Finite Group Databasebeta

← #27170395

Subgroups — #27170395

Order 12928.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265573C211yesmaximal, characteristic, complemented
2#20265462C211yesmaximal, characteristic, complemented
3#20265382C211yesmaximal, characteristic
4#20264923C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic
7D4×Q8×C101C211yesmaximal, characteristic
8C2×#13108062—21no—
9#15692255C2211yescharacteristic
10#15692186—22no—
11#15692186—21no—
12C4⋊#11717579C2211yescharacteristic
13C4⋊#11717579C2211yescharacteristic
14#15692179—21no—
15#15692178—21no—
16C4×#11717579C2211yescharacteristic
17C22×Q8×C101—21no—
18C4⋊Q8×C101—21no—
19C4⋊Q8×C101C2211yescharacteristic
20C22⋊Q8×C101—21no—
21C22⋊Q8×C101—22no—
22C4×Q8×C101C2211yescharacteristic
23C4×D4×C101—21no—
24C4×D4×C101C2211yescharacteristic
25C2×Dic202—21no—
26#13108062—42no—
27C2×#11717579—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—21no—
31C4⋊Dic101—21no—
32C2×Q8×C101—22no—
33C2×Q8×C101—21no—
34C2×Q8×C101—42no—
35C2×Q8×C101—42no—
36C2×Q8×C101D411yescharacteristic
37C2×D4×C101D411yescharacteristic
38C22×C4×C101—22no—
39C22×C4×C101—21no—
40C4⋊C4×C101—22no—
41C4⋊C4×C101—21no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101D411yescharacteristic
44C4⋊C4×C101D411yescharacteristic
45C4⋊C4×C101D412yes—
46C22⋊C4×C101—22no—
47C22⋊C4×C101—21no—
48C42×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51Q8×C101—21no—
52Q8×C101—44no—
53Q8×C101—41no—
54Q8×C101—41no—
55Q8×C101SD1612yescomplemented
56D4×C101—21no—
57D4×C101SD1612yes—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—22no—
61C2×C4×C101—21no—
62C2×C4×C101—44no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101C2×D411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×Dic101—41no—
69Dic202—41no—
70Dic202—41no—
71#11717579—22no—
72#11717579—41no—
73#11717579—41no—
74C22×C101—22no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101#12011yescharacteristic
78C4×C101—22no—
79C4×C101—21no—
80C4×C101—42no—
81C4×C101—41no—
82C4×C101—41no—
83C4×C101C2×SD1611yescharacteristic
84C4×C101C2×SD1611yescharacteristic
85C4×C101AGL(1,ℤ/8)11yescharacteristic
86C4×C101#13711yescharacteristic
87Dic101—81no—
88C2×C101—42no—
89C2×C101#44811yescharacteristic
90C2×C101#45011yescharacteristic
91C2×C101#45611yescharacteristic
92#1651—1011nomaximal, Sylow
93C101#165111yescharacteristic, complemented, Sylow
94D4×Q8D10111yescharacteristic
95#498—1011no—
96#475—1011no—
97#461—1011no—
98#332—1011no—
99Q8⋊C8—1011no—
100D4⋊C8—1011no—
101C22×Q8—21no—
102C2×SD16—2021no—
103C4⋊Q8—21no—
104C4⋊Q8—1011no—
105C4⋊Q8C2×D10111yescharacteristic
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C4×Q8C2×D10111yescharacteristic
109C4×D4—21no—
110C4×D4C2×D10111yescharacteristic
111#106—2021no—
112#106—2022no—
113C4⋊C8—1011no—
114C4⋊C8—1011no—
115#103—2021no—
116#103—2021no—
117C4×C8—1011no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—2021no—
123C2×Q8C22⋊D10111yescharacteristic
124C2×D4C22⋊D10111yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127SD16—4042no—
128C2×C8—2021no—
129C2×C8—2021no—
130C2×C8—2021no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2021no—
135C4⋊C4—2021no—
136C4⋊C4C22⋊D10111yescharacteristic
137C4⋊C4C22⋊D10111yescharacteristic
138C4⋊C4C22⋊D10112yes—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C42—21no—
142C42C22×D10111yescharacteristic
143C23—21no—
144Q8—21no—
145Q8—44no—
146Q8—41no—
147Q8—41no—
148Q8—4041no—
149Q8—4041no—
150Q8#1310806212yescomplemented
151D4—21no—
152D4#1310806112yes—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—4041no—
161C2×C4C2×C22⋊D10111yescharacteristic
162C2×C4C2×C22⋊D10111yescharacteristic
163C2×C4C2×C22⋊D10111yescharacteristic
164C8—4041no—
165C8—4041no—
166C8—2022no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1569222411yescharacteristic
171C4—22no—
172C4—21no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—8081no—
177C4C2×#1310806111yescharacteristic
178C4C2×#1310806211yescharacteristic
179C4#1569228111yescharacteristic
180C4#1569228611yescharacteristic
181C2—42no—
182C2#2026531611yescharacteristic
183C2#2026532811yescharacteristic
184C2#2026536011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4×Q8×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenterD4×Q8D101
hypocenterC101#1651
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20265573C2semidirect
2. #20265462C2semidirect
4. #20264923C2semidirect
5. Q8⋊#11717579C2semidirect
55. Q8×C101SD16semidirect
93. C101#1651semidirect
150. Q8#13108062semidirect

Maximal subgroups

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

subgroupindex
#202655732
#202654622
#202653822
#202649232
Q8⋊#117175792
D4⋊#117175792
D4×Q8×C1012
#1651101