Finite Group Databasebeta

← #27170368

Subgroups — #27170368

Order 12928.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265830C211yesmaximal, characteristic
2#20265580C211yesmaximal, characteristic, complemented
3#20265458C211yesmaximal, characteristic, complemented
4#20265383C211yesmaximal, characteristic, complemented
5#20264904C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7#332×C101C211yesmaximal, characteristic, complemented
8C2×#13108075—21no—
9C2×#13108063—21no—
10#15692255C2211yescharacteristic
11#15692240—22no—
12#15692231—21no—
13#15692230—21no—
14C4×D404C2211yescharacteristic
15C4×D404C2211yescharacteristic
16C4⋊C4×D101—21no—
17#15692189—21no—
18#15692188—22no—
19#15692180—21no—
20#15692177—21no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25#13108075—42no—
26#13108075—42no—
27C2×Dic202—21no—
28C2×D404—22no—
29C2×D404D411yescharacteristic
30C2×D404D411yescharacteristic
31C2×C4×D101—22no—
32C2×C4×D101—21no—
33C2×C4×D101—21no—
34#13108063—42no—
35C2×#11717579—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101D411yescharacteristic
39C4⋊Dic101D411yescharacteristic
40#13108053—22no—
41#13108051—22no—
42#13108051—21no—
43#13108051—21no—
44C2×Q8×C101—21no—
45C2×C8×C101—21no—
46C2×C8×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×C101D412yes—
49C42×C101C2311yescharacteristic
50C22×D101—21no—
51C22×D101—21no—
52Q8×C101—41no—
53Q8×C101—41no—
54C2×C4×C101—22no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—41no—
61D404—21no—
62D404—21no—
63D404—21no—
64D404—44no—
65D404D812yes—
66C4×D101—44no—
67C4×D101—41no—
68C4×D101—41no—
69C4×D101—41no—
70C4×D101—41no—
71Dic202—41no—
72Dic202—41no—
73C8×C101—41no—
74C8×C101—41no—
75#11717579—22no—
76C22×C101#12011yescharacteristic
77C2×D101—22no—
78C2×D101—41no—
79C2×D101—41no—
80C2×D101—41no—
81C2×D101—41no—
82C2×D101—41no—
83C4×C101—21no—
84C4×C101—42no—
85C4×C101C2×D811yescharacteristic
86C4×C101#13511yescharacteristic
87C4×C101AGL(1,ℤ/8)11yescharacteristic
88C4×C101#13711yescharacteristic
89Dic101—41no—
90Dic101—41no—
91Dic101—81no—
92C2×C101#44711yescharacteristic
93C2×C101#45211yescharacteristic
94C2×C101#45611yescharacteristic
95D101—42no—
96D101—81no—
97#1647—1011nomaximal, Sylow
98C101#164711yescharacteristic, complemented, Sylow
99#547—1011no—
100#500—1011no—
101#474—1011no—
102#461—1011no—
103#332D10111yescharacteristic, complemented
104D4⋊C8—1011no—
105D4⋊C8—1011no—
106C2×#41—2021no—
107C2×SD16—2021no—
108C4⋊Q8—1011no—
109C4⋊Q8C2×D10111yescharacteristic
110#123—2022no—
111C22⋊Q8—2021no—
112#121—2021no—
113C4×D4—1011no—
114C4×D4—1011no—
115C2×C4⋊C4—2021no—
116#107—2021no—
117#107—2022no—
118C4⋊C8C2×D10111yescharacteristic
119C4⋊C8C2×D10111yescharacteristic
120#103—2021no—
121#102—2021no—
122C4×C8—1011no—
123#41—4042no—
124#41—4042no—
125C2×Q8—21no—
126C2×Q8—2021no—
127C2×D4—1011no—
128C2×D4—1011no—
129C2×D4—2022no—
130C22×C4—2022no—
131C22×C4—2021no—
132C22×C4—2021no—
133SD16—4042no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—2021no—
137C4⋊C4—21no—
138C4⋊C4—1011no—
139C4⋊C4—1011no—
140C4⋊C4—2022no—
141C4⋊C4—2021no—
142C4⋊C4—2021no—
143C4⋊C4D40412yes—
144C22⋊C4—2022no—
145C22⋊C4—2021no—
146C22⋊C4—2021no—
147C42C22×D10111yescharacteristic
148C23—2021no—
149C23—2021no—
150Q8—41no—
151Q8—41no—
152Q8—4041no—
153Q8—4041no—
154D4—1012no—
155D4—4044no—
156D4—2021no—
157D4—2021no—
158D4—2021no—
159C2×C4—22no—
160C2×C4—4044no—
161C2×C4—4041no—
162C2×C4—4041no—
163C2×C4—4041no—
164C2×C4—4041no—
165C2×C4—4041no—
166C2×C4—2021no—
167C2×C4—2021no—
168C2×C4D4×D10111yescharacteristic
169C2×C4D4×D10111yescharacteristic
170C2×C4C2×D40411yescharacteristic
171C8—41no—
172C8—41no—
173C8—2022no—
174C22—4041no—
175C22—4041no—
176C22—4041no—
177C22—4041no—
178C22—4041no—
179C22—2022no—
180C22#1569222511yescharacteristic
181C4—21no—
182C4—42no—
183C4—8081no—
184C4—4041no—
185C4—4041no—
186C4D8×D10111yescharacteristic
187C4#1569227611yescharacteristic
188C4#1569228211yescharacteristic
189C4#1569228711yescharacteristic
190C2—8081no—
191C2—4042no—
192C2#2026531211yescharacteristic
193C2#2026534211yescharacteristic
194C2#2026536211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#332×C101C2
socleC22×C101#120
prime radical#20265830C2
hypercenter#332D101
hypocenterC101#1647
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20265580C2semidirect
3. #20265458C2semidirect
4. #20265383C2semidirect
5. #20264904C2semidirect
6. #20264904C2semidirect
7. #332×C101C2semidirect
98. C101#1647semidirect
103. #332D101semidirect

Maximal subgroups

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

subgroupindex
#202658302
#202655802
#202654582
#202653832
#202649042
#332×C1012
#1647101