Finite Group Databasebeta

← #44509471

Subgroups — #44509471

Order 30848.

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#29507899C211yesmaximal, characteristic, complemented
2#29507889C212yesmaximal, complemented
3#29507516C211yesmaximal, characteristic, complemented
4#29507514C212yesmaximal, complemented
5#555×C241C211yesmaximal, characteristic, complemented
6#21440690C2211yescharacteristic
7#21440673—21no—
8#21440668—22no—
9#21440570—22no—
10#21440570—21no—
11#21440561C2212yes—
12#21440561C2211yescharacteristic
13C4⋊Q8×C241C2211yescharacteristic
14#124×C241—21no—
15#124×C241C2212yes—
16#121×C241—22no—
17C4×D4×C241—21no—
18#117×C241—22no—
19#16838771—41no—
20#16838770—22no—
21#16838770—21no—
22Q8×D241—41no—
23#16838760—41no—
24#16838758—42no—
25#16838757—41no—
26#16838755—42no—
27#16838754—22no—
28#16838754—21no—
29C2×Q8×C241D411yescharacteristic
30C2×Q8×C241C2311yescharacteristic
31C2×D4×C241—21no—
32C2×D4×C241—21no—
33C2×D4×C241D412yes—
34C22×C4×C241—22no—
35C4⋊C4×C241—21no—
36C4⋊C4×C241—22no—
37C22⋊C4×C241—22no—
38C22⋊C4×C241—22no—
39C22⋊C4×C241—22no—
40C42×C241—21no—
41C42×C241D412yes—
42C42×C241D411yescharacteristic
43Dic482—22no—
44Dic482—21no—
45D964—22no—
46D964—21no—
47C22⋊D241—41no—
48C4×D241—42no—
49C4×D241—41no—
50#14515349—42no—
51#14515349—41no—
52C23×C241—22no—
53Q8×C241—41no—
54Q8×C241—41no—
55D4×C241—42no—
56D4×C241—42no—
57D4×C241—81no—
58C2×C4×C241—21no—
59C2×C4×C241—22no—
60C2×C4×C241—21no—
61C2×C4×C241—44no—
62C2×C4×C241C2×D411yescharacteristic
63C2×C4×C241C2×D412yes—
64C22×C241—42no—
65C22×C241—42no—
66C22×C241—42no—
67C22×C241#12011yescharacteristic
68C2×D241—41no—
69C4×C241—22no—
70C4×C241—22no—
71C4×C241—21no—
72C4×C241—42no—
73C4×C241—41no—
74Dic241—41no—
75C2×C241—21no—
76C2×C241—82no—
77C2×C241B(4,𝔽2)11yescharacteristic
78D241—81no—
79C241#220411yescharacteristic, complemented, Sylow
80#2204—2411nomaximal, Sylow
81#555D24111yescharacteristic, complemented
82#456—2411no—
83#454—2412no—
84#356—2411no—
85#355—2412no—
862−1+4—2411no—
87#137—4821no—
88AGL(1,ℤ/8)—4822no—
89C4⋊Q8C2×D24111yescharacteristic
90#124—21no—
91#124C2×D24112yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—4822no—
96#104—4821no—
97#101—2411no—
98#101—2412no—
99#41—9641no—
100#41—4822no—
101#41—4821no—
102C2×Q8—9641no—
103C2×Q8C22⋊D24111yescharacteristic
104C2×Q8C22×D24111yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D24112yes—
108C22×C4—22no—
109Q16—9641no—
110SD16—9642no—
111SD16—9641no—
112D8—9642no—
113M16—4822no—
114M16—4821no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D24112yes—
122C42C22⋊D24111yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—4822no—
127Q8—4821no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—9641no—
132D4—4822no—
133D4—4821no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—9642no—
139C2×C4—9641no—
140C2×C4C2×C22⋊D24111yescharacteristic
141C2×C4C2×C22⋊D24112yes—
142C8—9642no—
143C8—9641no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—9641no—
148C22#2144061111yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—9641no—
155C2—21no—
156C2—82no—
157C2—19281no—
158C2#2950790411yescharacteristic
subgroupG/H
centerC2#29507904
commutator, cosocleC2×Q8×C241C23
FrattiniC2×Q8C22×D241
Fitting#555×C241C2
socleC2×C241B(4,𝔽2)
hypercenter#555D241
hypocenterC241#2204
LCSG ▷ C2×Q8×C241 ▷ C22×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #29507899C2semidirect
2. #29507889C2semidirect
3. #29507516C2semidirect
4. #29507514C2semidirect
5. #555×C241C2semidirect
79. C241#2204semidirect
81. #555D241semidirect

Maximal subgroups

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

subgroupindex
#295078992
#295078892
#295075162
#295075142
#555×C2412
#2204241