Finite Group Databasebeta

← #46812578

Subgroups — #46812578

Order 34432.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 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#31771180C211yesmaximal, characteristic
2#31770588C212yesmaximal
3#31770404C211yesmaximal, characteristic, complemented
4#31770403C212yesmaximal, complemented
5#356×C269C211yesmaximal, characteristic, complemented
6C2×#16864403C2211yescharacteristic
7C2×Q8×D269C411yescharacteristic
8M16×D269—22no—
9#22575451C2211yescharacteristic
10#22575442C411yescharacteristic
11#22575441C411yescharacteristic
12#22575438—22no—
13#22575431—22no—
14#22575410C411yescharacteristic
15#22575372C2212yes—
16#22575369—22no—
17C4⋊Q8×C269C2211yescharacteristic
18#101×C269C2212yes—
19#16864403—42no—
20C2×Dic538—21no—
21C2×Dic538C2×C411yescharacteristic
22Q8×D269—42no—
23C2×D1076C2×C411yescharacteristic
24C2×D1076D412yes—
25C2×C4×D269—21no—
26C2×C4×D269C2×C411yescharacteristic
27C2×C4×D269D412yes—
28#16864386—22no—
29#16864385—42no—
30C8×D269—42no—
31#16864381—21no—
32#16864381—22no—
33#16864379—21no—
34#16864379—22no—
35#16864379—22no—
36C4×Dic269C2×C411yescharacteristic
37C2×Q8×C269C2×C411yescharacteristic
38C2×Q8×C269C2311yescharacteristic
39M16×C269—22no—
40C4⋊C4×C269—22no—
41C42×C269C2×C411yescharacteristic
42C22×D269C22⋊C412yes—
43C22×D269C22⋊C411yescharacteristic
44Dic538—42no—
45Dic538—41no—
46D1076—41no—
47D1076—42no—
48C2×Dic269—21no—
49C2×Dic269—21no—
50C2×Dic269C22⋊C411yescharacteristic
51C4×D269—22no—
52C4×D269—22no—
53C4×D269—22no—
54C4×D269—42no—
55#14524130—42no—
56Q8×C269—41no—
57Q8×C269—41no—
58C2×C4×C269—21no—
59C2×C4×C269—21no—
60C2×C4×C269C22×C411yescharacteristic
61C2×C4×C269C2×D412yes—
62C8×C269—42no—
63C22×C269C2×C22⋊C411yescharacteristic
64C2×D269—21no—
65C2×D269—22no—
66C2×D269—42no—
67C2×D269—41no—
68C4×C269—22no—
69C4×C269—21no—
70C4×C269—41no—
71C4×C269—41no—
72Dic269—41no—
73Dic269—41no—
74Dic269#9912yes—
75C2×C269—21no—
76C2×C269C2×#9911yescharacteristic
77D269—41no—
78D269—42no—
79C269#213511yescharacteristic, complemented, Sylow
80#2135—2691nomaximal, Sylow
81#536—2691no—
82#413—2692no—
83#356—2691no—
84#356—2692no—
85#356D26911yescharacteristic, complemented
86C2×#41—2691no—
87C22×Q8—2691no—
88C2×M16—5382no—
89C4⋊Q8—2691no—
90C4⋊Q8C2×D26911yescharacteristic
91#124—2691no—
92#124—2691no—
93#123—5382no—
94C22⋊Q8—5382no—
95#117—2691no—
96#101—2692no—
97#101C2×D26912yes—
98#100—5382no—
99#41—10762no—
100C2×Q8—2691no—
101C2×Q8—5381no—
102C2×Q8—10762no—
103C2×Q8C4×D26911yescharacteristic
104C2×Q8C22×D26911yescharacteristic
105C2×D4—2691no—
106C2×D4—2692no—
107C22×C4—2691no—
108C22×C4—2692no—
109C22×C4—5381no—
110M16—22no—
111M16—5382no—
112M16—10762no—
113C2×C8—10762no—
114C4⋊C4—22no—
115C4⋊C4—5381no—
116C4⋊C4—5382no—
117C22⋊C4—5381no—
118C22⋊C4—5382no—
119C22⋊C4—5382no—
120C42—2691no—
121C42C4×D26911yescharacteristic
122C23—2692no—
123C23—2691no—
124Q8—41no—
125Q8—41no—
126Q8—10761no—
127Q8—10762no—
128D4—10762no—
129D4—10761no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—2691no—
133C2×C4—5382no—
134C2×C4—5382no—
135C2×C4—5382no—
136C2×C4—5381no—
137C2×C4—5381no—
138C2×C4—10762no—
139C2×C4C2×C4×D26911yescharacteristic
140C2×C4D4×D26912yes—
141C8—42no—
142C8—10762no—
143C22—5381no—
144C22—5382no—
145C22—10762no—
146C22—10761no—
147C22C22⋊C4×D26911yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—2692no—
153C4—10761no—
154C4—10761no—
155C2—21no—
156C2—10761no—
157C2—10762no—
158C2#99×D26911yescharacteristic
subgroupG/H
centerC2#99×D269
commutatorC2×C4×C269C22×C4
FrattiniC2×Q8C22×D269
Fitting#356×C269C2
socleC2×C269C2×#99
prime radicalC2×#16864403C22
cosocleC2×Q8×C269C23
hypercenter#356D269
hypocenterC269#2135
LCSG ▷ C2×C4×C269 ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #31770404C2semidirect
4. #31770403C2semidirect
5. #356×C269C2semidirect
79. C269#2135semidirect
85. #356D269semidirect

Maximal subgroups

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

subgroupindex
#317711802
#317705882
#317704042
#317704032
#356×C2692
#2135269