Finite Group Databasebeta

← #27170366

Subgroups — #27170366

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#20265832C211yesmaximal, characteristic
2#20265580C211yesmaximal, characteristic, complemented
3#20265379C211yesmaximal, characteristic, complemented
4#20264922C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7#474×C101C211yesmaximal, characteristic, complemented
8C2×#13108074—21no—
9C2×#13108061—21no—
10#15692255C2211yescharacteristic
11#15692237—22no—
12#15692234—21no—
13#15692226—21no—
14C4×D404C2211yescharacteristic
15C4⋊C4×D101—21no—
16#15692188—22no—
17C4⋊#11717579C2211yescharacteristic
18#15692179—21no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×D4×C101C2211yescharacteristic
22#107×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#102×C101—21no—
25#13108074—42no—
26#13108074—42no—
27C2×Dic202—21no—
28C2×D404D411yescharacteristic
29C2×C22⋊D101—22no—
30C2×C4×D101—22no—
31C2×C4×D101—21no—
32#13108061—42no—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37C4⋊Dic101D412yes—
38#13108053—22no—
39C22⋊Dic101—22no—
40#13108051—21no—
41C2×Q8×C101—21no—
42C2×D4×C101D411yescharacteristic
43C22×C4×C101—21no—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51C22×D101—21no—
52Q8×C101—41no—
53Q8×C101—41no—
54D4×C101—21no—
55D4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C22⋊D101—44no—
64C2×Dic101—21no—
65C2×Dic101—22no—
66D404—21no—
67D404D812yes—
68C4×D101—44no—
69C4×D101—41no—
70C4×D101—41no—
71Dic202—41no—
72Dic202—41no—
73C8×C101—41no—
74#11717579—22no—
75#11717579—41no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101#12011yescharacteristic
80C2×D101—22no—
81C2×D101—41no—
82C2×D101—41no—
83C4×C101—21no—
84C4×C101—41no—
85C4×C101—81no—
86C4×C101C2×D811yescharacteristic
87C4×C101#13511yescharacteristic
88C4×C101AGL(1,ℤ/8)11yescharacteristic
89C4×C101#13711yescharacteristic
90Dic101—42no—
91Dic101—41no—
92C2×C101—81no—
93C2×C101#44711yescharacteristic
94C2×C101#45211yescharacteristic
95C2×C101#45611yescharacteristic
96D101—42no—
97#1647—1011nomaximal, Sylow
98C101#164711yescharacteristic, complemented, Sylow
99#547—1011no—
100#500—1011no—
101#474D10111yescharacteristic, complemented
102#461—1011no—
103#332—1011no—
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×D4C2×D10111yescharacteristic
115C2×C4⋊C4—2021no—
116#107—21no—
117#107—2022no—
118C4⋊C8—1011no—
119C4⋊C8C2×D10111yescharacteristic
120#103—2021no—
121#102—21no—
122C4×C8—1011no—
123#41—4042no—
124#41—4042no—
125C2×Q8—21no—
126C2×Q8—2021no—
127C2×D4—1011no—
128C2×D4—2022no—
129C2×D4C22⋊D10111yescharacteristic
130C22×C4—21no—
131C22×C4—2022no—
132C22×C4—2021no—
133SD16—4042no—
134C2×C8—21no—
135C2×C8—2021no—
136C2×C8—2021no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—1011no—
140C4⋊C4—1012no—
141C4⋊C4—2022no—
142C4⋊C4—2021no—
143C4⋊C4C22⋊D10111yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—2022no—
146C22⋊C4—2021no—
147C42C22×D10111yescharacteristic
148C23—21no—
149C23—2021no—
150Q8—41no—
151Q8—41no—
152Q8—4041no—
153Q8—4041no—
154D4—21no—
155D4—21no—
156D4—1012no—
157D4—4044no—
158D4—2021no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—4044no—
164C2×C4—4041no—
165C2×C4—4041no—
166C2×C4—2021no—
167C2×C4—2022no—
168C2×C4D4×D10111yescharacteristic
169C2×C4D4×D10111yescharacteristic
170C2×C4C2×C22⋊D10111yescharacteristic
171C8—41no—
172C8—4041no—
173C8—2022no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—4041no—
178C22—4041no—
179C22—2022no—
180C22#1569222311yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—4042no—
185C4—4041no—
186C4D8×D10111yescharacteristic
187C4#1569227511yescharacteristic
188C4#1569227911yescharacteristic
189C4#1569228611yescharacteristic
190C2—81no—
191C2—4042no—
192C2#2026530911yescharacteristic
193C2#2026533711yescharacteristic
194C2#2026535811yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#474×C101C2
socleC22×C101#120
prime radical#20265832C2
hypercenter#474D101
hypocenterC101#1647
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20265580C2semidirect
3. #20265379C2semidirect
4. #20264922C2semidirect
5. D4⋊#11717579C2semidirect
6. #20264904C2semidirect
7. #474×C101C2semidirect
98. C101#1647semidirect
101. #474D101semidirect

Maximal subgroups

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

subgroupindex
#202658322
#202655802
#202653792
#202649222
D4⋊#117175792
#202649042
#474×C1012
#1647101