Finite Group Databasebeta

← #30639112

Subgroups — #30639112

Order 16256.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#22550642C211yesmaximal, characteristic, complemented
2#22550426C211yesmaximal, characteristic
3#22549894C211yesmaximal, characteristic, complemented
4#22549892C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523248C2211yescharacteristic
10#16861217—22no—
11#16861215—22no—
12C2×#14523232C2211yescharacteristic
13M16×D127—22no—
14#16861209C411yescharacteristic, complemented
15#16861206C411yescharacteristic, complemented
16#16861197C411yescharacteristic, complemented
17#16861197C411yescharacteristic, complemented
18#16861183—22no—
19C4×D508—22no—
20#16861168C2211yescharacteristic
21#16861137—22no—
22#16861135—22no—
23#16861123—22no—
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254C2×C411yescharacteristic
36C2×Dic254C2×C411yescharacteristic
37C2×D508—22no—
38C2×D508C2×C411yescharacteristic
39C2×D508C2×C411yescharacteristic
40C2×C22⋊D127—21no—
41C2×C22⋊D127D412yes—
42C2×C4×D127—21no—
43C2×C4×D127D412yes—
44#14523232—21no—
45#14523232D412yescomplemented
46#14523231—24no—
47C2×C127⋊C8—21no—
48C8×D127—24no—
49C4⋊Dic127—22no—
50C4⋊Dic127—21no—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—21no—
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57M16×C127—21no—
58M16×C127D412yescomplemented
59C2×C8×C127—21no—
60C4⋊C4×C127—21no—
61C22⋊C4×C127—21no—
62C42×C127—21no—
63C42×C127—21no—
64C42×C127C2×C411yescharacteristic
65C42×C127C2×C411yescharacteristic
66C22×D127—21no—
67C22×D127C22⋊C412yes—
68Dic254—21no—
69Dic254—21no—
70Dic254—21no—
71Dic254—21no—
72D508—21no—
73D508—21no—
74D508—21no—
75D508—21no—
76D508—44no—
77C22⋊D127—22no—
78C22⋊D127—22no—
79C22⋊D127—41no—
80C22⋊D127—41no—
81C2×Dic127—21no—
82C2×Dic127—41no—
83C2×Dic127C22⋊C412yes—
84C4×D127—22no—
85C4×D127—22no—
86C4×D127—41no—
87C4×D127—41no—
88C127⋊C8—22no—
89C23×C127C2×D411yescharacteristic
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—42no—
93C2×C4×C127—41no—
94C2×C4×C127C22×C411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—22no—
101C22×C127—41no—
102C22×C127C2×C22⋊C411yescharacteristic
103C22×C127#12111yescharacteristic
104C22×C127#12111yescharacteristic
105C2×D127—22no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—41no—
109C4×C127—22no—
110C4×C127—41no—
111C4×C127—81no—
112C4×C127C4×D411yescharacteristic
113C4×C127C4×D411yescharacteristic
114C4×C127#12011yescharacteristic
115C4×C127#12311yescharacteristic
116Dic127—22no—
117Dic127—41no—
118Dic127—81no—
119C2×C127—21no—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127#38611yescharacteristic
123D127—42no—
124D127—81no—
125#1884—1271nomaximal, Sylow
126C127#188411yescharacteristic, complemented, Sylow
127#568—1271no—
128#532—1271no—
129#421—1271no—
130#421—1271no—
131#410—1271no—
132#339—1271no—
133#339D12711yescharacteristic, complemented
134C2×#41—1271no—
135C2×#41—1271no—
136#131—2542no—
137#131—2542no—
138C2×M16—1271no—
139C2×M16—2542no—
140C2×M16C2×D12711yescharacteristic
141C4⋊Q8—1271no—
142#127—1271no—
143#124—1271no—
144#124—1271no—
145#121—2542no—
146C4×D4—2542no—
147#117—1271no—
148#117C2×D12711yescharacteristic
149C2×C42C2×D12711yescharacteristic
150#104—2542no—
151#104—2542no—
152#99—2542no—
153#41—2541no—
154#41—2541no—
155#41—2541no—
156#41—2541no—
157#41—2541no—
158#41—2541no—
159#41—2541no—
160#41—2541no—
161C2×Q8—1271no—
162C2×Q8—1271no—
163C2×D4—1272no—
164C2×D4—1271no—
165C2×D4—1271no—
166C2×D4—2542no—
167C2×D4—2541no—
168C22×C4—21no—
169C22×C4—1272no—
170C22×C4—2541no—
171C22×C4C22×D12711yescharacteristic
172M16—21no—
173M16—1272no—
174M16—2544no—
175M16—2541no—
176M16D50812yescomplemented
177C2×C8—21no—
178C2×C8—2544no—
179C2×C8—2541no—
180C4⋊C4—21no—
181C4⋊C4—2542no—
182C4⋊C4—2541no—
183C22⋊C4—21no—
184C22⋊C4—2542no—
185C22⋊C4—2542no—
186C22⋊C4—2541no—
187C42—21no—
188C42—21no—
189C42—2541no—
190C42C4×D12711yescharacteristic
191C42C4×D12711yescharacteristic
192C23—1272no—
193C23—2541no—
194C23D4×D12711yescharacteristic
195Q8—2541no—
196Q8—2541no—
197Q8—2541no—
198Q8—2541no—
199D4—5084no—
200D4—5081no—
201D4—5081no—
202D4—2542no—
203D4—2542no—
204D4—2541no—
205D4—2541no—
206D4—2541no—
207D4—2541no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—1272no—
213C2×C4—5081no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—2541no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4C2×C4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×D50811yescharacteristic
223C2×C4#1452324811yescharacteristic
224C2×C4#1452324911yescharacteristic
225C8—22no—
226C8—2542no—
227C22—41no—
228C22—5082no—
229C22—5082no—
230C22—5081no—
231C22—2542no—
232C22C22⋊C4×D12711yescharacteristic
233C22#1686118211yescharacteristic
234C22#1686118411yescharacteristic
235C4—22no—
236C4—41no—
237C4—81no—
238C4—10161no—
239C4—5081no—
240C4—2542no—
241C4#1686117111yescharacteristic
242C4C4×D50811yescharacteristic
243C4#1686117911yescharacteristic
244C4#1686119411yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—10161no—
249C2—5082no—
250C2#2254975411yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC2×C127#386
prime radical#22550426C2
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1884
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #22550642C2semidirect
3. #22549894C2semidirect
4. #22549892C2semidirect
5. #22549849C2semidirect
6. #22549636C2semidirect
7. #339×C127C2semidirect
14. #16861209C4semidirect
15. #16861206C4semidirect
16. #16861197C4semidirect
17. #16861197C4semidirect
45. #14523232D4semidirect
58. M16×C127D4semidirect
126. C127#1884semidirect
133. #339D127semidirect
176. M16D508semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504262
#225498942
#225498922
#225498492
#225496362
#339×C1272
#1884127