4—4221no—
141C2×D4—4222no—
142C2×D4—4222no—
143C2×D4—4222no—
144C2×D4—4222no—
145C2×D4—4222no—
146C2×D4—4222no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—4222no—
152C22×C4—2112no—
153C22×C4C22⋊D21111yescharacteristic
154C22×C4C22⋊D21111yescharacteristic
155C4⋊C4—42no—
156C4⋊C4—4221no—
157C4⋊C4—4221no—
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—8444no—
161C22⋊C4—4222no—
162C22⋊C4—4222no—
163C22⋊C4—4222no—
164C22⋊C4—4222no—
165C22⋊C4D84412yescomplemented
166C23—41no—
167C23—41no—
168C23—8442no—
169C23—8442no—
170C23—4222no—
171C23—4222no—
172C23—4222no—
173C23D4×D21112yes—
174C23C2×C22⋊D21111yescharacteristic
175C23C2×D84411yescharacteristic
176C23#1680996211yescharacteristic
177C23#1680996312yes—
178D4—8442no—
179D4—8442no—
180D4—8442no—
181D4—8442no—
182D4—16881no—
183D4—16881no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—8442no—
197C2×C4—8442no—
198C2×C4—8442no—
199C2×C4—4222no—
200C2×C4—4222no—
201C2×C4—4222no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—8444no—
210C22—8442no—
211C22—8442no—
212C22—8442no—
213C22—16882no—
214C22#2026994511yescharacteristic
215C22#2026995111yescharacteristic
216C22#2026995212yes—
217C22#2026996212yes—
218C22#2026996611yescharacteristic
219C4—42no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—8442no—
224C4—16882no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—16882no—
230C2#2721306311yescharacteristic
231C2#2721333211yescharacteristic
232C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#385×C211C2
socleC22×C211#120
prime radical#27213708C2
hypercenter#385D211
hypocenterC211#2044
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#20269891C2semidirect
3. #27213025C2semidirect
4. #27212913C2semidirect
5. #385×C211C2semidirect
48. C22⋊C4×C211D4semidirect
116. C211#2044semidirect
121. #385D211semidirect
165. C22⋊C4D844semidirect

Maximal subgroups

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

subgroupindex
#272137082
C2×#202698912
#272130252
#272129132
#385×C2112
#2044211