Finite Group Databasebeta

← #30650185

Subgroups — #30650185

Order 16256.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#22550493C212yesmaximal, complemented
2#22550170C212yesmaximal, complemented
3#22550154C212yesmaximal, complemented
4#22550120C212yesmaximal, complemented
5#22550117C211yesmaximal, characteristic, complemented
6#22550097C212yesmaximal, complemented
7#22550089C212yesmaximal, complemented
8#22549902C211yesmaximal, characteristic, complemented
9#466×C127C211yesmaximal, characteristic
10C2×Dic508C2211yescharacteristic
11C2×#14523236C2212yes—
12C2×#14523232C2211yescharacteristic
13#16861208C2212yes—
14#16861202—22no—
15#16861196—22no—
16#16861196C2212yes—
17#16861190—22no—
18#16861188C2212yes—
19#16861186—22no—
20#16861174C2212yes—
21D4×Dic127C2212yes—
22#16861168C2211yescharacteristic
23#16861143C2211yescharacteristic
24#16861142C2212yes—
25Dic127⋊C8C2212yes—
26Dic127⋊C8C2212yes—
27#16861134C2212yes—
28#16861132C2212yes—
29#16861130C2212yes—
30C2×D8×C127C2211yescharacteristic
31C22×C8×C127C2211yescharacteristic
32#121×C127C2212yes—
33#107×C127C2211yescharacteristic
34#102×C127C2212yes—
35C2×Dic254—22no—
36C2×Dic254C2312yes—
37C22×Dic127—22no—
38Dic508—41no—
39#14523236—42no—
40#14523232—41no—
41C2×C127⋊C8C2312yes—
42C4⋊Dic127D411yescharacteristic
43C4⋊Dic127D411yescharacteristic
44C4⋊Dic127C2312yes—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48C22⋊Dic127—22no—
49C22⋊Dic127—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127—22no—
52C22⋊Dic127D412yes—
53C4×Dic127—22no—
54C4×Dic127C2312yes—
55C2×D4×C127—22no—
56C2×D4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58D8×C127—41no—
59C2×C8×C127—41no—
60C2×C8×C127C2311yescharacteristic
61C2×C8×C127C2311yescharacteristic
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—22no—
64Dic254—22no—
65Dic254—22no—
66Dic254—82no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—82no—
70C2×Dic127C2×D412yes—
71C2×Dic127C2×D412yes—
72C127⋊C8—22no—
73C23×C127—22no—
74C23×C127C2×D411yescharacteristic
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—82no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—22no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2411yescharacteristic
83C8×C127—21no—
84C8×C127—21no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127C22×D411yescharacteristic
92C4×C127—21no—
93C4×C127—42no—
94C4×C127H(5,𝔽2)11yescharacteristic
95C4×C1272−1+411yescharacteristic
96Dic127—22no—
97Dic127—22no—
98Dic127—42no—
99Dic127—42no—
100C2×C127—41no—
101C2×C127—82no—
102C2×C127#53711yescharacteristic
103C2×C127#57611yescharacteristic
104C2×C127#57811yescharacteristic
105#3262—1271nomaximal, Sylow
106C127#326211yescharacteristic, complemented, Sylow
107#542—1272no—
108#477—1272no—
109#474—1272no—
110#468—1272no—
111#467—1271no—
112#466D12711yescharacteristic
113#464—1272no—
114#463—1272no—
115#424—1271no—
116C2×Q16—1271no—
117C2×SD16—1272no—
118C2×D8C2×D12711yescharacteristic
119C2×M16—1271no—
120C22×C8C2×D12711yescharacteristic
121C4⋊Q8—1272no—
122#126—2542no—
123#124—1272no—
124#124—2542no—
125#123—2542no—
126C22⋊Q8—1272no—
127C22⋊Q8—2542no—
128#121C2×D12712yes—
129C4×Q8—1272no—
130C4×D4—1272no—
131#117—1271no—
132#107—1271no—
133#107—1272no—
134#107C2×D12711yescharacteristic
135C4⋊C8—1272no—
136C4⋊C8—1272no—
137#103—1272no—
138#103—1272no—
139#102—1272no—
140#102C2×D12712yes—
141C2×Q8—1272no—
142C2×Q8—2542no—
143C2×D4—22no—
144C2×D4C22×D12712yes—
145C22×C4—2542no—
146C22×C4C22×D12711yescharacteristic
147Q16—5081no—
148SD16—5082no—
149D8—41no—
150M16—5081no—
151C2×C8—41no—
152C2×C8—1272no—
153C2×C8C22×D12711yescharacteristic
154C2×C8C22×D12711yescharacteristic
155C4⋊C4—1271no—
156C4⋊C4—1271no—
157C4⋊C4—1272no—
158C4⋊C4—2542no—
159C4⋊C4—2542no—
160C4⋊C4—2542no—
161C4⋊C4C22×D12712yes—
162C22⋊C4—22no—
163C22⋊C4—1272no—
164C22⋊C4—2542no—
165C22⋊C4—2542no—
166C22⋊C4—2542no—
167C22⋊C4—2542no—
168C42—1272no—
169C42—2542no—
170C23—22no—
171C23D4×D12711yescharacteristic
172Q8—10162no—
173Q8—2542no—
174Q8—2542no—
175D4—22no—
176D4—22no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—1272no—
182C2×C4—1272no—
183C2×C4—10162no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4D4×D12711yescharacteristic
187C2×C4C23×D12711yescharacteristic
188C8—21no—
189C8—21no—
190C8—2542no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22C2×D4×D12711yescharacteristic
198C4—21no—
199C4—42no—
200C4—5082no—
201C4—5082no—
202C4—2542no—
203C4—2542no—
204C4#1686125411yescharacteristic
205C4#1686125611yescharacteristic
206C2—41no—
207C2—82no—
208C2#2255045311yescharacteristic
209C2#2255067711yescharacteristic
210C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3262
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22550493C2semidirect
2. #22550170C2semidirect
3. #22550154C2semidirect
4. #22550120C2semidirect
5. #22550117C2semidirect
6. #22550097C2semidirect
7. #22550089C2semidirect
8. #22549902C2semidirect
106. C127#3262semidirect

Maximal subgroups

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

subgroupindex
#225504932
#225501702
#225501542
#225501202
#225501172
#225500972
#225500892
#225499022
#466×C1272
#3262127