Finite Group Databasebeta

← #27170182

Subgroups — #27170182

Order 12928.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265765C211yesmaximal, characteristic
2#20265513C211yesmaximal, characteristic, complemented
3#20265496C211yesmaximal, characteristic, complemented
4#20265113C211yesmaximal, characteristic, complemented
5#20264915C211yesmaximal, characteristic, complemented
6#20264912C211yesmaximal, characteristic, complemented
7#323×C101C211yesmaximal, characteristic, complemented
8#15692248—21no—
9#15692241—21no—
10#15692237—21no—
11#15692234C2211yescharacteristic
12#15692229C2211yescharacteristic
13#15692228—21no—
14#15692224—21no—
15C4×C22⋊D101—21no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×#13108053—21no—
18C2×C22⋊Dic101—21no—
19#15692189—21no—
20#15692187—21no—
21#15692180—21no—
22#15692177—21no—
23#15692162C2211yescharacteristic
24C2×C22⋊C4×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26C22⋊C8×C101C2211yescharacteristic
27C2×Dic202D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C22×Dic101—21no—
33C22×Dic101—21no—
34C2×C4×D101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101D411yescharacteristic
38C4⋊Dic101D411yescharacteristic
39#13108053—21no—
40#13108053—21no—
41#13108053—42no—
42C22⋊Dic101—21no—
43C22⋊Dic101—21no—
44C22⋊Dic101—21no—
45C22⋊Dic101—42no—
46#13108051—21no—
47#13108051—21no—
48C4×Dic101—21no—
49C24×C101D411yescharacteristic
50C22×C4×C101D411yescharacteristic
51C22×C4×C101C2311yescharacteristic
52C2×C8×C101—21no—
53C2×C8×C101—21no—
54C22⋊C4×C101—21no—
55C22⋊C4×C101—21no—
56C23×C101—21no—
57C23×C101—21no—
58C23×C101—21no—
59C23×C101—41no—
60C23×C101—41no—
61C23×C101C2×D411yescharacteristic
62C22×D101—21no—
63C2×C4×C101—21no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C22⋊D101—41no—
69C22⋊D101—41no—
70C22⋊D101—81no—
71C22⋊D101—81no—
72C2×Dic101—22no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—41no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81D404—41no—
82C4×D101—81no—
83Dic202—41no—
84C8×C101—41no—
85C8×C101—41no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101—42no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—81no—
93C22×C101#12011yescharacteristic
94C22×C101#13511yescharacteristic
95C22×C101#13711yescharacteristic
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—41no—
99C4×C101—21no—
100C4×C101—21no—
101C4×C101—41no—
102Dic101—42no—
103Dic101—41no—
104Dic101—41no—
105Dic101—41no—
106Dic101—81no—
107C2×C101—21no—
108C2×C101—21no—
109C2×C101—81no—
110C2×C101#45211yescharacteristic
111C2×C101#45311yescharacteristic
112C2×C101#45811yescharacteristic
113D101—81no—
114#1618—1011nomaximal, Sylow
115C101#161811yescharacteristic, complemented, Sylow
116#538—1011no—
117#484—1011no—
118#481—1011no—
119#399—1011no—
120#328—1011no—
121#327—1011no—
122#323D10111yescharacteristic, complemented
123#126—2021no—
124#124—2021no—
125#123—2021no—
126C22⋊Q8—1011no—
127#121—1011no—
128#121—2021no—
129#120—2021no—
130C4×D4—2021no—
131C2×C4⋊C4—1011no—
132C2×C4⋊C4—2021no—
133C2×C22⋊C4—2021no—
134C2×C22⋊C4C2×D10111yescharacteristic
135#107—2021no—
136#106—2021no—
137#103—2021no—
138#102—2021no—
139C22⋊C8C2×D10111yescharacteristic
140C22⋊C8C2×D10111yescharacteristic
141#95—1011no—
142C24D40411yescharacteristic
143C2×Q8—1011no—
144C2×D4—1011no—
145C2×D4—2021no—
146C2×D4—2021no—
147C2×D4—2021no—
148C22×C4—2021no—
149C22×C4—2021no—
150C22×C4—2021no—
151C22×C4D40411yescharacteristic
152C22×C4C22×D10111yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—1011no—
156C4⋊C4—1011no—
157C4⋊C4—4042no—
158C4⋊C4—2021no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—4042no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C22⋊C4—2021no—
169C22⋊C4—2021no—
170C42—2021no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—2021no—
177C23C2×D40411yescharacteristic
178Q8—4041no—
179D4—8081no—
180D4—8081no—
181D4—4041no—
182D4—4041no—
183D4—4041no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—8081no—
188C2×C4—4041no—
189C2×C4—4041no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—2022no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4D4×D10111yescharacteristic
198C2×C4D4×D10111yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—4041no—
209C22—4041no—
210C22—4041no—
211C22#1569222511yescharacteristic
212C22#1569227711yescharacteristic
213C22#1569228811yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—8081no—
218C4—4042no—
219C4—4041no—
220C4—4041no—
221C4—4041no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—8081no—
226C2#2026534111yescharacteristic
227C2#2026534711yescharacteristic
228C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radical#20265765C2
hypercenter#323D101
hypocenterC101#1618
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265513C2semidirect
3. #20265496C2semidirect
4. #20265113C2semidirect
5. #20264915C2semidirect
6. #20264912C2semidirect
7. #323×C101C2semidirect
115. C101#1618semidirect
122. #323D101semidirect

Maximal subgroups

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

subgroupindex
#202657652
#202655132
#202654962
#202651132
#202649152
#202649122
#323×C1012
#1618101