Finite Group Databasebeta

← #30639232

Subgroups — #30639232

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#22550676C211yesmaximal, characteristic
2#22550344C211yesmaximal, characteristic, complemented
3#22549892C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523248C2211yescharacteristic
10#16861242—21no—
11#16861242C412yescomplemented
12#16861237—21no—
13#16861237C412yescomplemented
14#16861231—22no—
15#16861174—21no—
16#16861171—21no—
17(C22⋊D127)⋊C4—22no—
18#16861168C2211yescharacteristic
19#16861166—21no—
20C2×C4×Dic127C2211yescharacteristic
21C42×D127—21no—
22#16861135—22no—
23#16861135—21no—
24#16861135—21no—
25#16861123—22no—
26C2×M16×C127C2211yescharacteristic
27C2×M16×C127C2211yescharacteristic
28#117×C127C2211yescharacteristic
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248C2×C412yes—
36#14523248D412yes—
37C2×Dic254C2×C411yescharacteristic, complemented
38C2×Dic254D411yescharacteristic
39C2×D508C2×C411yescharacteristic, complemented
40C2×D508D411yescharacteristic
41C2×C22⋊D127—21no—
42C2×C22⋊D127—21no—
43C22×Dic127—21no—
44C2×C4×D127—22no—
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47Dic508—22no—
48#14523237—22no—
49#14523237—22no—
50D1016—22no—
51C4⋊Dic127—21no—
52Dic127⋊C4—22no—
53C22⋊Dic127—21no—
54(C2×D127)⋊C4—22no—
55C4×Dic127—22no—
56C4×Dic127—21no—
57C4×Dic127—21no—
58C4×Dic127D411yescharacteristic
59C4×Dic127D411yescharacteristic
60C22×C4×C127C2311yescharacteristic
61M16×C127—21no—
62M16×C127—21no—
63M16×C127—21no—
64M16×C127C2×C412yescomplemented
65C2×C8×C127—21no—
66C2×C8×C127—21no—
67C4⋊C4×C127—21no—
68C22⋊C4×C127D412yes—
69C42×C127—21no—
70C22×D127—21no—
71C22×D127—21no—
72Dic254—21no—
73Dic254—21no—
74Dic254—21no—
75Dic254C22⋊C412yes—
76D508—21no—
77D508—21no—
78D508—21no—
79D508C22⋊C412yes—
80C22⋊D127—22no—
81C22⋊D127—22no—
82C22⋊D127—41no—
83C22⋊D127—41no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—42no—
89C2×Dic127—41no—
90C4×D127—22no—
91C4×D127—22no—
92C4×D127—44no—
93C4×D127—41no—
94C4×D127—41no—
95C23×C127C2×D411yescharacteristic
96C2×C4×C127—22no—
97C2×C4×C127C22×C411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C8×C127—22no—
104C8×C127—41no—
105C22×C127—41no—
106C22×C127C4×D411yescharacteristic
107C22×C127#12011yescharacteristic
108C22×C127#12111yescharacteristic
109C2×D127—22no—
110C2×D127—42no—
111C2×D127—41no—
112C2×D127—81no—
113C4×C127—42no—
114C4×C127C2×C22⋊C411yescharacteristic
115C4×C127C4×D411yescharacteristic
116C4×C127#12111yescharacteristic
117C4×C127#12311yescharacteristic
118Dic127—22no—
119Dic127—22no—
120Dic127—41no—
121Dic127—41no—
122Dic127—81no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127#38611yescharacteristic
127D127—42no—
128D127—81no—
129#1906—1271nomaximal, Sylow
130C127#190611yescharacteristic, complemented, Sylow
131#575—1271no—
132C4×#41—1271no—
133#421—1271no—
134C2×#104—1271no—
135#410—1271no—
136#344D12711yescharacteristic, complemented
137#339—1271no—
138C2×#41—1271no—
139C2×#41—1271no—
140#137—1272no—
141#137—2541no—
142AGL(1,ℤ/8)—1272no—
143AGL(1,ℤ/8)—2541no—
144#135—2542no—
145C2×M16C2×D12711yescharacteristic
146C2×M16C2×D12711yescharacteristic
147C4×Q8—2541no—
148C4×D4—2541no—
149C4×D4—2542no—
150#117—1271no—
151#117—2541no—
152#117C2×D12711yescharacteristic
153C2×C42—1271no—
154C2×C42—2541no—
155#104—2541no—
156#104—2541no—
157#104—2542no—
158#99—2542no—
159#41—1272no—
160#41—1272no—
161#41—2541no—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165#41—2541no—
166#41—2541no—
167C2×Q8—1271no—
168C2×Q8—1271no—
169C2×D4—1271no—
170C2×D4—1271no—
171C2×D4—2541no—
172C2×D4—2541no—
173C22×C4—2542no—
174C22×C4—2541no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178Q16—2542no—
179SD16—2542no—
180SD16—2542no—
181D8—2542no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C4×D12712yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—2542no—
190C4⋊C4—2541no—
191C22⋊C4—2542no—
192C22⋊C4—2541no—
193C22⋊C4D50812yes—
194C42—21no—
195C42—1271no—
196C42—1271no—
197C42—2542no—
198C42—2541no—
199C42—2541no—
200C23—2541no—
201C23—2541no—
202C23C2×D50811yescharacteristic
203Q8—1272no—
204Q8—2541no—
205Q8—2541no—
206Q8—2541no—
207D4—1272no—
208D4—5081no—
209D4—5081no—
210D4—2542no—
211D4—2542no—
212D4—2541no—
213D4—2541no—
214D4—2541no—
215C2×C4—22no—
216C2×C4—5084no—
217C2×C4—5081no—
218C2×C4—5082no—
219C2×C4—5081no—
220C2×C4—5081no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4C2×C4×D12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4#1452324811yescharacteristic
232C2×C4#1452324911yescharacteristic
233C8—22no—
234C8—41no—
235C22—41no—
236C22—10161no—
237C22—5082no—
238C22—5081no—
239C22—2542no—
240C22C4×D50811yescharacteristic
241C22#1686117911yescharacteristic
242C22#1686118411yescharacteristic
243C4—42no—
244C4—10161no—
245C4—5081no—
246C4—5081no—
247C4—2542no—
248C4—2542no—
249C4C22⋊C4×D12711yescharacteristic
250C4#1686117111yescharacteristic
251C4#1686118211yescharacteristic
252C4#1686119411yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—10161no—
257C2—5082no—
258C2#2254975411yescharacteristic
subgroupG/H
centerC4#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127#386
prime radical#22550676C2
cosocleC22×C4×C127C23
hypercenter#344D127
hypocenterC127#1906
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #22550344C2semidirect
3. #22549892C2semidirect
4. C2×#16861135C2semidirect
5. #22549849C2semidirect
6. #22549637C2semidirect
7. #344×C127C2semidirect
11. #16861242C4semidirect
13. #16861237C4semidirect
37. C2×Dic254C2×C4semidirect
39. C2×D508C2×C4semidirect
64. M16×C127C2×C4semidirect
130. C127#1906semidirect
136. #344D127semidirect
185. M16C4×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503442
#225498922
C2×#168611352
#225498492
#225496372
#344×C1272
#1906127