Finite Group Databasebeta

← #36352633

Subgroups — #36352633

Order 22144.

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

normal subgroups only

This group has 236 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#26012198C211yesmaximal, characteristic, complemented
2#26011863C211yesmaximal, characteristic, complemented
3#26011688C211yesmaximal, characteristic, complemented
4#26011239C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic, complemented
7#459×C173C211yesmaximal, characteristic
8C2×#14583814—21no—
9C2×#14583814—22no—
10#19149679C2211yescharacteristic
11#19149670—21no—
12#19149658—21no—
13#19149656—22no—
14C4×Dic346C2211yescharacteristic
15D4×Dic173—21no—
16C2×C4⋊Dic173—21no—
17C4⋊#13063436C2211yescharacteristic
18#19149600—21no—
19C4×#13063436C2211yescharacteristic
20C2×D8×C173—21no—
21#127×C173C2211yescharacteristic
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25C2×Dic346D411yescharacteristic
26C2×Dic346D412yes—
27C22×Dic173—22no—
28#14583814—44no—
29#14583814—42no—
30C2×#13063436—21no—
31C2×#13063436—21no—
32C4⋊Dic173—22no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173D411yescharacteristic
36#14583805—22no—
37#14583805—21no—
38C22⋊Dic173—22no—
39C4×Dic173—21no—
40C2×D4×C173—21no—
41C2×D4×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173D411yescharacteristic
44C22×C4×C173—21no—
45D8×C173—42no—
46C2×C8×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50Dic346—21no—
51Dic346—21no—
52Dic346—42no—
53C2×Dic173—21no—
54C2×Dic173—22no—
55C2×Dic173—21no—
56C2×Dic173—82no—
57#13063436—22no—
58#13063436—41no—
59C23×C173—21no—
60C23×C173—41no—
61D4×C173—21no—
62D4×C173—42no—
63D4×C173—41no—
64D4×C173—41no—
65D4×C173—41no—
66D4×C173—41no—
67D4×C173SD1612yescomplemented
68C2×C4×C173—21no—
69C2×C4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C8×C173—41no—
75C22×C173—22no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—81no—
79C22×C173—81no—
80C22×C173—81no—
81C22×C173#12011yescharacteristic
82C4×C173—21no—
83C4×C173—41no—
84C4×C173C2×SD1611yescharacteristic
85C4×C173#13511yescharacteristic
86C4×C173AGL(1,ℤ/8)11yescharacteristic
87C4×C173#13711yescharacteristic
88Dic173—41no—
89Dic173—41no—
90Dic173—42no—
91C2×C173—42no—
92C2×C173—161no—
93C2×C173#44811yescharacteristic
94C2×C173#44911yescharacteristic
95C2×C173#45311yescharacteristic
96C173#162211yescharacteristic, complemented, Sylow
97#1622—1731nomaximal, Sylow
98#554—1731no—
99#492—1731no—
100#460—1731no—
101#459D17311yescharacteristic
102#333—1731no—
103Q8⋊C8—1731no—
104D4⋊C8—1731no—
105C2×SD16—3461no—
106C2×SD16—3462no—
107C2×D8—21no—
108C4⋊Q8—1731no—
109#127C2×D17311yescharacteristic
110#125—3461no—
111C22⋊Q8—3462no—
112C22⋊Q8—3461no—
113C4×Q8—1731no—
114C4×D4—3461no—
115C4×D4C2×D17311yescharacteristic
116C2×C4⋊C4—3461no—
117C4⋊C8—1731no—
118C4⋊C8C2×D17311yescharacteristic
119#102—21no—
120#102—3461no—
121C4×C8—1731no—
122C2×Q8—1731no—
123C2×Q8—1732no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C22⋊D17311yescharacteristic
128C22×C4—21no—
129C22×C4—3462no—
130SD16—6924no—
131SD16—6922no—
132D8—42no—
133C2×C8—21no—
134C2×C8—3461no—
135C2×C8—3461no—
136C4⋊C4—3462no—
137C4⋊C4—3462no—
138C4⋊C4—3461no—
139C4⋊C4—3461no—
140C4⋊C4—3461no—
141C4⋊C4—1731no—
142C4⋊C4C22⋊D17311yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—3462no—
145C42—3461no—
146C42C22×D17311yescharacteristic
147C23—21no—
148C23—41no—
149Q8—3461no—
150Q8—3461no—
151Q8—6922no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4#1458381412yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3461no—
163C2×C4—3461no—
164C2×C4—3462no—
165C2×C4—13842no—
166C2×C4D4×D17311yescharacteristic
167C2×C4D4×D17311yescharacteristic
168C2×C4C2×C22⋊D17311yescharacteristic
169C8—41no—
170C8—3462no—
171C8—6921no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#1914964711yescharacteristic
179C4—21no—
180C4—41no—
181C4—6921no—
182C4—6922no—
183C4—6921no—
184C4C2×#1458381411yescharacteristic
185C4#1914969811yescharacteristic
186C4#1914970311yescharacteristic
187C4#1914970911yescharacteristic
188C2—42no—
189C2—161no—
190C2#2601162911yescharacteristic
191C2#2601163311yescharacteristic
192C2#2601165711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting, prime radical#459×C173C2
socleC22×C173#120
hypercenter#459D173
hypocenterC173#1622
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #26012198C2semidirect
2. #26011863C2semidirect
3. #26011688C2semidirect
4. #26011239C2semidirect
5. #26011221C2semidirect
6. D4⋊#13063436C2semidirect
67. D4×C173SD16semidirect
96. C173#1622semidirect
158. D4#14583814semidirect

Maximal subgroups

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

subgroupindex
#260121982
#260118632
#260116882
#260112392
#260112212
D4⋊#130634362
#459×C1732
#1622173