Finite Group Databasebeta

← #36352690

Subgroups — #36352690

Order 22144.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#26011863C211yesmaximal, characteristic, complemented
2#26011786C211yesmaximal, characteristic, complemented
3#26011781C211yesmaximal, characteristic, complemented
4#26011235C211yesmaximal, characteristic, complemented
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic, complemented
7#550×C173C211yesmaximal, characteristic
8C2×#14583814—22no—
9#19149679C2211yescharacteristic
10#19149612—21no—
11#19149610—21no—
12C4⋊#13063436C2211yescharacteristic
13C4⋊#13063436C2211yescharacteristic
14#19149603—21no—
15#19149600—21no—
16C4×#13063436C2211yescharacteristic
17C2×#41×C173—21no—
18#127×C173—21no—
19#127×C173C2211yescharacteristic
20#121×C173—22no—
21#121×C173—21no—
22C4×Q8×C173C2211yescharacteristic
23C4×D4×C173—21no—
24C4×D4×C173C2211yescharacteristic
25C2×Dic346—21no—
26#14583814—44no—
27C2×#13063436—21no—
28C2×#13063436—21no—
29C2×#13063436—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173—21no—
32#41×C173—42no—
33#41×C173—42no—
34C2×Q8×C173D411yescharacteristic
35C2×D4×C173—22no—
36C2×D4×C173—22no—
37C2×D4×C173—21no—
38C2×D4×C173—21no—
39C2×D4×C173D411yescharacteristic
40C2×D4×C173D412yes—
41C22×C4×C173—22no—
42C22×C4×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173D411yescharacteristic
45C4⋊C4×C173D411yescharacteristic
46C22⋊C4×C173—22no—
47C22⋊C4×C173—21no—
48C42×C173—21no—
49C42×C173C2311yescharacteristic
50Dic346—42no—
51C2×Dic173—41no—
52#13063436—22no—
53#13063436—41no—
54#13063436—41no—
55C23×C173—21no—
56C23×C173—22no—
57Q8×C173—21no—
58Q8×C173SD1612yescomplemented
59D4×C173—21no—
60D4×C173—21no—
61D4×C173—42no—
62D4×C173—44no—
63D4×C173—41no—
64D4×C173—41no—
65D4×C173—41no—
66D4×C173—41no—
67D4×C173—82no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—21no—
71C2×C4×C173—44no—
72C2×C4×C173—41no—
73C2×C4×C173—41no—
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C2×C4×C173C2×D411yescharacteristic
77C22×C173—42no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—42no—
81C22×C173—42no—
82C22×C173—41no—
83C22×C173#12011yescharacteristic
84C4×C173—22no—
85C4×C173—21no—
86C4×C173—41no—
87C4×C173—41no—
88C4×C173C2×SD1611yescharacteristic
89C4×C173#13511yescharacteristic
90C4×C173AGL(1,ℤ/8)11yescharacteristic
91C4×C173#13711yescharacteristic
92Dic173—81no—
93C2×C173—81no—
94C2×C173—82no—
95C2×C173#45011yescharacteristic
96C2×C173#45211yescharacteristic
97C2×C173#45311yescharacteristic
98C173#163111yescharacteristic, complemented, Sylow
99#1631—1731nomaximal, Sylow
100#550D17311yescharacteristic
101#492—1731no—
102#477—1731no—
103#476—1731no—
104#331—1731no—
105Q8⋊C8—1731no—
106D4⋊C8—1731no—
107C2×#41—21no—
108C2×SD16—3462no—
109C4⋊Q8—1731no—
110#127—21no—
111#127C2×D17311yescharacteristic
112#121—22no—
113#121—21no—
114C4×Q8C2×D17311yescharacteristic
115C4×D4—21no—
116C4×D4C2×D17311yescharacteristic
117#107—3461no—
118#106—3461no—
119C4⋊C8—1731no—
120C4⋊C8—1731no—
121#103—3461no—
122#102—3461no—
123C4×C8—1731no—
124#41—42no—
125#41—42no—
126C2×Q8—3461no—
127C2×Q8C22⋊D17311yescharacteristic
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊D17311yescharacteristic
133C2×D4C22⋊D17312yes—
134C22×C4—22no—
135C22×C4—21no—
136SD16—6924no—
137C2×C8—3461no—
138C2×C8—3461no—
139C2×C8—3461no—
140C4⋊C4—21no—
141C4⋊C4—3461no—
142C4⋊C4—3461no—
143C4⋊C4C22⋊D17311yescharacteristic
144C4⋊C4C22⋊D17311yescharacteristic
145C22⋊C4—22no—
146C22⋊C4—21no—
147C42—21no—
148C42C22×D17311yescharacteristic
149C23—21no—
150C23—22no—
151Q8—21no—
152Q8—6922no—
153Q8#1458381412yescomplemented
154D4—21no—
155D4—21no—
156D4—42no—
157D4—44no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—82no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—6921no—
170C2×C4C2×C22⋊D17311yescharacteristic
171C2×C4C2×C22⋊D17311yescharacteristic
172C2×C4C2×C22⋊D17311yescharacteristic
173C8—3462no—
174C8—6921no—
175C8—6921no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22#1914964811yescharacteristic
183C4—21no—
184C4—22no—
185C4—41no—
186C4—41no—
187C4—13841no—
188C4C2×#1458381411yescharacteristic
189C4#1914969911yescharacteristic
190C4#1914970511yescharacteristic
191C4#1914970911yescharacteristic
192C2—81no—
193C2—82no—
194C2#2601164211yescharacteristic
195C2#2601165311yescharacteristic
196C2#2601165911yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting, prime radical#550×C173C2
socleC22×C173#120
hypercenter#550D173
hypocenterC173#1631
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #26011863C2semidirect
2. #26011786C2semidirect
3. #26011781C2semidirect
4. #26011235C2semidirect
5. Q8⋊#13063436C2semidirect
6. D4⋊#13063436C2semidirect
58. Q8×C173SD16semidirect
98. C173#1631semidirect
153. Q8#14583814semidirect

Maximal subgroups

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

subgroupindex
#260118632
#260117862
#260117812
#260112352
Q8⋊#130634362
D4⋊#130634362
#550×C1732
#1631173