Finite Group Databasebeta

← #27170236

Subgroups — #27170236

Order 12928.

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

normal subgroups only

This group has 250 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, complemented
2#20265525C211yesmaximal, characteristic, complemented
3#20265378C211yesmaximal, characteristic, complemented
4#20265373C211yesmaximal, characteristic
5#20264907C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7#332×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101—21no—
9C2×#13108063—21no—
10C2×D808—21no—
11#15692254—21no—
12#15692252C2211yescharacteristic
13#15692233—22no—
14#15692231—21no—
15C4×Dic202C2211yescharacteristic
16C4×D404C2211yescharacteristic
17#15692217—21no—
18#15692177—21no—
19#15692177—21no—
20#15692175—22no—
21C4×#11717579C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4⋊C8×C101C2211yescharacteristic
25C2×Dic202—22no—
26C2×Dic202D411yescharacteristic
27Q8×D101—42no—
28Q8×D101—42no—
29C2×D404—21no—
30C2×D404—21no—
31C2×D404—21no—
32C2×D404D411yescharacteristic
33C2×C4×D101—22no—
34C2×C4×D101—21no—
35#13108063—42no—
36D808—42no—
37C2×#11717579—21no—
38C4⋊Dic101D411yescharacteristic
39C4⋊Dic101D411yescharacteristic
40#13108053—22no—
41#13108053—21no—
42#13108051—22no—
43#13108051—21no—
44C4×Dic101—21no—
45C2×Q8×C101—21no—
46C2×C8×C101—21no—
47C2×C8×C101—21no—
48C4⋊C4×C101—21no—
49C4⋊C4×C101D412yes—
50C42×C101C2311yescharacteristic
51C22×D101—21no—
52C22×D101—41no—
53Q8×C101—41no—
54Q8×C101—41no—
55C2×C4×C101—22no—
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×Dic101—21no—
60C2×Dic101—21no—
61C2×Dic101—21no—
62D404—21no—
63D404—42no—
64D404—41no—
65D404—41no—
66D404—41no—
67D404—41no—
68D404SD1612yescomplemented
69C4×D101—44no—
70C4×D101—41no—
71C4×D101—41no—
72Dic202—21no—
73Dic202—44no—
74Dic202D812yescomplemented
75C8×C101—41no—
76C8×C101—41no—
77#11717579—41no—
78C22×C101#12011yescharacteristic
79C2×D101—22no—
80C2×D101—41no—
81C2×D101—41no—
82C2×D101—81no—
83C2×D101—81no—
84C2×D101—81no—
85C4×C101—21no—
86C4×C101—42no—
87C4×C101C2×D811yescharacteristic
88C4×C101C2×SD1611yescharacteristic
89C4×C101AGL(1,ℤ/8)11yescharacteristic
90C4×C101#13711yescharacteristic
91Dic101—22no—
92Dic101—41no—
93Dic101—41no—
94C2×C101#44711yescharacteristic
95C2×C101#44811yescharacteristic
96C2×C101#45411yescharacteristic
97D101—42no—
98D101—161no—
99#1627—1011nomaximal, Sylow
100C101#162711yescharacteristic, complemented, Sylow
101D4×Q8—1011no—
102#486—1011no—
103#460—1011no—
104#459—1011no—
105#332D10111yescharacteristic, complemented
106Q8⋊C8—1011no—
107D4⋊C8—1011no—
108C22×Q8—2021no—
109C2×SD16—2021no—
110C2×D8—2021no—
111C4⋊Q8—2021no—
112C4⋊Q8C2×D10111yescharacteristic
113#127—1011no—
114C22⋊Q8—2021no—
115C22⋊Q8—2022no—
116C4×Q8—1011no—
117C4×D4—1011no—
118C4×D4—2021no—
119C4⋊C8C2×D10111yescharacteristic
120C4⋊C8C2×D10111yescharacteristic
121#102—2021no—
122#102—2021no—
123#102—2022no—
124C4×C8—1011no—
125C2×Q8—21no—
126C2×Q8—1011no—
127C2×Q8—4042no—
128C2×Q8—4042no—
129C2×Q8—2022no—
130C2×D4—1011no—
131C2×D4—2021no—
132C2×D4—2021no—
133C2×D4—2021no—
134C22×C4—2022no—
135C22×C4—2021no—
136SD16—4042no—
137D8—4042no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2021no—
141C4⋊C4—21no—
142C4⋊C4—1011no—
143C4⋊C4—1011no—
144C4⋊C4—2022no—
145C4⋊C4—2021no—
146C4⋊C4D40412yes—
147C22⋊C4—2022no—
148C22⋊C4—2021no—
149C42—2021no—
150C42C22×D10111yescharacteristic
151C23—4041no—
152C23—2021no—
153Q8—41no—
154Q8—41no—
155Q8—1012no—
156Q8—4044no—
157Q8—2021no—
158D4—1012no—
159D4—4042no—
160D4—4041no—
161D4—4041no—
162D4—4041no—
163D4—4041no—
164D4—2021no—
165C2×C4—22no—
166C2×C4—4044no—
167C2×C4—4041no—
168C2×C4—4041no—
169C2×C4—2021no—
170C2×C4—2021no—
171C2×C4—2021no—
172C2×C4D4×D10111yescharacteristic
173C2×C4D4×D10111yescharacteristic
174C2×C4C2×D40411yescharacteristic
175C8—41no—
176C8—41no—
177C8—4041no—
178C22—8081no—
179C22—8081no—
180C22—8081no—
181C22—4041no—
182C22—4041no—
183C22—2022no—
184C22#1569222511yescharacteristic
185C4—21no—
186C4—42no—
187C4—4041no—
188C4—4041no—
189C4—2022no—
190C4D8×D10111yescharacteristic
191C4SD16×D10111yescharacteristic
192C4#1569228211yescharacteristic
193C4#1569228711yescharacteristic
194C2—16161no—
195C2—4042no—
196C2#2026531211yescharacteristic
197C2#2026531711yescharacteristic
198C2#2026535211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#332×C101C2
socleC22×C101#120
prime radical#20265373C2
hypercenter#332D101
hypocenterC101#1627
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #20265842C2semidirect
2. #20265525C2semidirect
3. #20265378C2semidirect
5. #20264907C2semidirect
6. #20264904C2semidirect
7. #332×C101C2semidirect
68. D404SD16semidirect
74. Dic202D8semidirect
100. C101#1627semidirect
105. #332D101semidirect

Maximal subgroups

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

subgroupindex
#202658422
#202655252
#202653782
#202653732
#202649072
#202649042
#332×C1012
#1627101