Finite Group Databasebeta

← #30637792

Subgroups — #30637792

Order 16256.

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

normal subgroups only

This group has 278 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#22550546C211yesmaximal, characteristic, complemented
2#22550073C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic, complemented
4#22549620C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7#493×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523236—21no—
10C2×#14523233—21no—
11#16861205—21no—
12#16861181—22no—
13#16861180—21no—
14C4×Dic254C2211yescharacteristic
15C4×D508C2211yescharacteristic
16D4×Dic127—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861130—21no—
20#16861130—21no—
21C2×D8×C127—22no—
22#127×C127C2211yescharacteristic
23#127×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—22no—
30C2×C22⋊D127—22no—
31C22×Dic127—22no—
32C2×C4×D127—21no—
33#14523236—42no—
34#14523233—42no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—21no—
40C22⋊Dic127—22no—
41(C2×D127)⋊C4—21no—
42C4×Dic127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127D412yes—
49D8×C127—44no—
50C2×C8×C127—21no—
51C42×C127C2311yescharacteristic
52C22×D127—21no—
53Dic254—21no—
54Dic254D812yescomplemented
55D508—21no—
56D508—21no—
57C22⋊D127—44no—
58C22⋊D127—82no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—44no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—41no—
66C127⋊C8—41no—
67C23×C127—22no—
68C23×C127—41no—
69D4×C127—42no—
70D4×C127—42no—
71D4×C127—41no—
72D4×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C8×C127—22no—
83C22×C127—42no—
84C22×C127—42no—
85C22×C127—42no—
86C22×C127—81no—
87C22×C127—81no—
88C22×C127—81no—
89C22×C127#12011yescharacteristic
90C2×D127—41no—
91C2×D127—41no—
92C2×D127—41no—
93C4×C127—21no—
94C4×C127C2×D811yescharacteristic
95C4×C127#13511yescharacteristic
96C4×C127AGL(1,ℤ/8)11yescharacteristic
97C4×C127AGL(1,ℤ/8)11yescharacteristic
98Dic127—22no—
99Dic127—41no—
100Dic127—41no—
101C2×C127—82no—
102C2×C127—161no—
103C2×C127#44711yescharacteristic
104C2×C127#44911yescharacteristic
105C2×C127#45311yescharacteristic
106D127—81no—
107#1621—1271nomaximal, Sylow
108C127#162111yescharacteristic, complemented, Sylow
109#550—1271no—
110#493D12711yescharacteristic, complemented
111#460—1271no—
112#459—1271no—
113#331—1271no—
114Q8⋊C8—1271no—
115D4⋊C8—1271no—
116C2×#41—2541no—
117C2×SD16—2541no—
118C2×D8—22no—
119C2×D8—2541no—
120#127—2541no—
121#127C2×D12711yescharacteristic
122#127C2×D12711yescharacteristic
123#121—2541no—
124#121—2542no—
125C4×Q8—1271no—
126C4×D4—1271no—
127C4×D4—2541no—
128C4⋊C8—1271no—
129C4⋊C8—1271no—
130#102—2541no—
131#102—2541no—
132C4×C8C2×D12711yescharacteristic
133#41—5082no—
134#41—5082no—
135C2×Q8—1271no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—1271no—
142C2×D4—2542no—
143C2×D4—2542no—
144C2×D4C22⋊D12712yes—
145C22×C4—2542no—
146C22×C4—2541no—
147SD16—5082no—
148D8—44no—
149D8—5082no—
150C2×C8—21no—
151C2×C8—2541no—
152C2×C8—2541no—
153C4⋊C4—1271no—
154C4⋊C4—1271no—
155C4⋊C4—2541no—
156C22⋊C4—2542no—
157C22⋊C4—2541no—
158C42—2541no—
159C42C22×D12711yescharacteristic
160C23—22no—
161C23—41no—
162C23—2541no—
163Q8—1272no—
164Q8—2541no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—10162no—
176D4—5084no—
177D4—2541no—
178D4—2541no—
179C2×C4—5084no—
180C2×C4—5081no—
181C2×C4—5081no—
182C2×C4—2541no—
183C2×C4—2541no—
184C2×C4—2541no—
185C2×C4D4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4C2×C22⋊D12711yescharacteristic
188C8—22no—
189C8—5081no—
190C8—5081no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—5081no—
198C22—5081no—
199C22—5081no—
200C22#1686117711yescharacteristic
201C4—21no—
202C4—5081no—
203C4—5081no—
204C4—2542no—
205C4D8×D12711yescharacteristic
206C4#1686122811yescharacteristic
207C4#1686123311yescharacteristic
208C4#1686123311yescharacteristic
209C2—82no—
210C2—161no—
211C2—10161no—
212C2#2255000811yescharacteristic
213C2#2255001811yescharacteristic
214C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#493×C127C2
socleC22×C127#120
hypercenter#493D127
hypocenterC127#1621
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #22550546C2semidirect
2. #22550073C2semidirect
3. #22550068C2semidirect
4. #22549620C2semidirect
5. #22549608C2semidirect
6. #22549605C2semidirect
7. #493×C127C2semidirect
54. Dic254D8semidirect
108. C127#1621semidirect
110. #493D127semidirect

Maximal subgroups

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

subgroupindex
#225505462
#225500732
#225500682
#225496202
#225496082
#225496052
#493×C1272
#1621127