Finite Group Databasebeta

← #30640423

Subgroups — #30640423

Order 16256.

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#22550451C211yesmaximal, characteristic
2#22549859C212yesmaximal
3#22549675C211yesmaximal, characteristic, complemented
4#22549674C212yesmaximal, complemented
5#356×C127C211yesmaximal, characteristic, complemented
6C2×#14523249C2211yescharacteristic
7C2×Q8×D127C411yescharacteristic
8M16×D127—22no—
9#16861207C2211yescharacteristic
10#16861198C411yescharacteristic
11#16861197C411yescharacteristic
12#16861194—22no—
13#16861187—22no—
14#16861166C411yescharacteristic
15#16861128C2212yes—
16#16861125—22no—
17C4⋊Q8×C127C2211yescharacteristic
18#101×C127C2212yes—
19#14523249—42no—
20C2×Dic254—21no—
21C2×Dic254C2×C411yescharacteristic
22Q8×D127—42no—
23C2×D508C2×C411yescharacteristic
24C2×D508D412yes—
25C2×C4×D127—21no—
26C2×C4×D127C2×C411yescharacteristic
27C2×C4×D127D412yes—
28#14523232—22no—
29#14523231—42no—
30C8×D127—42no—
31Dic127⋊C4—21no—
32Dic127⋊C4—22no—
33(C2×D127)⋊C4—22no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—21no—
36C4×Dic127C2×C411yescharacteristic
37C2×Q8×C127C2×C411yescharacteristic
38C2×Q8×C127C2311yescharacteristic
39M16×C127—22no—
40C4⋊C4×C127—22no—
41C42×C127C2×C411yescharacteristic
42C22×D127C22⋊C412yes—
43C22×D127C22⋊C411yescharacteristic
44Dic254—42no—
45Dic254—41no—
46D508—41no—
47D508—42no—
48C2×Dic127—21no—
49C2×Dic127—21no—
50C2×Dic127C22⋊C411yescharacteristic
51C4×D127—22no—
52C4×D127—22no—
53C4×D127—22no—
54C4×D127—42no—
55C127⋊C8—42no—
56Q8×C127—41no—
57Q8×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D412yes—
62C8×C127—42no—
63C22×C127C2×C22⋊C411yescharacteristic
64C2×D127—21no—
65C2×D127—22no—
66C2×D127—42no—
67C2×D127—41no—
68C4×C127—22no—
69C4×C127—21no—
70C4×C127—41no—
71C4×C127—41no—
72Dic127—41no—
73Dic127—41no—
74Dic127#9912yes—
75C2×C127—21no—
76C2×C127C2×#9911yescharacteristic
77D127—41no—
78D127—42no—
79#2135—1271nomaximal, Sylow
80C127#213511yescharacteristic, complemented, Sylow
81#536—1271no—
82#413—1272no—
83#356—1271no—
84#356—1272no—
85#356D12711yescharacteristic, complemented
86C2×#41—1271no—
87C22×Q8—1271no—
88C2×M16—2542no—
89C4⋊Q8—1271no—
90C4⋊Q8C2×D12711yescharacteristic
91#124—1271no—
92#124—1271no—
93#123—2542no—
94C22⋊Q8—2542no—
95#117—1271no—
96#101—1272no—
97#101C2×D12712yes—
98#100—2542no—
99#41—5082no—
100C2×Q8—1271no—
101C2×Q8—5082no—
102C2×Q8—2541no—
103C2×Q8C4×D12711yescharacteristic
104C2×Q8C22×D12711yescharacteristic
105C2×D4—1271no—
106C2×D4—1272no—
107C22×C4—1271no—
108C22×C4—1272no—
109C22×C4—2541no—
110M16—22no—
111M16—5082no—
112M16—2542no—
113C2×C8—5082no—
114C4⋊C4—22no—
115C4⋊C4—2541no—
116C4⋊C4—2542no—
117C22⋊C4—2541no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C42—1271no—
121C42C4×D12711yescharacteristic
122C23—1272no—
123C23—1271no—
124Q8—41no—
125Q8—41no—
126Q8—5081no—
127Q8—5082no—
128D4—5082no—
129D4—5081no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—1271no—
133C2×C4—5082no—
134C2×C4—2542no—
135C2×C4—2542no—
136C2×C4—2542no—
137C2×C4—2541no—
138C2×C4—2541no—
139C2×C4C2×C4×D12711yescharacteristic
140C2×C4D4×D12712yes—
141C8—42no—
142C8—5082no—
143C22—5082no—
144C22—5081no—
145C22—2541no—
146C22—2542no—
147C22C22⋊C4×D12711yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—1272no—
153C4—5081no—
154C4—5081no—
155C2—21no—
156C2—5081no—
157C2—5082no—
158C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
Fitting#356×C127C2
socleC2×C127C2×#99
prime radicalC2×#14523249C22
cosocleC2×Q8×C127C23
hypercenter#356D127
hypocenterC127#2135
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #22549675C2semidirect
4. #22549674C2semidirect
5. #356×C127C2semidirect
80. C127#2135semidirect
85. #356D127semidirect

Maximal subgroups

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

subgroupindex
#225504512
#225498592
#225496752
#225496742
#356×C1272
#2135127