Finite Group Databasebeta

← #27170124

Subgroups — #27170124

Order 12928.

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

normal subgroups only

This group has 228 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#20265508C211yesmaximal, characteristic, complemented
3#20265505C211yesmaximal, characteristic, complemented
4#20265104C211yesmaximal, characteristic
5#20264913C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic
7#328×C101C211yesmaximal, characteristic, complemented
8#15692235—21no—
9#15692234C2211yescharacteristic
10#15692233—21no—
11#15692233—21no—
12#15692232—21no—
13#15692231—21no—
14C2×C4⋊Dic101C2211yescharacteristic
15C2×#13108051—21no—
16C2×#13108051C2211yescharacteristic
17C22⋊C4×D101—21no—
18#15692187—21no—
19#15692186—21no—
20#15692180—21no—
21#15692179—21no—
22C22⋊#11717579C2211yescharacteristic
23#15692162—21no—
24C22⋊Q8×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202D411yescharacteristic
30C22×Dic101—21no—
31C22×Dic101D411yescharacteristic
32C2×C4×D101—21no—
33C2×C4×D101—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101D411yescharacteristic
39#13108053—21no—
40#13108053—21no—
41#13108053—21no—
42C22⋊Dic101—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—42no—
48C2×Q8×C101D411yescharacteristic
49C22×C4×C101—21no—
50C22×C4×C101C2311yescharacteristic
51C2×C8×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101D411yescharacteristic
54C22⋊C4×C101—21no—
55C23×C101C2×D411yescharacteristic
56C22×D101—21no—
57C22×D101—21no—
58C22×D101—41no—
59C22×D101—41no—
60C22×D101SD1612yes—
61Q8×C101—41no—
62C2×C4×C101—21no—
63C2×C4×C101—21no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101C2×D411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C2×Dic101—21no—
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—41no—
74C2×Dic101—41no—
75C2×Dic101—41no—
76C2×Dic101—41no—
77C4×D101—81no—
78C4×D101—81no—
79Dic202—41no—
80Dic202—81no—
81C8×C101—41no—
82#11717579—41no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101#12011yescharacteristic
86C22×C101C2×SD1611yescharacteristic
87C22×C101#13711yescharacteristic
88C2×D101—22no—
89C2×D101—41no—
90C2×D101—41no—
91C2×D101—42no—
92C2×D101—81no—
93C4×C101—21no—
94C4×C101—21no—
95C4×C101—41no—
96C4×C101—41no—
97C4×C101—81no—
98Dic101—41no—
99Dic101—41no—
100Dic101—41no—
101Dic101—81no—
102C2×C101—21no—
103C2×C101—21no—
104C2×C101#44811yescharacteristic
105C2×C101#45511yescharacteristic
106C2×C101#45811yescharacteristic
107D101—42no—
108#1608—1011nomaximal, Sylow
109C101#160811yescharacteristic, complemented, Sylow
110#543—1011no—
111#483—1011no—
112#483—1011no—
113#397—1011no—
114#328—1011no—
115#328D10111yescharacteristic, complemented
116#323—1011no—
117C22⋊Q8—1011no—
118C22⋊Q8—2021no—
119C22⋊Q8—2021no—
120C22⋊Q8—2021no—
121C22⋊Q8—2021no—
122C22⋊Q8—2021no—
123C22⋊Q8C2×D10111yescharacteristic
124C2×C4⋊C4—1011no—
125C2×C22⋊C4—1011no—
126C2×C22⋊C4—2021no—
127C2×C22⋊C4—2021no—
128#106—2021no—
129#106—2021no—
130#103—2021no—
131#103—2021no—
132C22⋊C8—1011no—
133C22⋊C8C2×D10111yescharacteristic
134#95—2021no—
135#95C2×D10111yescharacteristic
136C24—1011no—
137C2×Q8—1011no—
138C2×Q8—2021no—
139C2×Q8D40411yescharacteristic
140C22×C4—21no—
141C22×C4—1011no—
142C22×C4—2021no—
143C22×C4—2021no—
144C22×C4—2021no—
145C22×C4C22×D10111yescharacteristic
146C2×C8—21no—
147C2×C8—2021no—
148C4⋊C4—21no—
149C4⋊C4—1011no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4—2021no—
154C4⋊C4—2021no—
155C4⋊C4—2021no—
156C4⋊C4D40411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—4042no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C23—1012no—
165C23—4041no—
166C23—4041no—
167C23—2021no—
168C23—2021no—
169C23D4×D10111yescharacteristic
170Q8—41no—
171Q8—8081no—
172Q8—4041no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—8081no—
180C2×C4—8081no—
181C2×C4—4041no—
182C2×C4—4041no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—2021no—
186C2×C4—2021no—
187C2×C4—2021no—
188C2×C4D4×D10111yescharacteristic
189C2×C4C2×D40411yescharacteristic
190C8—41no—
191C8—4041no—
192C22—21no—
193C22—21no—
194C22—8081no—
195C22—4042no—
196C22—4041no—
197C22—4041no—
198C22—2022no—
199C22#1569222511yescharacteristic
200C22SD16×D10111yescharacteristic
201C22#1569228711yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—8081no—
208C4—4041no—
209C4—4041no—
210C4—4041no—
211C2—21no—
212C2—21no—
213C2—4042no—
214C2#2026531711yescharacteristic
215C2#2026535611yescharacteristic
216C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#328×C101C2
socleC22×C101#120
prime radicalC23×D101D4
hypercenter#328D101
hypocenterC101#1608
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. #20265508C2semidirect
3. #20265505C2semidirect
5. #20264913C2semidirect
7. #328×C101C2semidirect
109. C101#1608semidirect
115. #328D101semidirect

Maximal subgroups

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

subgroupindex
#202658022
#202655082
#202655052
#202651042
#202649132
#202649002
#328×C1012
#1608101