Finite Group Databasebeta

← #27170090

Subgroups — #27170090

Order 12928.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265802C211yesmaximal, characteristic
2#20265335C211yesmaximal, characteristic, complemented
3#20265332C211yesmaximal, characteristic, complemented
4#20265090C211yesmaximal, characteristic
5#20264913C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic
7#328×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C2211yescharacteristic
9C2×Dic404—21no—
10C2×#13108064—21no—
11#15692235—21no—
12#15692234C2211yescharacteristic
13#15692233—21no—
14#15692233—21no—
15#15692232—21no—
16#15692231—21no—
17C2×#13108051—21no—
18C2×#13108051C2211yescharacteristic
19C22⋊C4×D101—21no—
20#15692180—21no—
21#15692178—21no—
22C22⋊#11717579C2211yescharacteristic
23#15692162—21no—
24C22⋊Q8×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×Dic202—42no—
32C2×Dic202D411yescharacteristic
33C22×Dic101—21no—
34C22×Dic101D411yescharacteristic
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37Dic404—42no—
38#13108064—42no—
39C2×#11717579—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101D411yescharacteristic
42#13108053—21no—
43#13108053—21no—
44#13108053—21no—
45C22⋊Dic101—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—42no—
51C2×Q8×C101D411yescharacteristic
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C4⋊C4×C101—21no—
56C4⋊C4×C101D411yescharacteristic
57C22⋊C4×C101—21no—
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60C22×D101—21no—
61C22×D101—41no—
62C22×D101—41no—
63C22×D101Q1612yescomplemented
64Q8×C101—41no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×Dic101—22no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C4×D101—81no—
81C4×D101—81no—
82Dic202—22no—
83Dic202—22no—
84Dic202—41no—
85Dic202—41no—
86Dic202—41no—
87Dic202—81no—
88C8×C101—41no—
89#11717579—41no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101#12011yescharacteristic
93C22×C101C2×Q1611yescharacteristic
94C22×C101#13711yescharacteristic
95C2×D101—22no—
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—42no—
99C2×D101—81no—
100C4×C101—21no—
101C4×C101—21no—
102C4×C101—41no—
103C4×C101—41no—
104C4×C101—81no—
105Dic101—41no—
106Dic101—41no—
107Dic101—42no—
108Dic101—41no—
109C2×C101—21no—
110C2×C101—21no—
111C2×C101#45111yescharacteristic
112C2×C101#45511yescharacteristic
113C2×C101B(4,𝔽2)11yescharacteristic
114D101—42no—
115#1602—1011nomaximal, Sylow
116C101#160211yescharacteristic, complemented, Sylow
117#543—1011no—
118#451—1011no—
119#451—1011no—
120#393—1011no—
121#328—1011no—
122#328D10111yescharacteristic, complemented
123#323—1011no—
124C22×Q8—1011no—
125C2×Q16—2021no—
126C2×Q16—2021no—
127C22⋊Q8—1011no—
128C22⋊Q8—2021no—
129C22⋊Q8—2021no—
130C22⋊Q8—2021no—
131C22⋊Q8—2021no—
132C22⋊Q8—2021no—
133C22⋊Q8C2×D10111yescharacteristic
134C2×C22⋊C4—1011no—
135C2×C22⋊C4—2021no—
136C2×C22⋊C4—2021no—
137#103—2021no—
138#103—2021no—
139C22⋊C8—1011no—
140C22⋊C8C2×D10111yescharacteristic
141#95—2021no—
142#95C2×D10111yescharacteristic
143C24—1011no—
144C2×Q8—1011no—
145C2×Q8—4042no—
146C2×Q8—2021no—
147C2×Q8—2021no—
148C2×Q8—2021no—
149C2×Q8D40411yescharacteristic
150C22×C4—21no—
151C22×C4—1011no—
152C22×C4—2021no—
153C22×C4—2021no—
154C22×C4—2021no—
155C22×C4C22×D10111yescharacteristic
156Q16—4042no—
157Q16—4042no—
158C2×C8—21no—
159C2×C8—2021no—
160C4⋊C4—21no—
161C4⋊C4—1011no—
162C4⋊C4—2021no—
163C4⋊C4—2021no—
164C4⋊C4—2021no—
165C4⋊C4—2021no—
166C4⋊C4D40411yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—4042no—
169C22⋊C4—2021no—
170C22⋊C4—2021no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C23—1012no—
175C23—4041no—
176C23—4041no—
177C23—2021no—
178C23—2021no—
179C23D4×D10111yescharacteristic
180Q8—41no—
181Q8—8081no—
182Q8—4041no—
183Q8—4041no—
184Q8—4041no—
185Q8—2022no—
186Q8—2022no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—8081no—
194C2×C4—8081no—
195C2×C4—4041no—
196C2×C4—4041no—
197C2×C4—4041no—
198C2×C4—2022no—
199C2×C4—2021no—
200C2×C4—2021no—
201C2×C4—2021no—
202C2×C4D4×D10111yescharacteristic
203C2×C4C2×D40411yescharacteristic
204C8—41no—
205C8—4041no—
206C22—21no—
207C22—21no—
208C22—8081no—
209C22—4042no—
210C22—4041no—
211C22—4041no—
212C22—2022no—
213C22#1569222511yescharacteristic
214C22Q16×D10111yescharacteristic
215C22#1569228411yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—4041no—
222C4—4041no—
223C4—4041no—
224C4—4042no—
225C2—21no—
226C2—21no—
227C2—4042no—
228C2#2026533411yescharacteristic
229C2#2026535611yescharacteristic
230C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#328×C101C2
socleC22×C101#120
prime radicalC23×D101D4
hypercenter#328D101
hypocenterC101#1602
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20265335C2semidirect
3. #20265332C2semidirect
5. #20264913C2semidirect
7. #328×C101C2semidirect
63. C22×D101Q16semidirect
116. C101#1602semidirect
122. #328D101semidirect

Maximal subgroups

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

subgroupindex
#202658022
#202653352
#202653322
#202650902
#202649132
#202649002
#328×C1012
#1602101