Finite Group Databasebeta

← #30651403

Subgroups — #30651403

Order 16256.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic254C211yesmaximal, characteristic
2#22550601C211yesmaximal, characteristic
3#22550226C211yesmaximal, characteristic
4#22550225C212yesmaximal
5#22550172C211yesmaximal, characteristic
6#22550169C212yesmaximal
7#22550084C212yesmaximal
8#22550000C211yesmaximal, characteristic
9Q16×Dic127C212yesmaximal
10C4×#14523238C211yesmaximal, characteristic
11#462×C127C211yesmaximal, characteristic
12C2×#14523238C2211yescharacteristic
13#16861209—21no—
14#16861209C2211yescharacteristic
15#16861208—22no—
16#16861208C2212yes—
17#16861207C2212yes—
18#16861200—21no—
19#16861199—22no—
20Q8×Dic127C2212yes—
21Q8×Dic127C2212yes—
22C4×Dic254—21no—
23C4×Dic254C2211yescharacteristic
24C4×Dic254C2211yescharacteristic
25#16861174—21no—
26#16861143C2212yes—
27#16861142C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861133C2212yes—
31#16861133C2211yescharacteristic
32#16861133C2212yes—
33#16861132C2211yescharacteristic
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C2×Q16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254—21no—
43C2×Dic254D411yescharacteristic
44C2×Dic254C2311yescharacteristic
45#14523238—22no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2311yescharacteristic
48C4⋊Dic127—22no—
49C4⋊Dic127—22no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—22no—
55Dic127⋊C4—22no—
56Dic127⋊C4—22no—
57Dic127⋊C4—21no—
58Dic127⋊C4D412yes—
59C4×Dic127—22no—
60C4×Dic127—22no—
61C4×Dic127—21no—
62C4×Dic127C2312yes—
63C2×Q8×C127C2312yes—
64C2×Q8×C127C2311yescharacteristic
65Q16×C127—24no—
66C2×C8×C127C2312yes—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2311yescharacteristic
71C42×C127—21no—
72C42×C127C2311yescharacteristic
73Dic254—21no—
74Dic254—21no—
75Dic254—82no—
76Dic254—81no—
77Dic254Q1614yescomplemented
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C127⋊C8—21no—
84C127⋊C8—21no—
85Q8×C127—21no—
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127#4112yes—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2411yescharacteristic
95C8×C127—22no—
96C22×C127C22×D411yescharacteristic
97C4×C127—22no—
98C4×C127—21no—
99C4×C127—42no—
100C4×C127—41no—
101C4×C127C2×Q1612yes—
102C4×C127C2×#4111yescharacteristic
103C4×C1272−1+411yescharacteristic
104Dic127—22no—
105Dic127—41no—
106Dic127—41no—
107Dic127—42no—
108Dic127C2×Q1614yes—
109C2×C127#54811yescharacteristic
110C2×C127C22×Q1611yescharacteristic
111C2×C127#57711yescharacteristic
112#3363—1271nomaximal, Sylow
113C127#336311yescharacteristic, complemented, Sylow
114Q82—1271no—
115#557—1271no—
116#487—1271no—
117#487—1272no—
118#477—1271no—
119#477—1272no—
120#462—1272no—
121#462D12711yescharacteristic
122C8×Q8—1271no—
123C4×Q16—1272no—
124C4×Q16—1271no—
125C2×Q16—1271no—
126C2×Q16C2×D12712yes—
127C4⋊Q8—1272no—
128C4⋊Q8—1272no—
129C4⋊Q8—1271no—
130C4⋊Q8—2541no—
131C4⋊Q8—2542no—
132C4⋊Q8C2×D12711yescharacteristic
133#125—2542no—
134#125—2541no—
135C4×Q8—1271no—
136C4×Q8—1272no—
137C4×Q8—1272no—
138C4×Q8—1271no—
139C4×Q8—2541no—
140C4×Q8—2541no—
141C4×Q8C2×D12711yescharacteristic
142#107—1272no—
143#107—1271no—
144C4⋊C8—1272no—
145C4⋊C8C2×D12711yescharacteristic
146#103—1271no—
147#103—1272no—
148#103—1272no—
149#103—1272no—
150#103—1271no—
151#103C2×D12712yes—
152C4×C8—1272no—
153C4×C8—1271no—
154C2×Q8—1271no—
155C2×Q8—1271no—
156C2×Q8—2542no—
157C2×Q8—2541no—
158C2×Q8C22×D12712yes—
159C2×Q8C22×D12711yescharacteristic
160Q16—24no—
161Q16—2542no—
162C2×C8—1271no—
163C2×C8—1271no—
164C2×C8C22×D12712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—1272no—
169C4⋊C4—1271no—
170C4⋊C4—1272no—
171C4⋊C4—1271no—
172C4⋊C4—2542no—
173C4⋊C4—2542no—
174C4⋊C4—2542no—
175C4⋊C4—2542no—
176C4⋊C4—2542no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C4⋊C4C22×D12711yescharacteristic
180C42—21no—
181C42—1272no—
182C42—2542no—
183C42—2542no—
184C42—2541no—
185C42C22×D12711yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—1274no—
190Q8—10161no—
191Q8—10162no—
192Q8—2541no—
193Q8—2541no—
194Q8#1452324712yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—1272no—
199C2×C4—1272no—
200C2×C4—2541no—
201C2×C4—2542no—
202C2×C4—2541no—
203C2×C4D4×D12711yescharacteristic
204C2×C4D4×D12711yescharacteristic
205C2×C4C23×D12711yescharacteristic
206C8—22no—
207C8—2541no—
208C8—2541no—
209C22C2×D4×D12711yescharacteristic
210C4—22no—
211C4—21no—
212C4—42no—
213C4—41no—
214C4—1274no—
215C4—5081no—
216C4—5082no—
217C4—5081no—
218C4—2542no—
219C4Q16×D12712yes—
220C4C2×#1452324711yescharacteristic
221C4#1686125611yescharacteristic
222C2#2255053211yescharacteristic
223C2C2×Q16×D12711yescharacteristic
224C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#462×C127C2
socle, prime radicalC22×C127C22×D4
hypercenter#462D127
hypocenterC127#3363
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
77. Dic254Q16semidirect
113. C127#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225506012
#225502262
#225502252
#225501722
#225501692
#225500842
#225500002
Q16×Dic1272
C4×#145232382
#462×C1272
#3363127