Finite Group Databasebeta

← #27170147

Subgroups — #27170147

Order 12928.

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

normal subgroups only

This group has 232 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#20265764C211yesmaximal, characteristic
2#20265508C211yesmaximal, characteristic, complemented
3#20265488C211yesmaximal, characteristic, complemented
4#20265104C211yesmaximal, characteristic, complemented
5#20264913C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8#15692250—21no—
9#15692241—21no—
10#15692239—21no—
11#15692234C2211yescharacteristic
12#15692227—21no—
13#15692223—21no—
14D4×Dic101—21no—
15C2×C4⋊Dic101C2211yescharacteristic
16C2×#13108051—21no—
17C2×#13108051C2211yescharacteristic
18#15692188—21no—
19#15692187—21no—
20#15692180—21no—
21#15692176—21no—
22C22⋊#11717579C2211yescharacteristic
23#15692162—21no—
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×Dic202D411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C22×Dic101—21no—
32C22×Dic101—21no—
33C22×Dic101D411yescharacteristic
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101D411yescharacteristic
39#13108053—21no—
40C22⋊Dic101—21no—
41C22⋊Dic101—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—42no—
47C4×Dic101—21no—
48C2×D4×C101—21no—
49C2×D4×C101D411yescharacteristic
50C22×C4×C101—21no—
51C22×C4×C101C2311yescharacteristic
52C2×C8×C101—21no—
53C4⋊C4×C101D411yescharacteristic
54C22⋊C4×C101—21no—
55C23×C101—21no—
56C23×C101C2×D411yescharacteristic
57C22×D101—21no—
58C22×D101—21no—
59C22×D101—21no—
60C22×D101—41no—
61C22×D101—41no—
62D4×C101—41no—
63D4×C101—41no—
64D4×C101—41no—
65C2×C4×C101—21no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C22⋊D101—81no—
73C22⋊D101—81no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—81no—
82Dic202—41no—
83C8×C101—41no—
84#11717579—41no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—41no—
88C22×C101—41no—
89C22×C101—41no—
90C22×C101#12011yescharacteristic
91C22×C101#13511yescharacteristic
92C22×C101AGL(1,ℤ/8)11yescharacteristic
93C2×D101—42no—
94C2×D101—41no—
95C2×D101—41no—
96C2×D101—41no—
97C2×D101—81no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—41no—
101C4×C101—81no—
102Dic101—41no—
103Dic101—41no—
104Dic101—41no—
105Dic101—81no—
106C2×C101—21no—
107C2×C101—21no—
108C2×C101—81no—
109C2×C101#44911yescharacteristic
110C2×C101#45511yescharacteristic
111C2×C101#45811yescharacteristic
112D101—81no—
113#1612—1011nomaximal, Sylow
114C101#161211yescharacteristic, complemented, Sylow
115#538—1011no—
116#483—1011no—
117#480—1011no—
118#397—1011no—
119#328—1011no—
120#327D10111yescharacteristic, complemented
121#323—1011no—
122#126—2021no—
123#124—2021no—
124#123—2021no—
125C22⋊Q8—1011no—
126#121—2021no—
127#121C2×D10111yescharacteristic
128#120—2021no—
129C4×D4—2021no—
130C2×C4⋊C4—1011no—
131C2×C22⋊C4—1011no—
132C2×C22⋊C4—2021no—
133#107—2021no—
134#106—2021no—
135#103—2021no—
136#102—2021no—
137C22⋊C8—1011no—
138C22⋊C8C2×D10111yescharacteristic
139#95—2021no—
140#95C2×D10111yescharacteristic
141C24—1011no—
142C2×Q8—1011no—
143C2×D4—21no—
144C2×D4—2021no—
145C2×D4—2021no—
146C2×D4D40411yescharacteristic
147C22×C4—21no—
148C22×C4—1011no—
149C22×C4—2021no—
150C22×C4—2021no—
151C22×C4C22×D10111yescharacteristic
152C2×C8—21no—
153C2×C8—2021no—
154C4⋊C4—1011no—
155C4⋊C4—2021no—
156C4⋊C4—2021no—
157C4⋊C4—2021no—
158C4⋊C4—2021no—
159C4⋊C4D40411yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—4042no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C22⋊C4—2021no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C42—2021no—
169C23—21no—
170C23—4041no—
171C23—4041no—
172C23—2021no—
173C23—2021no—
174C23—2021no—
175C23D4×D10111yescharacteristic
176Q8—4041no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—8081no—
181D4—8081no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—8081no—
188C2×C4—4041no—
189C2×C4—4041no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—2021no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4D4×D10111yescharacteristic
196C2×C4C2×D40411yescharacteristic
197C8—41no—
198C8—4041no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—8081no—
205C22—4042no—
206C22—4041no—
207C22—4041no—
208C22—4041no—
209C22#1569222511yescharacteristic
210C22#1569227611yescharacteristic
211C22#1569227911yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—8081no—
217C4—4041no—
218C4—4041no—
219C4—4041no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—8081no—
224C2#2026532311yescharacteristic
225C2#2026535611yescharacteristic
226C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101#120
prime radical#20265764C2
hypercenter#327D101
hypocenterC101#1612
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265508C2semidirect
3. #20265488C2semidirect
4. #20265104C2semidirect
5. #20264913C2semidirect
6. #20264900C2semidirect
7. #327×C101C2semidirect
114. C101#1612semidirect
120. #327D101semidirect

Maximal subgroups

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

subgroupindex
#202657642
#202655082
#202654882
#202651042
#202649132
#202649002
#327×C1012
#1612101