Finite Group Databasebeta

← #44509470

Subgroups — #44509470

Order 30848.

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#29508433C211yesmaximal, characteristic, complemented
2#29507890C212yesmaximal, complemented
3#29507516C211yesmaximal, characteristic, complemented
4#29507513C212yesmaximal, complemented
5#456×C241C211yesmaximal, characteristic, complemented
6#21440667—22no—
7#21440631—21no—
8#21440631C2212yes—
9#21440617—22no—
10C4×D964—21no—
11#21440600—22no—
12#21440569—22no—
13#21440561C2212yes—
142−1+4×C241C2211yescharacteristic
15#137×C241—21no—
16C4⋊Q8×C241C2211yescharacteristic
17#104×C241—21no—
18#101×C241C2211yescharacteristic
19C2×D964—21no—
20C2×D964—21no—
21C2×D964D412yes—
22C2×C4×D241—22no—
23#16838757—42no—
24#16838755—42no—
25#16838754—22no—
26C4⋊Dic241—21no—
27#16838747—22no—
28#16838747—22no—
29#16838747—22no—
30C4×Dic241—21no—
31C4×Dic241D412yes—
32#41×C241—21no—
33#41×C241—22no—
34#41×C241—41no—
35C2×Q8×C241—41no—
36C2×Q8×C241D411yescharacteristic
37C2×Q8×C241C2311yescharacteristic
38Q16×C241—41no—
39SD16×C241—41no—
40M16×C241—21no—
41C4⋊C4×C241—22no—
42C42×C241D411yescharacteristic
43C22×D241—22no—
44D964—42no—
45D964—42no—
46D964—81no—
47C2×Dic241—22no—
48C4×D241—44no—
49#14515349—42no—
50Q8×C241—21no—
51Q8×C241—22no—
52Q8×C241—41no—
53Q8×C241—41no—
54D4×C241—21no—
55D4×C241—22no—
56D4×C241—41no—
57C2×C4×C241—21no—
58C2×C4×C241—21no—
59C2×C4×C241—41no—
60C2×C4×C241—42no—
61C2×C4×C241C2×D412yes—
62C2×C4×C241C2×D411yescharacteristic
63C8×C241—41no—
64C22×C241—41no—
65C22×C241#12011yescharacteristic
66C2×D241—42no—
67C2×D241—42no—
68C2×D241—42no—
69C4×C241—22no—
70C4×C241—21no—
71C4×C241—22no—
72C4×C241—41no—
73C4×C241—41no—
74Dic241—42no—
75C2×C241—21no—
76C2×C241—81no—
77C2×C241B(4,𝔽2)11yescharacteristic
78D241—82no—
79C241#220411yescharacteristic, complemented, Sylow
80#2204—2411nomaximal, Sylow
81#555—2411no—
82#456D24111yescharacteristic, complemented
83#454—2412no—
84#356—2411no—
85#355—2412no—
862−1+4C2×D24111yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—4822no—
89C4⋊Q8C2×D24111yescharacteristic
90#124—4821no—
91#124—2412no—
92#121—4822no—
93C4×D4—4821no—
94#117—4822no—
95#104—21no—
96#104—4822no—
97#101—2412no—
98#101C2×D24111yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D24111yescharacteristic
104C2×Q8C22×D24111yescharacteristic
105C2×D4—4821no—
106C2×D4—4821no—
107C2×D4—2412no—
108C22×C4—4822no—
109Q16—41no—
110SD16—41no—
111SD16—9642no—
112D8—9642no—
113M16—21no—
114M16—4822no—
115C4⋊C4—22no—
116C4⋊C4—4821no—
117C22⋊C4—4822no—
118C22⋊C4—4822no—
119C22⋊C4—4822no—
120C42—4821no—
121C42—2412no—
122C42C22⋊D24111yescharacteristic
123C23—4822no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—19281no—
132D4—9642no—
133D4—9642no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—9644no—
139C2×C4—4822no—
140C2×C4D4×D24112yes—
141C2×C4C2×C22⋊D24111yescharacteristic
142C8—41no—
143C8—9642no—
144C22—41no—
145C22—9642no—
146C22—9642no—
147C22—9642no—
148C22#2144061011yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—9642no—
155C2—21no—
156C2—81no—
157C2—19282no—
158C2#2950790311yescharacteristic
subgroupG/H
centerC2#29507903
commutator, cosocleC2×Q8×C241C23
FrattiniC2×Q8C22×D241
Fitting#456×C241C2
socleC2×C241B(4,𝔽2)
hypercenter#456D241
hypocenterC241#2204
LCSG ▷ C2×Q8×C241 ▷ C22×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #29508433C2semidirect
2. #29507890C2semidirect
3. #29507516C2semidirect
4. #29507513C2semidirect
5. #456×C241C2semidirect
79. C241#2204semidirect
82. #456D241semidirect

Maximal subgroups

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

subgroupindex
#295084332
#295078902
#295075162
#295075132
#456×C2412
#2204241