Finite Group Databasebeta

← #36352731

Subgroups — #36352731

Order 22144.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic346C211yesmaximal, characteristic
2#26011842C211yesmaximal, characteristic, complemented
3#26011787C211yesmaximal, characteristic, complemented
4#26011781C211yesmaximal, characteristic
5#26011238C211yesmaximal, characteristic, complemented
6#26011223C211yesmaximal, characteristic, complemented
7D4⋊C8×C173C211yesmaximal, characteristic
8C22×Dic346—21no—
9#19149679C2211yescharacteristic
10#19149679C2211yescharacteristic
11#19149678—21no—
12#19149658—21no—
13#19149656—22no—
14C4×Dic346C2211yescharacteristic
15D4×Dic173—21no—
16#19149613—21no—
17#19149610—21no—
18C4⋊#13063436C2211yescharacteristic
19#19149604—22no—
20#19149604—21no—
21#19149600—21no—
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic
25C2×Dic346—22no—
26C2×Dic346—21no—
27C2×Dic346—21no—
28C2×Dic346—42no—
29C2×Dic346—42no—
30C2×Dic346D411yescharacteristic
31C22×Dic173—22no—
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D412yes—
37C4⋊Dic173D411yescharacteristic
38#14583805—22no—
39#14583805—21no—
40C22⋊Dic173—22no—
41C4×Dic173—21no—
42C2×D4×C173D411yescharacteristic
43C22×C4×C173—21no—
44C2×C8×C173—21no—
45C2×C8×C173—21no—
46C4⋊C4×C173D411yescharacteristic
47C22⋊C4×C173—21no—
48C42×C173C2311yescharacteristic
49Dic346—21no—
50Dic346—44no—
51Dic346—41no—
52Dic346—41no—
53Dic346—42no—
54Dic346SD1612yes—
55C2×Dic173—21no—
56C2×Dic173—22no—
57C2×Dic173—21no—
58C2×Dic173—44no—
59C2×Dic173—41no—
60#13063436—41no—
61C23×C173—21no—
62D4×C173—21no—
63D4×C173Q1612yes—
64C2×C4×C173—21no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173C2×D411yescharacteristic
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C8×C173—41no—
71C8×C173—41no—
72C22×C173—22no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173#12011yescharacteristic
76C4×C173—21no—
77C4×C173—41no—
78C4×C173C2×SD1611yescharacteristic
79C4×C173C2×Q1611yescharacteristic
80C4×C173AGL(1,ℤ/8)11yescharacteristic
81C4×C173#13711yescharacteristic
82Dic173—22no—
83Dic173—42no—
84Dic173—41no—
85Dic173—81no—
86C2×C173—42no—
87C2×C173#45011yescharacteristic
88C2×C173#45111yescharacteristic
89C2×C173#45411yescharacteristic
90C173#163711yescharacteristic, complemented, Sylow
91#1637—1731nomaximal, Sylow
92D4×Q8—1731no—
93#487—1731no—
94#477—1731no—
95#476—1731no—
96#332—1731no—
97Q8⋊C8—1731no—
98D4⋊C8D17311yescharacteristic
99C22×Q8—3461no—
100C4⋊Q8—3461no—
101C4⋊Q8—1731no—
102C4⋊Q8—1731no—
103C22⋊Q8—3462no—
104C22⋊Q8—3461no—
105C4×Q8—1731no—
106C4×D4—3461no—
107C4×D4C2×D17311yescharacteristic
108#107—3461no—
109#106—3461no—
110C4⋊C8—1731no—
111C4⋊C8C2×D17311yescharacteristic
112#103—3461no—
113#103—3462no—
114#102—3461no—
115C4×C8C2×D17311yescharacteristic
116C2×Q8—3462no—
117C2×Q8—3461no—
118C2×Q8—3461no—
119C2×Q8—1731no—
120C2×Q8—6922no—
121C2×Q8—6922no—
122C2×D4D69211yescharacteristic
123C22×C4—21no—
124C22×C4—3462no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3461no—
128C4⋊C4—3462no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4—1732no—
134C4⋊C4—1731no—
135C4⋊C4D69211yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3462no—
138C42—3461no—
139C42C22×D17311yescharacteristic
140C23—21no—
141Q8—3461no—
142Q8—1732no—
143Q8—6922no—
144Q8—6924no—
145Q8—6921no—
146Q8—6921no—
147D4—21no—
148D4Dic69212yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—3462no—
155C2×C4—6924no—
156C2×C4—6921no—
157C2×C4D4×D17311yescharacteristic
158C2×C4D4×D17311yescharacteristic
159C2×C4C2×D69211yescharacteristic
160C8—41no—
161C8—41no—
162C8—6921no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1914964911yescharacteristic
167C4—21no—
168C4—41no—
169C4—3462no—
170C4—13841no—
171C4—6921no—
172C4—6922no—
173C4SD16×D17311yescharacteristic
174C4C2×Dic69211yescharacteristic
175C4#1914970311yescharacteristic
176C4#1914971211yescharacteristic
177C2—42no—
178C2#2601164311yescharacteristic
179C2#2601164911yescharacteristic
180C2#2601166511yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
FittingD4⋊C8×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenterD4⋊C8D173
hypocenterC173#1637
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #26011842C2semidirect
3. #26011787C2semidirect
5. #26011238C2semidirect
6. #26011223C2semidirect
90. C173#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3462
#260118422
#260117872
#260117812
#260112382
#260112232
D4⋊C8×C1732
#1637173