Finite Group Databasebeta

← #30648382

Subgroups — #30648382

Order 16256.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14523238C211yesmaximal, characteristic, complemented
2C2×#16861216C211yesmaximal, characteristic
3#22550447C211yesmaximal, characteristic
4#22550132C214yesmaximal, complemented
5#22550115C214yesmaximal, complemented
6#22549912C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549884C211yesmaximal, characteristic, complemented
9#536×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11C2×#14523238—24no—
12C2×#14523238C2212yes—
13C2×#14523238C2212yes—
14#16861216—24no—
15C2×#14523232C2212yes—
16C2×#14523232C2211yescharacteristic
17C22×C127⋊C8C2212yes—
18C22×C127⋊C8C2211yescharacteristic
19#16861207—21no—
20#16861196—21no—
21#16861190—22no—
22#16861188C2214yes—
23#16861168C2211yescharacteristic
24#16861142C2212yes—
25#16861140C2212yes—
26#16861133C2214yes—
27#16861132C2214yes—
28C2×#41×C127C2211yescharacteristic
29C22×Q8×C127C2211yescharacteristic
30C4⋊Q8×C127—21no—
31#124×C127—21no—
32#123×C127—22no—
33C22⋊Q8×C127C2214yes—
34#117×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254—22no—
37C2×Dic254—22no—
38C2×Dic254C2312yes—
39C22×Dic127—21no—
40#14523238—24no—
41#14523238—24no—
42#14523232—24no—
43#14523232—22no—
44C2×C127⋊C8—24no—
45C2×C127⋊C8D414yes—
46C2×C127⋊C8C2312yes—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127C2312yes—
50Dic127⋊C4—24no—
51C22⋊Dic127—24no—
52C22⋊Dic127—21no—
53C4×Dic127—21no—
54#41×C127—22no—
55#41×C127—22no—
56C2×Q8×C127—21no—
57C2×Q8×C127—22no—
58C2×Q8×C127—22no—
59C2×Q8×C127D411yescharacteristic
60C2×Q8×C127C2312yes—
61C2×D4×C127D412yes—
62C2×D4×C127D411yescharacteristic
63C22×C4×C127—21no—
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—24no—
67C4⋊C4×C127C2312yes—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—24no—
70C42×C127—21no—
71Dic254—22no—
72Dic254—22no—
73Dic254—42no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—42no—
77C127⋊C8—22no—
78C127⋊C8C2×D412yes—
79C127⋊C8C2×D412yes—
80C23×C127C2×D412yes—
81C23×C127C2×D411yescharacteristic
82Q8×C127—22no—
83Q8×C127—22no—
84Q8×C127—22no—
85Q8×C127—42no—
86D4×C127—24no—
87D4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127#4112yes—
97C2×C4×C127C2411yescharacteristic
98C22×C127—22no—
99C22×C127—22no—
100C22×C127—24no—
101C22×C127#12112yes—
102C22×C127C22×D411yescharacteristic
103C4×C127—22no—
104C4×C127—42no—
105C4×C127—42no—
106C4×C127#12112yes—
107C4×C127C22×D411yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—42no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127—42no—
113C2×C127C2×#12111yescharacteristic
114C2×C127#57812yes—
115#3059—1271nomaximal, Sylow
116C127#305911yescharacteristic, complemented, Sylow
117C22×Q16—1271no—
118C2×#131—1271no—
119#536—1271no—
120#536D12711yescharacteristic
121#470—1274no—
122#467—1274no—
123#427—1271no—
124#419—1271no—
125#419—1271no—
126C2×#41C2×D12711yescharacteristic
127C22×Q8—1271no—
128C22×Q8C2×D12711yescharacteristic
129C2×Q16—1272no—
130C2×Q16—1272no—
131C2×Q16—2544no—
132#131—2544no—
133C2×M16—1271no—
134C2×M16—1272no—
135C22×C8—1271no—
136C22×C8—1272no—
137C4⋊Q8—21no—
138C4⋊Q8—2541no—
139#124—21no—
140#124—2541no—
141#123—22no—
142#123—2542no—
143C22⋊Q8—1274no—
144C22⋊Q8C2×D12714yes—
145#117—1271no—
146#117C2×D12711yescharacteristic
147#107—1272no—
148#106—1272no—
149#103—1274no—
150#103—1274no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1272no—
157C2×Q8—2542no—
158C2×Q8—2542no—
159C2×Q8—2541no—
160C2×Q8C22⋊D12711yescharacteristic
161C2×Q8C22×D12712yes—
162C2×D4C22⋊D12712yes—
163C2×D4C22⋊D12711yescharacteristic
164C22×C4—21no—
165C22×C4—2541no—
166C22×C4C22×D12712yes—
167C22×C4C22×D12711yescharacteristic
168Q16—2544no—
169Q16—2544no—
170M16—2544no—
171M16—2542no—
172C2×C8—1274no—
173C2×C8—1271no—
174C2×C8—1271no—
175C2×C8—1272no—
176C2×C8—2544no—
177C4⋊C4—24no—
178C4⋊C4—1272no—
179C4⋊C4—2544no—
180C4⋊C4C22×D12712yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—2541no—
184C22⋊C4—2544no—
185C42—21no—
186C42—2541no—
187C23C2×C22⋊D12712yes—
188C23C2×C22⋊D12711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—5082no—
194Q8—2542no—
195Q8—2542no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—5082no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4D4×D12712yes—
207C2×C4C2×C22⋊D12712yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4#1452324712yes—
210C2×C4C23×D12711yescharacteristic
211C8—1272no—
212C8—1272no—
213C8—2542no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1686118112yes—
218C22C22×C22⋊D12711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—5082no—
223C4—5082no—
224C4#1686118112yes—
225C4C2×D4×D12711yescharacteristic
226C4C2×#1452324711yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1686118111yescharacteristic
230C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenter#536D127
hypocenterC127#3059
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
4. #22550132C2semidirect
5. #22550115C2semidirect
6. #22549912C2semidirect
7. #22549885C2semidirect
8. #22549884C2semidirect
116. C127#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×#168612162
#225504472
#225501322
#225501152
#225499122
#225498852
#225498842
#536×C1272
#3059127