Finite Group Databasebeta

← #30639972

Subgroups — #30639972

Order 16256.

318 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#16861197C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic, complemented
3C2×#16861134C211yesmaximal, characteristic, complemented
4C2×#16861131C211yesmaximal, characteristic
5C2×#16861131C211yesmaximal, characteristic
6#22549774C211yesmaximal, characteristic, complemented
7#336×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861197—22no—
11#16861197—22no—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17#16861134—22no—
18#16861134—22no—
19#16861131—22no—
20#16861131—22no—
21C127⋊#95—21no—
22C22×C8×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24C2×C42×C127C2211yescharacteristic
25C23×D127—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254D412yes—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508D412yes—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C22×Dic127—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127D412yes—
45Dic127⋊C4—42no—
46(C2×D127)⋊C4—24no—
47(C2×D127)⋊C4—24no—
48C22×C4×C127D411yescharacteristic
49C22×C4×C127D411yescharacteristic
50C22×C4×C127C2311yescharacteristic
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C42×C127—21no—
58C42×C127—21no—
59C22×D127—22no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66Dic254—22no—
67Dic254—22no—
68Dic254—41no—
69Dic254—41no—
70D508—22no—
71D508—22no—
72D508—41no—
73D508—41no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—41no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C23×C127C2×D411yescharacteristic
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C8×C127—41no—
97C8×C127—41no—
98C22×C127#12011yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12411yescharacteristic
101C22×C127#13511yescharacteristic
102C22×C127#13511yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C22×C127#13711yescharacteristic
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—81no—
109C2×D127—81no—
110C2×D127—81no—
111C2×D127—81no—
112C4×C127—41no—
113C4×C127—41no—
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12311yescharacteristic
117C4×C127#12311yescharacteristic
118Dic127—42no—
119Dic127—42no—
120Dic127—81no—
121C2×C127#39411yescharacteristic
122C2×C127#44911yescharacteristic
123C2×C127#45211yescharacteristic
124C2×C127#46311yescharacteristic
125C2×C127#46411yescharacteristic
126C2×C127#48811yescharacteristic
127C2×C127#48911yescharacteristic
128D127—82no—
129#2042—1271nomaximal, Sylow
130C127#204211yescharacteristic, complemented, Sylow
131C2×#124—1271no—
132C2×#103—1271no—
133C2×#103—1271no—
134C2×#102—1271no—
135C2×#102—1271no—
136#389—1271no—
137#336D12711yescharacteristic, complemented
138C22×Q8—1271no—
139C22×D4—1271no—
140C22×C8C2×D12711yescharacteristic
141C22×C8C2×D12711yescharacteristic
142#124—2542no—
143#124—2542no—
144C2×C4⋊C4—1271no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4—2541no—
147C2×C22⋊C4—2541no—
148C2×C22⋊C4—2541no—
149C2×C42C2×D12711yescharacteristic
150#103—2542no—
151#103—2542no—
152#102—2542no—
153#102—2542no—
154#95—2541no—
155C24—2541no—
156C2×Q8—1272no—
157C2×Q8—2541no—
158C2×Q8—2541no—
159C2×Q8—2541no—
160C2×Q8—2541no—
161C2×Q8—2541no—
162C2×D4—1272no—
163C2×D4—2541no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C22×C4—2541no—
169C22×C4—2541no—
170C22×C4—2541no—
171C22×C4D50811yescharacteristic
172C22×C4D50811yescharacteristic
173C22×C4C22×D12711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C4⋊C4—1272no—
181C4⋊C4—5082no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C42—21no—
188C42—21no—
189C23—5081no—
190C23—5081no—
191C23—5081no—
192C23—5081no—
193C23—5081no—
194C23—5081no—
195C23—2542no—
196C23C2×D50811yescharacteristic
197Q8—5081no—
198Q8—5081no—
199Q8—2542no—
200Q8—2542no—
201D4—5081no—
202D4—5081no—
203D4—2542no—
204D4—2542no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—2542no—
219C2×C4—2542no—
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4D4×D12711yescharacteristic
223C2×C4#1452324811yescharacteristic
224C2×C4#1452324811yescharacteristic
225C2×C4#1452324911yescharacteristic
226C8—41no—
227C8—41no—
228C22—10161no—
229C22—10161no—
230C22—10161no—
231C22—10161no—
232C22—5082no—
233C22—5082no—
234C22—5082no—
235C22#1686117911yescharacteristic
236C22#1686118411yescharacteristic
237C22#1686119711yescharacteristic
238C22#1686123111yescharacteristic
239C22#1686123111yescharacteristic
240C22#1686123711yescharacteristic
241C22#1686124211yescharacteristic
242C4—41no—
243C4—41no—
244C4—10161no—
245C4—5082no—
246C4—5082no—
247C4#1686118211yescharacteristic
248C4#1686118211yescharacteristic
249C4#1686119411yescharacteristic
250C4#1686119411yescharacteristic
251C2—10162no—
252C2#2254979411yescharacteristic
253C2#2255002311yescharacteristic
254C2#2255004011yescharacteristic
255C2#2255009211yescharacteristic
256C2#2255009911yescharacteristic
257C2#2255023111yescharacteristic
258C2#2255023611yescharacteristic
subgroupG/H
centerC23C2×D508
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#336×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenter#336D127
hypocenterC127#2042
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
2. C2×#16861134C2semidirect
3. C2×#16861134C2semidirect
6. #22549774C2semidirect
7. #336×C127C2semidirect
130. C127#2042semidirect
137. #336D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611972
C2×#168611342
C2×#168611312
#225497742
#336×C1272
#2042127