Finite Group Databasebeta

← #11799434

Subgroups — #11799434

Order 1152.

240 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 39 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#10593838C211yesmaximal, characteristic
2C2×#63736C211yesmaximal, characteristic, complemented
3#10591683C212yesmaximal
4#10590070C211yesmaximal, characteristic, complemented
5#10590066C212yesmaximal, complemented
6#64157C2211yescharacteristic
7C3×#951—22no—
8C2×#3718C2212yes—
9#63773—21no—
10#63773C2211yescharacteristic
11C2×#3708—21no—
12C2×#3708C2211yescharacteristic
13#63736—21no—
14#63736—21no—
15#63578C2212yes—
16#63560—22no—
17#5204—62no—
18C32×#41D411yescharacteristic
19C2×C3×Dic6—22no—
20#3748—21no—
21#3748D411yescharacteristic
22C2×C3⋊(C3⋊C8)—21no—
23C2×C3⋊(C3⋊C8)C2311yescharacteristic
24C3×#216—42no—
25C3×#214—42no—
26C3×#208—22no—
27#3718—22no—
28#3718—22no—
29#3708—21no—
30#3708D411yescharacteristic
31#3708D411yescharacteristic
32#2222—91nomaximal, Sylow
33C3×#137—62no—
34#951—22no—
35#948—62no—
36C2×#216—62no—
37#836—62no—
38#824—62no—
39#822—62no—
40C32×Q8D811yescharacteristic
41C32×D4—21no—
42C32×D4D811yescharacteristic
43C2×C32×C4—21no—
44C2×C32×C4C2×D411yescharacteristic
45C2×C3×Dic3—42no—
46C3×Dic6—42no—
47C3×Dic6—42no—
48C3⋊(C3⋊C8)—21no—
49C3⋊(C3⋊C8)C2×D411yescharacteristic
50C3⋊(C3⋊C8)#4111yescharacteristic
51C3×C3⋊C8—42no—
52C2×Q32—91no—
53#504—91no—
54#473—92no—
55#368—91no—
56#362—92no—
57C3×#41—22no—
58C2×C3×Q8—62no—
59C2×Dic6—22no—
60C3×Q16—122no—
61C3×SD16—122no—
62C3×M16—62no—
63#216—42no—
64#216—122no—
65#216—122no—
66#214—42no—
67#208—22no—
68#208—62no—
69#208—122no—
70C2×C3⋊C8—62no—
71C2×C3⋊C8—122no—
72C22×C32—21no—
73C22×C32#13511yescharacteristic
74C32×C4—21no—
75C32×C4#12111yescharacteristic
76C32×C4C2×D811yescharacteristic
77C3×Dic3—82no—
78#137—62no—
79C2×Q16—92no—
80#131—91no—
81Q32—181no—
82Q32—181no—
83M32—91no—
84M32—181no—
85C2×C16—91no—
86C2×C16—181no—
87#108—92no—
88#101—182no—
89C3×Q8—22no—
90C3×Q8—122no—
91C3×Q8—122no—
92C3×D4—22no—
93C3×D4—42no—
94C2×C3×C4—22no—
95C2×C3×C4—42no—
96C2×C3×C4—122no—
97C2×Dic3—42no—
98Dic6—42no—
99Dic6—42no—
100C3×C8—122no—
101C3⋊C8—42no—
102C3⋊C8—62no—
103C3⋊C8—62no—
104C3⋊C8—122no—
105C2×C32—21no—
106C2×C32—41no—
107C2×C32#46611yescharacteristic
108#41GO+(4,𝔽2)11yescharacteristic
109C2×Q8—62no—
110Q16—122no—
111Q16—182no—
112Q16—182no—
113SD16—122no—
114M16—62no—
115M16—91no—
116M16—181no—
117C2×C8—91no—
118C2×C8—181no—
119C16—91no—
120C16—91no—
121C16—181no—
122C22×C3—22no—
123C22×C3—42no—
124C3×C4—22no—
125C3×C4—22no—
126C3×C4—42no—
127C3×C4—242no—
128Dic3—82no—
129C32#222211yescharacteristic, complemented, Sylow
130Q8—122no—
131Q8—122no—
132Q8#377411yescharacteristic
133D4—21no—
134D4#377411yescharacteristic
135C2×C4—21no—
136C2×C4—122no—
137C2×C4C2×GO+(4,𝔽2)11yescharacteristic
138C8—91no—
139C8—91no—
140C8—122no—
141C8—181no—
142C2×C3—22no—
143C2×C3—42no—
144C2×C3—82no—
145C22—21no—
146C22#6423711yescharacteristic
147C4—21no—
148C4—242no—
149C4C2×#377411yescharacteristic
150C4#6424211yescharacteristic
151C3—22no—
152C2—21no—
153C2—41no—
154C2#1059352011yescharacteristic
subgroupG/H
centerC2#10593520
commutator, cosocleC2×C3⋊(C3⋊C8)C23
FrattiniC2×C4C2×GO+(4,𝔽2)
Fitting, prime radicalC32×#41D4
socleC2×C32#466
hypercenter#41GO+(4,𝔽2)
hypocenterC32#2222
LCSG ▷ C2×C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×#63736C2semidirect
4. #10590070C2semidirect
5. #10590066C2semidirect
129. C32#2222semidirect

Maximal subgroups

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

subgroupindex
#105938382
C2×#637362
#105916832
#105900702
#105900662
#22229