Finite Group Databasebeta

← #37574545

Subgroups — #37574545

Order 24448.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133379C211yesmaximal, characteristic, complemented
2#27132836C212yesmaximal, complemented
3#27132462C211yesmaximal, characteristic, complemented
4#27132459C212yesmaximal, complemented
5#456×C191C211yesmaximal, characteristic, complemented
6#20259347—22no—
7#20259311—21no—
8#20259311C2212yes—
9#20259297—22no—
10C4×D764—21no—
11#20259280—22no—
12#20259249—22no—
13#20259241C2212yes—
142−1+4×C191C2211yescharacteristic
15#137×C191—21no—
16C4⋊Q8×C191C2211yescharacteristic
17#104×C191—21no—
18#101×C191C2211yescharacteristic
19C2×D764—21no—
20C2×D764—21no—
21C2×D764D412yes—
22C2×C4×D191—22no—
23#15690274—42no—
24#15690272—42no—
25#15690271—22no—
26C4⋊Dic191—21no—
27(C2×D191)⋊C4—22no—
28(C2×D191)⋊C4—22no—
29(C2×D191)⋊C4—22no—
30C4×Dic191—21no—
31C4×Dic191D412yes—
32#41×C191—22no—
33#41×C191—21no—
34#41×C191—41no—
35C2×Q8×C191—41no—
36C2×Q8×C191D411yescharacteristic
37C2×Q8×C191C2311yescharacteristic
38Q16×C191—41no—
39SD16×C191—41no—
40M16×C191—21no—
41C4⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43C22×D191—22no—
44D764—42no—
45D764—42no—
46D764—81no—
47C2×Dic191—22no—
48C4×D191—44no—
49C191⋊C8—42no—
50Q8×C191—21no—
51Q8×C191—22no—
52Q8×C191—41no—
53Q8×C191—41no—
54D4×C191—21no—
55D4×C191—22no—
56D4×C191—41no—
57C2×C4×C191—21no—
58C2×C4×C191—21no—
59C2×C4×C191—41no—
60C2×C4×C191—42no—
61C2×C4×C191C2×D412yes—
62C2×C4×C191C2×D411yescharacteristic
63C8×C191—41no—
64C22×C191—41no—
65C22×C191#12011yescharacteristic
66C2×D191—42no—
67C2×D191—42no—
68C2×D191—42no—
69C4×C191—22no—
70C4×C191—21no—
71C4×C191—22no—
72C4×C191—41no—
73C4×C191—41no—
74Dic191—42no—
75C2×C191—21no—
76C2×C191—81no—
77C2×C191B(4,𝔽2)11yescharacteristic
78D191—82no—
79C191#220411yescharacteristic, complemented, Sylow
80#2204—1911nomaximal, Sylow
81#555—1911no—
82#456D19111yescharacteristic, complemented
83#454—1912no—
84#356—1911no—
85#355—1912no—
862−1+4C2×D19111yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—3822no—
89C4⋊Q8C2×D19111yescharacteristic
90#124—3821no—
91#124—1912no—
92#121—3822no—
93C4×D4—3821no—
94#117—3822no—
95#104—21no—
96#104—3822no—
97#101—1912no—
98#101C2×D19111yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D19111yescharacteristic
104C2×Q8C22×D19111yescharacteristic
105C2×D4—3821no—
106C2×D4—3821no—
107C2×D4—1912no—
108C22×C4—3822no—
109Q16—41no—
110SD16—41no—
111SD16—7642no—
112D8—7642no—
113M16—21no—
114M16—3822no—
115C4⋊C4—22no—
116C4⋊C4—3821no—
117C22⋊C4—3822no—
118C22⋊C4—3822no—
119C22⋊C4—3822no—
120C42—3821no—
121C42—1912no—
122C42C22⋊D19111yescharacteristic
123C23—3822no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—15281no—
132D4—7642no—
133D4—7642no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—3822no—
139C2×C4—7644no—
140C2×C4D4×D19112yes—
141C2×C4C2×C22⋊D19111yescharacteristic
142C8—41no—
143C8—7642no—
144C22—41no—
145C22—7642no—
146C22—7642no—
147C22—7642no—
148C22#2025929011yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—7642no—
155C2—21no—
156C2—81no—
157C2—15282no—
158C2#2713284911yescharacteristic
subgroupG/H
centerC2#27132849
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#456×C191C2
socleC2×C191B(4,𝔽2)
hypercenter#456D191
hypocenterC191#2204
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #27133379C2semidirect
2. #27132836C2semidirect
3. #27132462C2semidirect
4. #27132459C2semidirect
5. #456×C191C2semidirect
79. C191#2204semidirect
82. #456D191semidirect

Maximal subgroups

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

subgroupindex
#271333792
#271328362
#271324622
#271324592
#456×C1912
#2204191