Finite Group Databasebeta

← #36352678

Subgroups — #36352678

Order 22144.

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

normal subgroups only

This group has 314 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×D692C211yesmaximal, characteristic
2#26011865C211yesmaximal, characteristic, complemented
3#26011778C211yesmaximal, characteristic, complemented
4#26011234C211yesmaximal, characteristic, complemented
5D4⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#476×C173C211yesmaximal, characteristic, complemented
8C22×D692—21no—
9C2×D4×D173—21no—
10C2×#14583813—22no—
11#19149676C2211yescharacteristic
12#19149654—21no—
13#19149653—21no—
14#19149649—22no—
15C4×D692C2211yescharacteristic
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic, complemented
18#19149599—21no—
19C4×#13063436C2211yescharacteristic, complemented
20C4⋊Q8×C173C2211yescharacteristic, complemented
21C4×D4×C173C2211yescharacteristic
22#106×C173—21no—
23C4⋊C8×C173C2211yescharacteristic, complemented
24#102×C173—21no—
25C23×D173—22no—
26C2×D692—22no—
27C2×D692—21no—
28C2×D692—21no—
29C2×D692—42no—
30C2×D692—42no—
31C2×D692D411yescharacteristic
32C2×D692D412yes—
33C2×C22⋊D173—22no—
34D4×D173—42no—
35D4×D173—42no—
36C2×C4×D173—21no—
37#14583813—44no—
38C2×#13063436—21no—
39C2×#13063436—21no—
40C4⋊Dic173D411yescharacteristic
41#14583803—22no—
42#14583803—21no—
43C2×Q8×C173—21no—
44C2×D4×C173D411yescharacteristic
45C22×C4×C173—21no—
46C2×C8×C173—21no—
47C4⋊C4×C173—21no—
48C4⋊C4×C173—21no—
49C4⋊C4×C173D411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173C2311yescharacteristic
52C22×D173—21no—
53C22×D173—22no—
54C22×D173—44no—
55C22×D173—44no—
56C22×D173—82no—
57D692—21no—
58D692—42no—
59D692—44no—
60D692—41no—
61D692—41no—
62D692—41no—
63D692—41no—
64D692SD1612yescomplemented
65C22⋊D173—44no—
66C2×Dic173—21no—
67C4×D173—41no—
68C4×D173—41no—
69#13063436—22no—
70#13063436—41no—
71C23×C173—21no—
72Q8×C173—42no—
73D4×C173—21no—
74D4×C173SD1612yescomplemented
75C2×C4×C173—21no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173—41no—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C8×C173—41no—
83C22×C173—22no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173#12011yescharacteristic
87C2×D173—22no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173—42no—
91C2×D173—42no—
92C2×D173—42no—
93C2×D173—88no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173—81no—
97C4×C173C2×SD1611yescharacteristic
98C4×C173C2×SD1611yescharacteristic
99C4×C173AGL(1,ℤ/8)11yescharacteristic
100C4×C173AGL(1,ℤ/8)11yescharacteristic
101Dic173—41no—
102C2×C173—42no—
103C2×C173#45011yescharacteristic
104C2×C173#45011yescharacteristic
105C2×C173#45311yescharacteristic
106D173—42no—
107D173—82no—
108C173#162911yescharacteristic, complemented, Sylow
109#1629—1731nomaximal, Sylow
110D42—1731no—
111#492—1731no—
112#476—1731no—
113#476D17311yescharacteristic, complemented
114#331—1731no—
115D4⋊C8—1731no—
116D4⋊C8—1731no—
117C22×D4—3461no—
118C22×D4—3461no—
119C2×SD16—3462no—
120C4⋊Q8C2×D17311yescharacteristic, complemented
121#127—1731no—
122#121—3461no—
123#121—3461no—
124#120—3462no—
125C4×D4—1731no—
126C4×D4C2×D17311yescharacteristic
127#106—21no—
128#106—3461no—
129C4⋊C8—1731no—
130C4⋊C8C2×D17311yescharacteristic, complemented
131#102—21no—
132#102—3461no—
133C4×C8—1731no—
134C24—3462no—
135C2×Q8—21no—
136C2×D4—3462no—
137C2×D4—3462no—
138C2×D4—3461no—
139C2×D4—3461no—
140C2×D4—1731no—
141C2×D4—1732no—
142C2×D4—6922no—
143C2×D4—6922no—
144C2×D4—6922no—
145C2×D4—6922no—
146C2×D4C22⋊D17311yescharacteristic
147C22×C4—21no—
148C22×C4—3461no—
149SD16—6924no—
150C2×C8—21no—
151C2×C8—3461no—
152C2×C8—3461no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—1731no—
156C4⋊C4C22⋊D17311yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3462no—
159C22⋊C4—3461no—
160C42C22×D17311yescharacteristic
161C23—21no—
162C23—3461no—
163C23—3462no—
164C23—13842no—
165C23—6924no—
166C23—6924no—
167Q8—42no—
168D4—21no—
169D4—3461no—
170D4—1732no—
171D4—6922no—
172D4—6924no—
173D4—6924no—
174D4—6921no—
175D4—6921no—
176D4—6921no—
177D4—6921no—
178D4#1458381312yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3461no—
184C2×C4—6921no—
185C2×C4—6921no—
186C2×C4D4×D17311yescharacteristic
187C2×C4D4×D17311yescharacteristic
188C2×C4C2×C22⋊D17311yescharacteristic
189C8—41no—
190C8—3462no—
191C8—6921no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—3462no—
196C22—13848no—
197C22—6922no—
198C22—6922no—
199C22—6922no—
200C22—6921no—
201C22—6921no—
202C22#1914964711yescharacteristic
203C4—21no—
204C4—41no—
205C4—81no—
206C4—6921no—
207C4SD16×D17311yescharacteristic
208C4C2×#1458381311yescharacteristic
209C4#1914970411yescharacteristic
210C4#1914970611yescharacteristic
211C2—42no—
212C2—13842no—
213C2—6922no—
214C2#2601164011yescharacteristic
215C2#2601164111yescharacteristic
216C2#2601165711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#476×C173C2
socleC22×C173#120
prime radicalD4×D692C2
hypercenter#476D173
hypocenterC173#1629
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #26011865C2semidirect
3. #26011778C2semidirect
4. #26011234C2semidirect
5. D4⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #476×C173C2semidirect
17. C4⋊#13063436C22semidirect
19. C4×#13063436C22semidirect
20. C4⋊Q8×C173C22semidirect
23. C4⋊C8×C173C22semidirect
64. D692SD16semidirect
74. D4×C173SD16semidirect
108. C173#1629semidirect
113. #476D173semidirect
120. C4⋊Q8C2×D173semidirect
130. C4⋊C8C2×D173semidirect
178. D4#14583813semidirect

Maximal subgroups

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

subgroupindex
D4×D6922
#260118652
#260117782
#260112342
D4⋊#130634362
#260112182
#476×C1732
#1629173