Finite Group Databasebeta

← #36352824

Subgroups — #36352824

Order 22144.

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

normal subgroups only

This group has 234 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#26011887C211yesmaximal, characteristic, complemented
3#26011773C211yesmaximal, characteristic, complemented
4#26011236C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic
7#461×C173C211yesmaximal, characteristic
8C22×Dic346—21no—
9#19149679C2211yescharacteristic
10#19149678—21no—
11#19149658—21no—
12#19149656—22no—
13C4×Dic346C2211yescharacteristic
14D4×Dic173—21no—
15#19149610—21no—
16#19149610—22no—
17C4⋊#13063436C2211yescharacteristic
18#19149602—21no—
19C4×#13063436C2211yescharacteristic
20C2×SD16×C173—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#103×C173—21no—
25C2×Dic346—22no—
26C2×Dic346—21no—
27C2×Dic346—42no—
28C2×Dic346—42no—
29C2×Dic346D411yescharacteristic
30C22×Dic173—22no—
31C2×#13063436—21no—
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173D411yescharacteristic
35C4⋊Dic173D412yes—
36#14583805—22no—
37#14583805—21no—
38C22⋊Dic173—22no—
39C4×Dic173—21no—
40C2×Q8×C173—21no—
41C2×D4×C173D411yescharacteristic
42C22×C4×C173—21no—
43SD16×C173—42no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50Dic346—21no—
51Dic346—44no—
52Dic346—41no—
53Dic346—41no—
54Dic346SD1612yescomplemented
55C2×Dic173—21no—
56C2×Dic173—22no—
57C2×Dic173—21no—
58C2×Dic173—44no—
59#13063436—22no—
60#13063436—41no—
61C23×C173—21no—
62Q8×C173—41no—
63Q8×C173—41no—
64D4×C173—21no—
65D4×C173SD1612yes—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C8×C173—41no—
74C22×C173—22no—
75C22×C173—41no—
76C22×C173—41no—
77C22×C173#12011yescharacteristic
78C4×C173—21no—
79C4×C173—41no—
80C4×C173—81no—
81C4×C173C2×SD1611yescharacteristic
82C4×C173C2×SD1611yescharacteristic
83C4×C173AGL(1,ℤ/8)11yescharacteristic
84C4×C173#13711yescharacteristic
85Dic173—22no—
86Dic173—42no—
87Dic173—41no—
88C2×C173—42no—
89C2×C173#44811yescharacteristic
90C2×C173#45011yescharacteristic
91C2×C173#45611yescharacteristic
92C173#165111yescharacteristic, complemented, Sylow
93#1651—1731nomaximal, Sylow
94D4×Q8—1731no—
95#498—1731no—
96#475—1731no—
97#461D17311yescharacteristic
98#332—1731no—
99Q8⋊C8—1731no—
100D4⋊C8—1731no—
101C22×Q8—3461no—
102C2×SD16—21no—
103C4⋊Q8—3461no—
104C4⋊Q8—1731no—
105C4⋊Q8C2×D17311yescharacteristic
106C22⋊Q8—3462no—
107C22⋊Q8—3461no—
108C4×Q8—1731no—
109C4×D4—3461no—
110C4×D4C2×D17311yescharacteristic
111#106—3461no—
112#106—3462no—
113C4⋊C8—1731no—
114C4⋊C8C2×D17311yescharacteristic
115#103—21no—
116#103—3461no—
117C4×C8—1731no—
118C2×Q8—21no—
119C2×Q8—3462no—
120C2×Q8—3461no—
121C2×Q8—1731no—
122C2×Q8—6922no—
123C2×Q8—6922no—
124C2×D4C22⋊D17311yescharacteristic
125C22×C4—21no—
126C22×C4—3462no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—3461no—
130C2×C8—3461no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—3462no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4—1731no—
137C4⋊C4—1732no—
138C4⋊C4C22⋊D17311yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3462no—
141C42—3461no—
142C42C22×D17311yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—3461no—
147Q8—1732no—
148Q8—6924no—
149Q8—6921no—
150Q8—6921no—
151D4—21no—
152D4#1458381412yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—3461no—
158C2×C4—3461no—
159C2×C4—3462no—
160C2×C4—6924no—
161C2×C4D4×D17311yescharacteristic
162C2×C4D4×D17311yescharacteristic
163C2×C4C2×C22⋊D17311yescharacteristic
164C8—41no—
165C8—3462no—
166C8—6921no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1914964711yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—3462no—
175C4—6921no—
176C4—6922no—
177C4SD16×D17311yescharacteristic
178C4C2×#1458381411yescharacteristic
179C4#1914970611yescharacteristic
180C4#1914970911yescharacteristic
181C2—42no—
182C2#2601162911yescharacteristic
183C2#2601164011yescharacteristic
184C2#2601167211yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#461×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenter#461D173
hypocenterC173#1651
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #26011887C2semidirect
3. #26011773C2semidirect
4. #26011236C2semidirect
5. #26011221C2semidirect
54. Dic346SD16semidirect
92. C173#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3462
#260118872
#260117732
#260112362
#260112212
D4⋊#130634362
#461×C1732
#1651173