Finite Group Databasebeta

← #37574546

Subgroups — #37574546

Order 24448.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#27132845C211yesmaximal, characteristic, complemented
2#27132835C212yesmaximal, complemented
3#27132462C211yesmaximal, characteristic, complemented
4#27132460C212yesmaximal, complemented
5#555×C191C211yesmaximal, characteristic, complemented
6#20259370C2211yescharacteristic
7#20259353—21no—
8#20259348—22no—
9#20259250—22no—
10#20259250—21no—
11#20259241C2211yescharacteristic
12#20259241C2212yes—
13C4⋊Q8×C191C2211yescharacteristic
14#124×C191—21no—
15#124×C191C2212yes—
16#121×C191—22no—
17C4×D4×C191—21no—
18#117×C191—22no—
19#15690288—41no—
20#15690287—22no—
21#15690287—21no—
22Q8×D191—41no—
23#15690277—41no—
24#15690275—42no—
25#15690274—41no—
26#15690272—42no—
27#15690271—22no—
28#15690271—21no—
29C2×Q8×C191D411yescharacteristic
30C2×Q8×C191C2311yescharacteristic
31C2×D4×C191—21no—
32C2×D4×C191—21no—
33C2×D4×C191D412yes—
34C22×C4×C191—22no—
35C4⋊C4×C191—21no—
36C4⋊C4×C191—22no—
37C22⋊C4×C191—22no—
38C22⋊C4×C191—22no—
39C22⋊C4×C191—22no—
40C42×C191—21no—
41C42×C191D412yes—
42C42×C191D411yescharacteristic
43Dic382—22no—
44Dic382—21no—
45D764—22no—
46D764—21no—
47C22⋊D191—41no—
48C4×D191—42no—
49C4×D191—41no—
50C191⋊C8—42no—
51C191⋊C8—41no—
52C23×C191—22no—
53Q8×C191—41no—
54Q8×C191—41no—
55D4×C191—42no—
56D4×C191—42no—
57D4×C191—81no—
58C2×C4×C191—21no—
59C2×C4×C191—22no—
60C2×C4×C191—21no—
61C2×C4×C191—44no—
62C2×C4×C191C2×D411yescharacteristic
63C2×C4×C191C2×D412yes—
64C22×C191—42no—
65C22×C191—42no—
66C22×C191—42no—
67C22×C191#12011yescharacteristic
68C2×D191—41no—
69C4×C191—22no—
70C4×C191—22no—
71C4×C191—21no—
72C4×C191—42no—
73C4×C191—41no—
74Dic191—41no—
75C2×C191—21no—
76C2×C191—82no—
77C2×C191B(4,𝔽2)11yescharacteristic
78D191—81no—
79C191#220411yescharacteristic, complemented, Sylow
80#2204—1911nomaximal, Sylow
81#555D19111yescharacteristic, complemented
82#456—1911no—
83#454—1912no—
84#356—1911no—
85#355—1912no—
862−1+4—1911no—
87#137—3821no—
88AGL(1,ℤ/8)—3822no—
89C4⋊Q8C2×D19111yescharacteristic
90#124—21no—
91#124C2×D19112yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—3822no—
96#104—3821no—
97#101—1911no—
98#101—1912no—
99#41—3822no—
100#41—3821no—
101#41—7641no—
102C2×Q8—7641no—
103C2×Q8C22⋊D19111yescharacteristic
104C2×Q8C22×D19111yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D19112yes—
108C22×C4—22no—
109Q16—7641no—
110SD16—7642no—
111SD16—7641no—
112D8—7642no—
113M16—3822no—
114M16—3821no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D19112yes—
122C42C22⋊D19111yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—3822no—
127Q8—3821no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—3822no—
132D4—3821no—
133D4—7641no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—7642no—
139C2×C4—7641no—
140C2×C4C2×C22⋊D19111yescharacteristic
141C2×C4C2×C22⋊D19112yes—
142C8—7642no—
143C8—7641no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—7641no—
148C22#2025929111yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—7641no—
155C2—21no—
156C2—82no—
157C2—15281no—
158C2#2713285011yescharacteristic
subgroupG/H
centerC2#27132850
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#555×C191C2
socleC2×C191B(4,𝔽2)
hypercenter#555D191
hypocenterC191#2204
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #27132845C2semidirect
2. #27132835C2semidirect
3. #27132462C2semidirect
4. #27132460C2semidirect
5. #555×C191C2semidirect
79. C191#2204semidirect
81. #555D191semidirect

Maximal subgroups

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

subgroupindex
#271328452
#271328352
#271324622
#271324602
#555×C1912
#2204191