Finite Group Databasebeta

← #27170283

Subgroups — #27170283

Order 12928.

234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 61 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#20265529C211yesmaximal, characteristic, complemented
2#20265467C211yesmaximal, characteristic, complemented
3#20265467C211yesmaximal, characteristic, complemented
4#20264923C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#547×C101C211yesmaximal, characteristic
8#15692255C2211yescharacteristic
9#15692186—21no—
10#15692186—21no—
11C4⋊#11717579C2211yescharacteristic
12C4⋊#11717579C2211yescharacteristic
13#15692178—22no—
14#15692176—21no—
15#15692176—21no—
16C4×#11717579C2211yescharacteristic
17C2×#41×C101—21no—
18C4⋊Q8×C101C2211yescharacteristic
19#123×C101—22no—
20C22⋊Q8×C101—21no—
21#121×C101—21no—
22C4×D4×C101C2211yescharacteristic
23C4×D4×C101C2211yescharacteristic
24C2×C4⋊C4×C101—21no—
25C2×Dic202—21no—
26C2×#11717579—21no—
27C2×#11717579—21no—
28C2×#11717579—21no—
29C4⋊Dic101—21no—
30C4⋊Dic101—21no—
31#41×C101—42no—
32#41×C101—42no—
33C2×Q8×C101—21no—
34C2×D4×C101—22no—
35C2×D4×C101D411yescharacteristic
36C2×D4×C101D411yescharacteristic
37C22×C4×C101—22no—
38C22×C4×C101—21no—
39C22×C4×C101—21no—
40C4⋊C4×C101—22no—
41C4⋊C4×C101—21no—
42C4⋊C4×C101D411yescharacteristic
43C4⋊C4×C101D411yescharacteristic
44C4⋊C4×C101D412yes—
45C22⋊C4×C101—22no—
46C22⋊C4×C101—21no—
47C22⋊C4×C101—21no—
48C42×C101C2311yescharacteristic
49C23×C101—21no—
50C23×C101—21no—
51Q8×C101—41no—
52Q8×C101—41no—
53D4×C101—21no—
54D4×C101—21no—
55D4×C101—21no—
56D4×C101—44no—
57D4×C101SD1612yes—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—22no—
61C2×C4×C101—44no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C2×Dic101—41no—
70Dic202—42no—
71#11717579—41no—
72#11717579—41no—
73#11717579—41no—
74C22×C101—22no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101—41no—
80C22×C101#12011yescharacteristic
81C4×C101—21no—
82C4×C101—41no—
83C4×C101—42no—
84C4×C101—41no—
85C4×C101C2×SD1611yescharacteristic
86C4×C101#13511yescharacteristic
87C4×C101AGL(1,ℤ/8)11yescharacteristic
88C4×C101#13711yescharacteristic
89Dic101—81no—
90C2×C101—42no—
91C2×C101—81no—
92C2×C101#45011yescharacteristic
93C2×C101#45211yescharacteristic
94C2×C101#45411yescharacteristic
95#1635—1011nomaximal, Sylow
96C101#163511yescharacteristic, complemented, Sylow
97#547D10111yescharacteristic
98#487—1011no—
99#476—1011no—
100#476—1011no—
101#332—1011no—
102D4⋊C8—1011no—
103D4⋊C8—1011no—
104C2×#41—21no—
105C4⋊Q8—1011no—
106C4⋊Q8C2×D10111yescharacteristic
107#123—22no—
108C22⋊Q8—21no—
109#121—21no—
110C4×D4C2×D10111yescharacteristic
111C4×D4C2×D10111yescharacteristic
112C2×C4⋊C4—21no—
113#106—2021no—
114#106—2021no—
115C4⋊C8—1011no—
116C4⋊C8—1011no—
117#103—2022no—
118#102—2021no—
119#102—2021no—
120C4×C8—1011no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×Q8—2021no—
125C2×D4—22no—
126C2×D4C22⋊D10111yescharacteristic
127C2×D4C22⋊D10111yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C2×C8—2021no—
132C2×C8—2021no—
133C2×C8—2021no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—2021no—
137C4⋊C4—2021no—
138C4⋊C4C22⋊D10112yes—
139C4⋊C4C22⋊D10111yescharacteristic
140C4⋊C4C22⋊D10111yescharacteristic
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C42C22×D10111yescharacteristic
145C23—21no—
146C23—21no—
147Q8—41no—
148Q8—41no—
149Q8—4042no—
150D4—21no—
151D4—21no—
152D4—21no—
153D4—44no—
154D4#1310806212yes—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—4041no—
164C2×C4C2×C22⋊D10111yescharacteristic
165C2×C4C2×C22⋊D10111yescharacteristic
166C2×C4C2×C22⋊D10111yescharacteristic
167C8—4041no—
168C8—4041no—
169C8—4041no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#1569222411yescharacteristic
177C4—21no—
178C4—41no—
179C4—42no—
180C4—41no—
181C4—8081no—
182C4C2×#1310806211yescharacteristic
183C4#1569227511yescharacteristic
184C4#1569228111yescharacteristic
185C4#1569228511yescharacteristic
186C2—42no—
187C2—81no—
188C2#2026532811yescharacteristic
189C2#2026533911yescharacteristic
190C2#2026535011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radical#547×C101C2
socleC22×C101#120
hypercenter#547D101
hypocenterC101#1635
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20265529C2semidirect
2. #20265467C2semidirect
3. #20265467C2semidirect
4. #20264923C2semidirect
5. D4⋊#11717579C2semidirect
6. D4⋊#11717579C2semidirect
96. C101#1635semidirect

Maximal subgroups

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

subgroupindex
#202655292
#202654672
#202649232
D4⋊#117175792
#547×C1012
#1635101