Finite Group Databasebeta

← #27170360

Subgroups — #27170360

Order 12928.

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

normal subgroups only

This group has 212 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#20265527C211yesmaximal, characteristic, complemented
2#20265457C211yesmaximal, characteristic, complemented
3#20265382C211yesmaximal, characteristic, complemented
4#20264926C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#555×C101C211yesmaximal, characteristic
8C2×#13108062—21no—
9#15692255C2211yescharacteristic
10#15692188—21no—
11C4⋊#11717579C2211yescharacteristic
12C4⋊#11717579C2211yescharacteristic
13#15692179—22no—
14#15692178—21no—
15#15692176—21no—
16C4×#11717579C2211yescharacteristic
17C4⋊Q8×C101C2211yescharacteristic
18#124×C101—22no—
19#121×C101—21no—
20#121×C101—21no—
21C4×D4×C101C2211yescharacteristic
22C4×D4×C101C2211yescharacteristic
23#117×C101—21no—
24#117×C101—21no—
25C2×Dic202—21no—
26#13108062—42no—
27C2×#11717579—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—21no—
31C4⋊Dic101—21no—
32C2×Q8×C101D412yes—
33C2×D4×C101—22no—
34C2×D4×C101D411yescharacteristic
35C2×D4×C101D411yescharacteristic
36C22×C4×C101—21no—
37C22×C4×C101—21no—
38C4⋊C4×C101—21no—
39C4⋊C4×C101—21no—
40C4⋊C4×C101D411yescharacteristic
41C4⋊C4×C101D411yescharacteristic
42C22⋊C4×C101—22no—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—21no—
45C22⋊C4×C101—21no—
46C42×C101—22no—
47C42×C101C2311yescharacteristic
48C23×C101—21no—
49C23×C101—21no—
50Q8×C101—42no—
51D4×C101—21no—
52D4×C101—21no—
53D4×C101—21no—
54D4×C101—21no—
55D4×C101—82no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—22no—
59C2×C4×C101—41no—
60C2×C4×C101—41no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×Dic101—41no—
67Dic202—41no—
68Dic202—41no—
69#11717579—41no—
70#11717579—41no—
71#11717579—41no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101#12011yescharacteristic
79C4×C101—21no—
80C4×C101—41no—
81C4×C101—41no—
82C4×C101—42no—
83C4×C101#13511yescharacteristic
84C4×C101#13511yescharacteristic
85C4×C101AGL(1,ℤ/8)11yescharacteristic
86C4×C101#13711yescharacteristic
87Dic101—81no—
88C2×C101—81no—
89C2×C101—81no—
90C2×C101#44911yescharacteristic
91C2×C101#45211yescharacteristic
92C2×C101#45511yescharacteristic
93#1646—1011nomaximal, Sylow
94C101#164611yescharacteristic, complemented, Sylow
95#555D10111yescharacteristic
96#487—1011no—
97#474—1011no—
98#461—1011no—
99#333—1011no—
100D4⋊C8—1011no—
101D4⋊C8—1011no—
102C2×SD16—2021no—
103C4⋊Q8—1011no—
104C4⋊Q8C2×D10111yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D10111yescharacteristic
109C4×D4C2×D10111yescharacteristic
110#117—21no—
111#117—21no—
112#107—2021no—
113C4⋊C8—1011no—
114C4⋊C8—1011no—
115#103—2022no—
116#103—2021no—
117#102—2021no—
118C4×C8—1011no—
119C2×Q8—2021no—
120C2×Q8C22⋊D10112yes—
121C2×D4—22no—
122C2×D4C22⋊D10111yescharacteristic
123C2×D4C22⋊D10111yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—4042no—
127C2×C8—2021no—
128C2×C8—2021no—
129C2×C8—2021no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—2021no—
133C4⋊C4—2021no—
134C4⋊C4C22⋊D10111yescharacteristic
135C4⋊C4C22⋊D10111yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D10111yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—4041no—
146Q8—4041no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—4041no—
160C2×C4C2×C22⋊D10111yescharacteristic
161C2×C4C2×C22⋊D10111yescharacteristic
162C2×C4C2×C22⋊D10111yescharacteristic
163C8—4041no—
164C8—4041no—
165C8—4041no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1569222411yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—8081no—
178C4#1569227511yescharacteristic
179C4#1569227511yescharacteristic
180C4#1569228011yescharacteristic
181C4#1569228511yescharacteristic
182C2—81no—
183C2—81no—
184C2#2026532211yescharacteristic
185C2#2026533911yescharacteristic
186C2#2026535511yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radical#555×C101C2
socleC22×C101#120
hypercenter#555D101
hypocenterC101#1646
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20265527C2semidirect
2. #20265457C2semidirect
3. #20265382C2semidirect
4. #20264926C2semidirect
5. D4⋊#11717579C2semidirect
6. D4⋊#11717579C2semidirect
94. C101#1646semidirect

Maximal subgroups

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

subgroupindex
#202655272
#202654572
#202653822
#202649262
D4⋊#117175792
#555×C1012
#1646101