Finite Group Databasebeta

← #37571627

Subgroups — #37571627

Order 24448.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#27133387C211yesmaximal, characteristic, complemented
2#27132860C211yesmaximal, characteristic, complemented
3#27132855C211yesmaximal, characteristic
4#27132409C211yesmaximal, characteristic, complemented
5#27132395C211yesmaximal, characteristic, complemented
6#27132392C211yesmaximal, characteristic, complemented
7#486×C191C211yesmaximal, characteristic, complemented
8C2×#15690275—21no—
9C2×#15690272—21no—
10#20259317—22no—
11#20259311—21no—
12#20259302—21no—
13Q8×Dic191—21no—
14C4×Dic382C2211yescharacteristic
15C4×D764C2211yescharacteristic
16#20259280—21no—
17C4⋊(C191⋊C8)C2211yescharacteristic
18C4⋊(C191⋊C8)C2211yescharacteristic
19#20259243—21no—
20#20259243—21no—
21C4⋊Q8×C191C2211yescharacteristic
22#127×C191C2211yescharacteristic
23#102×C191—22no—
24C4×C8×C191C2211yescharacteristic
25C2×Dic382D411yescharacteristic
26C2×D764D411yescharacteristic
27C2×C4×D191—21no—
28#15690275—42no—
29#15690272—42no—
30C2×C191⋊C8—21no—
31C2×C191⋊C8—21no—
32C4⋊Dic191—22no—
33C4⋊Dic191D411yescharacteristic
34C4⋊Dic191D411yescharacteristic
35Dic191⋊C4—21no—
36(C2×D191)⋊C4—22no—
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—21no—
39C4×Dic191—22no—
40C4×Dic191—21no—
41C2×Q8×C191—21no—
42C2×D4×C191—21no—
43C2×D4×C191—21no—
44C2×D4×C191—21no—
45C2×C8×C191—21no—
46C4⋊C4×C191—21no—
47C4⋊C4×C191D412yes—
48C42×C191C2311yescharacteristic
49C22×D191—21no—
50Dic382—21no—
51Dic382—21no—
52D764—21no—
53D764—21no—
54C2×Dic191—21no—
55C2×Dic191—21no—
56C2×Dic191—21no—
57C4×D191—41no—
58C4×D191—41no—
59C191⋊C8—41no—
60C191⋊C8—41no—
61C23×C191—41no—
62Q8×C191—41no—
63Q8×C191—41no—
64D4×C191—42no—
65D4×C191—41no—
66D4×C191—41no—
67D4×C191—41no—
68D4×C191—41no—
69C2×C4×C191—22no—
70C2×C4×C191C2×D411yescharacteristic
71C2×C4×C191C2×D411yescharacteristic
72C2×C4×C191C2×D411yescharacteristic
73C8×C191—41no—
74C22×C191—81no—
75C22×C191—81no—
76C22×C191—81no—
77C22×C191#12011yescharacteristic
78C2×D191—41no—
79C2×D191—41no—
80C2×D191—41no—
81C4×C191—21no—
82C4×C191—42no—
83C4×C191#13511yescharacteristic
84C4×C191#13511yescharacteristic
85C4×C191AGL(1,ℤ/8)11yescharacteristic
86C4×C191AGL(1,ℤ/8)11yescharacteristic
87Dic191—41no—
88Dic191—41no—
89Dic191—41no—
90C2×C191—161no—
91C2×C191#44911yescharacteristic
92C2×C191#44911yescharacteristic
93C2×C191#45411yescharacteristic
94D191—81no—
95C191#162811yescharacteristic, complemented, Sylow
96#1628—1911nomaximal, Sylow
97#556—1911no—
98#486D19111yescharacteristic, complemented
99#460—1911no—
100#459—1911no—
101#332—1911no—
102Q8⋊C8—1911no—
103D4⋊C8—1911no—
104C2×SD16—3821no—
105C2×D8—3821no—
106C4⋊Q8C2×D19111yescharacteristic
107#127C2×D19111yescharacteristic
108#126—3822no—
109#124—3821no—
110C22⋊Q8—3821no—
111C4×Q8—3821no—
112C4×Q8—1911no—
113C4×D4—1911no—
114#117—3821no—
115C4⋊C8—1911no—
116C4⋊C8—1911no—
117#102—22no—
118#102—3821no—
119#102—3821no—
120C4×C8C2×D19111yescharacteristic
121C2×Q8—21no—
122C2×Q8—1911no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1911no—
127C22×C4—3821no—
128SD16—7642no—
129D8—7642no—
130C2×C8—21no—
131C2×C8—3821no—
132C2×C8—3821no—
133C4⋊C4—21no—
134C4⋊C4—3822no—
135C4⋊C4—3821no—
136C4⋊C4—1911no—
137C4⋊C4—1911no—
138C4⋊C4C22⋊D19112yes—
139C22⋊C4—3822no—
140C22⋊C4—3822no—
141C22⋊C4—3821no—
142C42—3822no—
143C42—3821no—
144C42C22×D19111yescharacteristic
145C23—41no—
146C23—3821no—
147Q8—41no—
148Q8—41no—
149Q8—3821no—
150Q8—3821no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3821no—
157D4—3821no—
158C2×C4—22no—
159C2×C4—3821no—
160C2×C4—3821no—
161C2×C4—3821no—
162C2×C4—7641no—
163C2×C4—7641no—
164C2×C4D4×D19111yescharacteristic
165C2×C4D4×D19111yescharacteristic
166C2×C4C2×C22⋊D19111yescharacteristic
167C8—41no—
168C8—7641no—
169C8—7641no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—7641no—
174C22—7641no—
175C22—7641no—
176C22#2025929011yescharacteristic
177C4—21no—
178C4—42no—
179C4—7641no—
180C4—7641no—
181C4—7641no—
182C4#2025934111yescharacteristic
183C4#2025934311yescharacteristic
184C4#2025934611yescharacteristic
185C4#2025934911yescharacteristic
186C2—161no—
187C2—15281no—
188C2#2713280511yescharacteristic
189C2#2713280711yescharacteristic
190C2#2713283511yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#486×C191C2
socleC22×C191#120
prime radical#27132855C2
hypercenter#486D191
hypocenterC191#1628
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #27133387C2semidirect
2. #27132860C2semidirect
4. #27132409C2semidirect
5. #27132395C2semidirect
6. #27132392C2semidirect
7. #486×C191C2semidirect
95. C191#1628semidirect
98. #486D191semidirect

Maximal subgroups

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

subgroupindex
#271333872
#271328602
#271328552
#271324092
#271323952
#271323922
#486×C1912
#1628191