Finite Group Databasebeta

← #30638965

Subgroups — #30638965

Order 16256.

286 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 97 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#22550359C211yesmaximal, characteristic
2#22549918C211yesmaximal, characteristic, complemented
3#22549877C211yesmaximal, characteristic
4#22549742C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127—21no—
10D4×Dic127—22no—
11D4×Dic127—21no—
12#16861168—21no—
13#16861168C2211yescharacteristic
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4⋊Dic127C2211yescharacteristic
17#16861143—21no—
18#16861141—21no—
19#16861133—21no—
20#16861130—21no—
21C127⋊#95—22no—
22C2×#41×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×C42×C127C2211yescharacteristic
25#104×C127C414yescomplemented
26C22×Dic127—22no—
27C22×Dic127—21no—
28C22×Dic127D412yes—
29#14523232—21no—
30#14523232—21no—
31C2×C127⋊C8—21no—
32C4⋊Dic127—22no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39C22⋊Dic127—22no—
40C22⋊Dic127—21no—
41C4×Dic127—22no—
42C4×Dic127—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127C2×C412yes—
46#41×C127Q812yes—
47C2×Q8×C127D411yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127C2×C412yescomplemented
55C2×C8×C127—21no—
56C42×C127—21no—
57C42×C127C2×C412yescomplemented
58C2×Dic127—21no—
59C2×Dic127—22no—
60C2×Dic127—21no—
61C2×Dic127—42no—
62C2×Dic127—42no—
63C2×Dic127—41no—
64C2×Dic127—82no—
65C127⋊C8—41no—
66C23×C127—21no—
67C23×C127C2×D411yescharacteristic
68Q8×C127—21no—
69Q8×C127C4⋊C412yes—
70D4×C127—22no—
71D4×C127—22no—
72D4×C127—21no—
73D4×C127C4⋊C412yes—
74C2×C4×C127—21no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×Q811yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12011yescharacteristic
94C22×C127C22⋊Q811yescharacteristic
95C4×C127—22no—
96C4×C127—42no—
97C4×C127C2×C4⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127C22⋊Q811yescharacteristic
100C4×C127#12311yescharacteristic
101Dic127—41no—
102Dic127—42no—
103Dic127—41no—
104Dic127—81no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—42no—
108C2×C127#38511yescharacteristic
109C2×C127#45411yescharacteristic
110C2×C127#45511yescharacteristic
111#1860—1271nomaximal, Sylow
112C127#186011yescharacteristic, complemented, Sylow
113#520—1271no—
114#428—1271no—
115C2×#104D12711yescharacteristic
116#417—1271no—
117#382—1271no—
118#343—1271no—
119#339—1271no—
120C2×#41C2×D12711yescharacteristic
121C2×M16—1271no—
122C2×M16C2×D12711yescharacteristic
123C4×Q8—2541no—
124C4×D4—2542no—
125C4×D4—2541no—
126#117—1271no—
127#117—2541no—
128C2×C4⋊C4—1271no—
129C2×C4⋊C4—1271no—
130C2×C4⋊C4—2541no—
131C2×C42C2×D12711yescharacteristic
132#107—2541no—
133#106—2541no—
134#104Dic12714yescomplemented
135#103—2541no—
136#102—2541no—
137#95—2542no—
138#41—21no—
139#41—21no—
140#41C2×Dic12712yes—
141#41Dic25412yes—
142C2×Q8D50811yescharacteristic
143C2×D4—21no—
144C2×D4D50811yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—1272no—
148C22×C4—2542no—
149C22×C4—2541no—
150C22×C4C22×D12711yescharacteristic
151M16—21no—
152M16—2541no—
153M16—2541no—
154M16C2×Dic12712yescomplemented
155C2×C8—21no—
156C2×C8—2541no—
157C4⋊C4—1271no—
158C4⋊C4—1271no—
159C4⋊C4—2542no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C22⋊C4—2542no—
165C22⋊C4—2541no—
166C42—21no—
167C42—2542no—
168C42—2541no—
169C42C2×Dic12712yescomplemented
170C23—21no—
171C23D4×D12711yescharacteristic
172Q8—21no—
173Q8C4⋊Dic12712yes—
174D4—22no—
175D4—22no—
176D4—21no—
177D4C4⋊Dic12712yes—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—10162no—
185C2×C4—5082no—
186C2×C4—5081no—
187C2×C4—5082no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2542no—
191C2×C4C22×Dic12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4C2×D50811yescharacteristic
194C2×C4C2×Dic25411yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324711yescharacteristic
197C8—22no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22D4×Dic12711yescharacteristic
205C22#1686117911yescharacteristic
206C22#1686118611yescharacteristic
207C4—22no—
208C4—42no—
209C4—10161no—
210C4—5081no—
211C4—5081no—
212C4—5082no—
213C4C2×C4⋊Dic12711yescharacteristic
214C4D4×Dic12711yescharacteristic
215C4#1686118611yescharacteristic
216C4#1686119311yescharacteristic
217C2—21no—
218C2—21no—
219C2—42no—
220C2#2254975111yescharacteristic
221C2#2255005011yescharacteristic
222C2#2255005511yescharacteristic
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1860
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22549918C2semidirect
4. #22549742C2semidirect
5. #22549644C2semidirect
6. #22549636C2semidirect
25. #104×C127C4semidirect
54. M16×C127C2×C4semidirect
57. C42×C127C2×C4semidirect
112. C127#1860semidirect
134. #104Dic127semidirect
154. M16C2×Dic127semidirect
169. C42C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503592
#225499182
#225498772
#225497422
#225496442
#225496362
C2×#104×C1272
#1860127