Finite Group Databasebeta

← #36352826

Subgroups — #36352826

Order 22144.

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#26011887C211yesmaximal, characteristic, complemented
2#26011776C211yesmaximal, characteristic, complemented
3#26011696C211yesmaximal, characteristic
4#26011237C211yesmaximal, characteristic, complemented
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic
7D4×Q8×C173C211yesmaximal, characteristic
8C2×#14583814—21no—
9#19149679C2211yescharacteristic
10#19149610—21no—
11#19149610—22no—
12C4⋊#13063436C2211yescharacteristic
13C4⋊#13063436C2211yescharacteristic
14#19149603—21no—
15#19149602—21no—
16C4×#13063436C2211yescharacteristic
17C22×Q8×C173—21no—
18C4⋊Q8×C173—21no—
19C4⋊Q8×C173C2211yescharacteristic
20C22⋊Q8×C173—21no—
21C22⋊Q8×C173—22no—
22C4×Q8×C173C2211yescharacteristic
23C4×D4×C173—21no—
24C4×D4×C173C2211yescharacteristic
25C2×Dic346—21no—
26#14583814—42no—
27C2×#13063436—21no—
28C2×#13063436—21no—
29C2×#13063436—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173—21no—
32C2×Q8×C173—22no—
33C2×Q8×C173—21no—
34C2×Q8×C173—42no—
35C2×Q8×C173—42no—
36C2×Q8×C173D411yescharacteristic
37C2×D4×C173D411yescharacteristic
38C22×C4×C173—22no—
39C22×C4×C173—21no—
40C4⋊C4×C173—22no—
41C4⋊C4×C173—21no—
42C4⋊C4×C173—21no—
43C4⋊C4×C173D411yescharacteristic
44C4⋊C4×C173D412yes—
45C4⋊C4×C173D411yescharacteristic
46C22⋊C4×C173—22no—
47C22⋊C4×C173—21no—
48C42×C173—21no—
49C42×C173C2311yescharacteristic
50Dic346—41no—
51Dic346—41no—
52C2×Dic173—41no—
53#13063436—22no—
54#13063436—41no—
55#13063436—41no—
56C23×C173—21no—
57Q8×C173—21no—
58Q8×C173—44no—
59Q8×C173—41no—
60Q8×C173—41no—
61Q8×C173SD1612yescomplemented
62D4×C173—21no—
63D4×C173SD1612yes—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—22no—
67C2×C4×C173—21no—
68C2×C4×C173—44no—
69C2×C4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C22×C173—22no—
75C22×C173—41no—
76C22×C173—41no—
77C22×C173#12011yescharacteristic
78C4×C173—22no—
79C4×C173—21no—
80C4×C173—42no—
81C4×C173—41no—
82C4×C173—41no—
83C4×C173C2×SD1611yescharacteristic
84C4×C173C2×SD1611yescharacteristic
85C4×C173AGL(1,ℤ/8)11yescharacteristic
86C4×C173#13711yescharacteristic
87Dic173—81no—
88C2×C173—42no—
89C2×C173#44811yescharacteristic
90C2×C173#45011yescharacteristic
91C2×C173#45611yescharacteristic
92C173#165111yescharacteristic, complemented, Sylow
93#1651—1731nomaximal, Sylow
94D4×Q8D17311yescharacteristic
95#498—1731no—
96#475—1731no—
97#461—1731no—
98#332—1731no—
99Q8⋊C8—1731no—
100D4⋊C8—1731no—
101C22×Q8—21no—
102C2×SD16—3461no—
103C4⋊Q8—21no—
104C4⋊Q8—1731no—
105C4⋊Q8C2×D17311yescharacteristic
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C4×Q8C2×D17311yescharacteristic
109C4×D4—21no—
110C4×D4C2×D17311yescharacteristic
111#106—3461no—
112#106—3462no—
113C4⋊C8—1731no—
114C4⋊C8—1731no—
115#103—3461no—
116#103—3461no—
117C4×C8—1731no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—3461no—
123C2×Q8C22⋊D17311yescharacteristic
124C2×D4C22⋊D17311yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127SD16—6922no—
128C2×C8—3461no—
129C2×C8—3461no—
130C2×C8—3461no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4C22⋊D17311yescharacteristic
137C4⋊C4C22⋊D17311yescharacteristic
138C4⋊C4C22⋊D17312yes—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C42—21no—
142C42C22×D17311yescharacteristic
143C23—21no—
144Q8—21no—
145Q8—44no—
146Q8—41no—
147Q8—41no—
148Q8—6921no—
149Q8—6921no—
150Q8#1458381412yescomplemented
151D4—21no—
152D4#1458381312yes—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—6921no—
161C2×C4C2×C22⋊D17311yescharacteristic
162C2×C4C2×C22⋊D17311yescharacteristic
163C2×C4C2×C22⋊D17311yescharacteristic
164C8—3462no—
165C8—6921no—
166C8—6921no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1914964811yescharacteristic
171C4—21no—
172C4—22no—
173C4—41no—
174C4—42no—
175C4—41no—
176C4—13841no—
177C4C2×#1458381311yescharacteristic
178C4C2×#1458381411yescharacteristic
179C4#1914970511yescharacteristic
180C4#1914971011yescharacteristic
181C2—42no—
182C2#2601163011yescharacteristic
183C2#2601164211yescharacteristic
184C2#2601167411yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
FittingD4×Q8×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenterD4×Q8D173
hypocenterC173#1651
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #26011887C2semidirect
2. #26011776C2semidirect
4. #26011237C2semidirect
5. Q8⋊#13063436C2semidirect
61. Q8×C173SD16semidirect
92. C173#1651semidirect
150. Q8#14583814semidirect

Maximal subgroups

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

subgroupindex
#260118872
#260117762
#260116962
#260112372
Q8⋊#130634362
D4⋊#130634362
D4×Q8×C1732
#1651173