Finite Group Databasebeta

← #27170396

Subgroups — #27170396

Order 12928.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265842C211yesmaximal, characteristic
2#20265573C211yesmaximal, characteristic, complemented
3#20265463C211yesmaximal, characteristic, complemented
4#20265383C211yesmaximal, characteristic
5#20264907C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#332×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101—21no—
9C2×#13108063—21no—
10#15692255C2211yescharacteristic
11#15692254—21no—
12#15692233—22no—
13#15692231—21no—
14C4×Dic202C2211yescharacteristic
15C4×D404C2211yescharacteristic
16#15692217—21no—
17#15692187—21no—
18#15692186—22no—
19#15692180—21no—
20#15692180—21no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25C2×Dic202—22no—
26C2×Dic202—21no—
27C2×Dic202D411yescharacteristic
28Q8×D101—42no—
29Q8×D101—42no—
30C2×D404D411yescharacteristic
31C2×C4×D101—22no—
32C2×C4×D101—21no—
33#13108063—42no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101D411yescharacteristic
38C4⋊Dic101D411yescharacteristic
39#13108053—22no—
40#13108053—21no—
41#13108051—22no—
42#13108051—21no—
43C4×Dic101—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—
51Q8×C101—41no—
52Q8×C101—41no—
53C2×C4×C101—22no—
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—41no—
61D404—21no—
62D404SD1612yes—
63C4×D101—44no—
64C4×D101—41no—
65C4×D101—41no—
66Dic202—21no—
67Dic202—44no—
68Dic202—41no—
69Dic202—41no—
70Dic202SD1612yescomplemented
71C8×C101—41no—
72C8×C101—41no—
73#11717579—22no—
74C22×C101#12011yescharacteristic
75C2×D101—22no—
76C2×D101—41no—
77C2×D101—41no—
78C4×C101—21no—
79C4×C101—42no—
80C4×C101C2×SD1611yescharacteristic
81C4×C101C2×SD1611yescharacteristic
82C4×C101AGL(1,ℤ/8)11yescharacteristic
83C4×C101#13711yescharacteristic
84Dic101—22no—
85Dic101—41no—
86Dic101—41no—
87Dic101—81no—
88C2×C101#44811yescharacteristic
89C2×C101#45011yescharacteristic
90C2×C101#45611yescharacteristic
91D101—42no—
92#1651—1011nomaximal, Sylow
93C101#165111yescharacteristic, complemented, Sylow
94D4×Q8—1011no—
95#498—1011no—
96#475—1011no—
97#461—1011no—
98#332D10111yescharacteristic, complemented
99Q8⋊C8—1011no—
100D4⋊C8—1011no—
101C22×Q8—2021no—
102C2×SD16—2021no—
103C4⋊Q8—1011no—
104C4⋊Q8—2021no—
105C4⋊Q8C2×D10111yescharacteristic
106C22⋊Q8—2021no—
107C22⋊Q8—2022no—
108C4×Q8—1011no—
109C4×D4—1011no—
110C4×D4—2021no—
111#106—2021no—
112#106—2022no—
113C4⋊C8C2×D10111yescharacteristic
114C4⋊C8C2×D10111yescharacteristic
115#103—2021no—
116#103—2021no—
117C4×C8—1011no—
118C2×Q8—21no—
119C2×Q8—1011no—
120C2×Q8—4042no—
121C2×Q8—4042no—
122C2×Q8—2022no—
123C2×Q8—2021no—
124C2×D4—1011no—
125C22×C4—2022no—
126C22×C4—2021no—
127SD16—4042no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2021no—
131C4⋊C4—21no—
132C4⋊C4—1011no—
133C4⋊C4—1011no—
134C4⋊C4—2022no—
135C4⋊C4—2021no—
136C4⋊C4—2021no—
137C4⋊C4—2021no—
138C4⋊C4D40412yes—
139C22⋊C4—2022no—
140C22⋊C4—2021no—
141C42—2021no—
142C42C22×D10111yescharacteristic
143C23—2021no—
144Q8—41no—
145Q8—41no—
146Q8—1012no—
147Q8—4044no—
148Q8—4041no—
149Q8—4041no—
150Q8—2021no—
151D4—1012no—
152D4—2021no—
153C2×C4—22no—
154C2×C4—4044no—
155C2×C4—4041no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—2021no—
159C2×C4—2021no—
160C2×C4—2021no—
161C2×C4D4×D10111yescharacteristic
162C2×C4D4×D10111yescharacteristic
163C2×C4C2×D40411yescharacteristic
164C8—41no—
165C8—41no—
166C8—2022no—
167C22—4041no—
168C22—4041no—
169C22—2022no—
170C22#1569222511yescharacteristic
171C4—21no—
172C4—42no—
173C4—8081no—
174C4—4041no—
175C4—4041no—
176C4—2022no—
177C4SD16×D10111yescharacteristic
178C4SD16×D10111yescharacteristic
179C4#1569227911yescharacteristic
180C4#1569228711yescharacteristic
181C2—4042no—
182C2#2026531711yescharacteristic
183C2#2026532911yescharacteristic
184C2#2026536211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#332×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#332D101
hypocenterC101#1651
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. #20265573C2semidirect
3. #20265463C2semidirect
5. #20264907C2semidirect
7. #332×C101C2semidirect
70. Dic202SD16semidirect
93. C101#1651semidirect
98. #332D101semidirect

Maximal subgroups

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

subgroupindex
#202658422
#202655732
#202654632
#202653832
#202649072
#202649042
#332×C1012
#1651101