Finite Group Databasebeta

← #30648027

Subgroups — #30648027

Order 16256.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861256C211yesmaximal, characteristic
2C2×#16861242C211yesmaximal, characteristic, complemented
3C22×Dic508C211yesmaximal, characteristic, complemented
4#22550447C211yesmaximal, characteristic, complemented
5#22550040C214yesmaximal, complemented
6#22550034C214yesmaximal, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549873C211yesmaximal, characteristic, complemented
9#408×C127C211yesmaximal, characteristic, complemented
10#16861256—24no—
11#16861256—24no—
12C2×#14523248C2212yes—
13C2×#14523248C2211yescharacteristic
14C2×#14523247—22no—
15C2×#14523247—21no—
16C22×Dic254C2212yes—
17C22×Dic254C2211yescharacteristic
18C22×Dic254C2211yescharacteristic
19C2×Q8×D127—21no—
20#16861242—24no—
21C2×Dic508—24no—
22C2×Dic508C2212yes—
23C2×Dic508C2214yes—
24C2×#14523237C2212yes—
25#16861207—21no—
26#16861196—21no—
27#16861190—22no—
28#16861188C2214yes—
29#16861168C2211yescharacteristic
30#16861134C2212yes—
31#16861134C2214yes—
32#16861131C2212yes—
33C2×#41×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35C22×C8×C127C2211yescharacteristic
36C22⋊C8×C127C2214yes—
37#14523248—24no—
38#14523248—24no—
39#14523248—22no—
40#14523248D414yes—
41#14523247—216no—
42#14523247—28no—
43C2×Dic254—24no—
44C2×Dic254—24no—
45C2×Dic254—22no—
46C2×Dic254—21no—
47C2×Dic254—22no—
48C2×Dic254—21no—
49C2×Dic254—22no—
50C2×Dic254—22no—
51C2×Dic254D414yes—
52C2×Dic254C2312yes—
53C2×Dic254C2311yescharacteristic
54C2×Dic254C2312yes—
55Q8×D127—28no—
56C2×D508C2311yescharacteristic
57C2×C22⋊D127—22no—
58C2×C22⋊D127—21no—
59C22×Dic127—22no—
60C22×Dic127—21no—
61C22×Dic127—21no—
62C2×C4×D127—22no—
63C2×C4×D127—21no—
64Dic508—28no—
65Dic508—24no—
66#14523237—24no—
67C4⋊Dic127C2312yes—
68Dic127⋊C4—24no—
69C22⋊Dic127—24no—
70C22⋊Dic127—21no—
71C4×Dic127—21no—
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127D411yescharacteristic
75C2×D4×C127D412yes—
76C2×D4×C127D411yescharacteristic
77C22×C4×C127C2312yes—
78C22×C4×C127C2311yescharacteristic
79M16×C127—22no—
80C2×C8×C127—22no—
81C2×C8×C127C2312yes—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84Dic254—28no—
85Dic254—24no—
86Dic254—21no—
87Dic254—22no—
88Dic254—22no—
89Dic254—22no—
90Dic254—42no—
91Dic254C2×D412yes—
92Dic254C2×D414yes—
93D508—21no—
94D508C2×D412yes—
95C22⋊D127—28no—
96C22⋊D127—24no—
97C2×Dic127—28no—
98C2×Dic127—24no—
99C2×Dic127—22no—
100C2×Dic127—21no—
101C2×Dic127—22no—
102C2×Dic127—22no—
103C2×Dic127—42no—
104C4×D127—28no—
105C4×D127—24no—
106C23×C127C2×D412yes—
107C23×C127C2×D411yescharacteristic
108Q8×C127—22no—
109D4×C127—24no—
110D4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D412yes—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C8×C127—22no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—24no—
123C22×C127#12012yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—42no—
127C4×C127—22no—
128C4×C127#12012yes—
129C4×C127C22×D411yescharacteristic
130C4×C127C22×D411yescharacteristic
131Dic127—24no—
132Dic127—22no—
133Dic127—42no—
134Dic127—42no—
135C2×C127—22no—
136C2×C127—42no—
137C2×C127C2×#12011yescharacteristic
138C2×C127#57812yes—
139D127—42no—
140#3011—1271nomaximal, Sylow
141C127#301111yescharacteristic, complemented, Sylow
142C2×2−1+4—1271no—
143C2×#137—1271no—
144C22×Q16—1271no—
145#536—1271no—
146#452—1274no—
147#451—1274no—
148#419—1271no—
149#416—1271no—
150#408D12711yescharacteristic, complemented
1512−1+4—2544no—
1522−1+4—2544no—
153C2×#41—1271no—
154C2×#41—1272no—
155C2×#41—2542no—
156C2×#41—2541no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8—1271no—
159C22×Q8—1271no—
160C22×Q8—1272no—
161C22×Q8—2541no—
162#137—2544no—
163C2×Q16—1274no—
164C2×Q16—1272no—
165C2×Q16—2544no—
166C2×SD16—1272no—
167C2×M16C2×D12711yescharacteristic
168C22×C8C2×D12711yescharacteristic
169C4⋊Q8—2541no—
170#124—2541no—
171#123—2542no—
172C22⋊Q8—1274no—
173#117—1271no—
174#103—1272no—
175#103—1274no—
176#102—1272no—
177C22⋊C8C2×D12714yes—
178#41—22no—
179#41—22no—
180#41—1274no—
181#41—2544no—
182#41—2544no—
183#41—25416no—
184#41—2548no—
185#41—2542no—
186C2×Q8—1274no—
187C2×Q8—1272no—
188C2×Q8—1271no—
189C2×Q8—1272no—
190C2×Q8—2544no—
191C2×Q8—2544no—
192C2×Q8—2542no—
193C2×Q8—2542no—
194C2×Q8—2548no—
195C2×Q8—2542no—
196C2×Q8—2541no—
197C2×Q8—2542no—
198C2×Q8—2541no—
199C2×Q8D50811yescharacteristic
200C2×D4—1271no—
201C2×D4—2542no—
202C2×D4—2541no—
203C2×D4D50812yes—
204C2×D4D50811yescharacteristic
205C22×C4—2541no—
206C22×C4—2542no—
207C22×C4—2541no—
208C22×C4—2542no—
209C22×C4—2541no—
210C22×C4C22×D12712yes—
211C22×C4C22×D12711yescharacteristic
212Q16—2548no—
213Q16—2544no—
214SD16—2544no—
215M16—22no—
216C2×C8—22no—
217C2×C8C22×D12712yes—
218C2×C8C22×D12712yes—
219C4⋊C4—1272no—
220C4⋊C4—2544no—
221C22⋊C4—2541no—
222C22⋊C4—2544no—
223C42—2541no—
224C23—2541no—
225C23C2×D50812yes—
226C23C2×D50811yescharacteristic
227Q8—22no—
228Q8—1274no—
229Q8—1272no—
230Q8—5082no—
231Q8—2548no—
232Q8—2542no—
233Q8—2542no—
234Q8—2542no—
235Q8—2541no—
236Q8—2544no—
237D4—24no—
238D4—22no—
239D4—1272no—
240D4—2548no—
241D4—2544no—
242D4—2541no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—5082no—
246C2×C4—2548no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2548no—
250C2×C4—2542no—
251C2×C4—2541no—
252C2×C4—2544no—
253C2×C4—2544no—
254C2×C4D4×D12712yes—
255C2×C4D4×D12712yes—
256C2×C4C2×D50812yes—
257C2×C4C2×D50811yescharacteristic
258C2×C4C23×D12711yescharacteristic
259C8—22no—
260C8—22no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—5082no—
265C22—2542no—
266C22#1686117912yes—
267C22C22×D50811yescharacteristic
268C4—22no—
269C4—5082no—
270C4—5082no—
271C4—2544no—
272C4—2542no—
273C4#1686117912yes—
274C4C2×D4×D12711yescharacteristic
275C4C2×D4×D12711yescharacteristic
276C2—22no—
277C2—42no—
278C2—5082no—
279C2C2×#1686117911yescharacteristic
280C2#2255069312yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#408×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861256C2
hypercenter#408D127
hypocenterC127#3011
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C22×Dic508C2semidirect
4. #22550447C2semidirect
5. #22550040C2semidirect
6. #22550034C2semidirect
7. #22549886C2semidirect
8. #22549873C2semidirect
9. #408×C127C2semidirect
141. C127#3011semidirect
150. #408D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612422
C22×Dic5082
#225504472
#225500402
#225500342
#225498862
#225498732
#408×C1272
#3011127