Finite Group Databasebeta

← #30638667

Subgroups — #30638667

Order 16256.

612 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14523247C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic, complemented
3C2×#16861132C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic, complemented
5C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
6#22549733C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—22no—
9C2×#14523247—22no—
10C2×#14523247—22no—
11C2×#14523247—22no—
12C2×#14523247—22no—
13C2×#14523247C414yescomplemented
14C22×Dic254C2211yescharacteristic
15C22×C22⋊D127—21no—
16C23×Dic127—21no—
17C22×C4×D127C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4—21no—
21#16861134—22no—
22#16861132—22no—
23#16861130—22no—
24(C2×D127)⋊C8—22no—
25C127⋊#95—21no—
26C22×D4×C127C2211yescharacteristic
27C22×C8×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29#102×C127—22no—
30C23×D127D411yescharacteristic
31#14523247—28no—
32#14523247—28no—
33#14523247—28no—
34#14523247—28no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254C2×C412yes—
39C2×Dic254D412yes—
40C2×Dic254D412yes—
41C2×C22⋊D127—24no—
42C2×C22⋊D127—24no—
43C2×C22⋊D127—24no—
44C22×Dic127—24no—
45C22×Dic127—24no—
46C22×Dic127—24no—
47C22×Dic127—21no—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C2×C4×D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127—21no—
53C2×C4×D127—21no—
54C2×C4×D127C2×C412yes—
55C2×C4×D127D412yes—
56C2×C127⋊C8—21no—
57C2×C127⋊C8—21no—
58C2×C127⋊C8—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127—21no—
61(C2×D127)⋊C4—42no—
62C24×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127C2×C412yes—
67C2×D4×C127D412yes—
68C2×D4×C127D412yes—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71C2×C8×C127—21no—
72C2×C8×C127—21no—
73C2×C8×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127—21no—
76C22×D127—21no—
77C22×D127—21no—
78C22×D127—21no—
79C22×D127—21no—
80C22×D127—21no—
81C22×D127—21no—
82C22×D127C22⋊C412yes—
83Dic254—22no—
84Dic254—22no—
85Dic254—22no—
86Dic254C22⋊C414yescomplemented
87C22⋊D127—216no—
88C2×Dic127—216no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127—41no—
95C2×Dic127—41no—
96C2×Dic127—41no—
97C2×Dic127C22⋊C412yes—
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—22no—
102C127⋊C8—41no—
103C23×C127—22no—
104C23×C127—22no—
105C23×C127—22no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127C2×D411yescharacteristic
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—22no—
114D4×C127C22⋊C414yescomplemented
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127C22×C411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D411yescharacteristic
124C8×C127—41no—
125C22×C127—24no—
126C22×C127—42no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127—42no—
130C22×C127—42no—
131C22×C127—42no—
132C22×C127C2×C22⋊C411yescharacteristic
133C22×C127#12011yescharacteristic
134C22×C127#12011yescharacteristic
135C22×C127#13511yescharacteristic
136C22×C127#13511yescharacteristic
137C22×C127AGL(1,ℤ/8)11yescharacteristic
138C22×C127#13711yescharacteristic
139C2×D127—22no—
140C2×D127—22no—
141C2×D127—22no—
142C2×D127—41no—
143C2×D127—41no—
144C2×D127—41no—
145C2×D127—41no—
146C4×C127—81no—
147C4×C127C2×C22⋊C411yescharacteristic
148C4×C127C2×C22⋊C411yescharacteristic
149C4×C127#12011yescharacteristic
150C4×C127#12011yescharacteristic
151Dic127—22no—
152Dic127—24no—
153Dic127—81no—
154C2×C127—44no—
155C2×C127#37911yescharacteristic
156C2×C127#41611yescharacteristic
157C2×C127#41711yescharacteristic
158C2×C127#44911yescharacteristic
159C2×C127#44911yescharacteristic
160C2×C127#45211yescharacteristic
161C2×C127#45211yescharacteristic
162D127—42no—
163#1788—1271nomaximal, Sylow
164C127#178811yescharacteristic, complemented, Sylow
165C22×#41—1271no—
166C2×#103—1271no—
167C2×#103—1271no—
168C2×#102—1271no—
169C2×#102D12711yescharacteristic, complemented
170C2×C22⋊C8—1271no—
171#380—1271no—
172C2×#41—1274no—
173C2×#41—2542no—
174C2×#41—2542no—
175C2×#41—2542no—
176C2×#41—2542no—
177C2×#41—2542no—
178C22×Q8—1271no—
179C22×D4—2541no—
180C22×D4C2×D12711yescharacteristic
181C23×C4—1271no—
182C23×C4—2541no—
183C22×C8—1271no—
184C22×C8C2×D12711yescharacteristic
185C2×C4⋊C4—1271no—
186C2×C4⋊C4C2×D12711yescharacteristic
187C2×C22⋊C4—2541no—
188#103—2542no—
189#103—2542no—
190#102—22no—
191#102—2542no—
192C22⋊C8—2542no—
193#95—2541no—
194C24—21no—
195C24—1271no—
196#41—2548no—
197#41—2548no—
198#41—2548no—
199#41—2548no—
200C2×Q8—1272no—
201C2×Q8—1272no—
202C2×Q8—1272no—
203C2×Q8—2541no—
204C2×Q8—2541no—
205C2×Q8—2541no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—2544no—
210C2×D4—2544no—
211C2×D4—2544no—
212C2×D4C4×D12712yes—
213C2×D4C22⋊D12712yes—
214C2×D4D50812yes—
215C22×C4—21no—
216C22×C4—1271no—
217C22×C4—1272no—
218C22×C4—1272no—
219C22×C4—2544no—
220C22×C4—2541no—
221C22×C4—2544no—
222C22×C4—2544no—
223C22×C4—2541no—
224C22×C4—2541no—
225C22×C4—2541no—
226C22×C4—2541no—
227C22×C4—2541no—
228C22×C4C22×D12711yescharacteristic
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—2541no—
233C2×C8—2541no—
234C2×C8—2541no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—2541no—
238C4⋊C4—2541no—
239C22⋊C4—5082no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—1272no—
248C23—2541no—
249C23—2541no—
250C23—2541no—
251C23—2541no—
252C23—2541no—
253C23—2541no—
254C23D4×D12711yescharacteristic
255Q8—1274no—
256Q8—2542no—
257Q8—2542no—
258Q8—2542no—
259D4—22no—
260D4—22no—
261D4—22no—
262D4—25416no—
263D4(C2×D127)⋊C414yescomplemented
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—1272no—
268C2×C4—5081no—
269C2×C4—5081no—
270C2×C4—5081no—
271C2×C4—25416no—
272C2×C4—2542no—
273C2×C4—2541no—
274C2×C4—2542no—
275C2×C4—2541no—
276C2×C4—2542no—
277C2×C4—2542no—
278C2×C4—2542no—
279C2×C4—2542no—
280C2×C4—2542no—
281C2×C4C2×C4×D12711yescharacteristic
282C2×C4D4×D12711yescharacteristic
283C2×C4D4×D12711yescharacteristic
284C2×C4D4×D12711yescharacteristic
285C2×C4C2×C22⋊D12711yescharacteristic
286C2×C4C2×D50811yescharacteristic
287C8—41no—
288C8—5081no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—5081no—
297C22—5081no—
298C22—5081no—
299C22—5081no—
300C22—2542no—
301C22—2542no—
302C22—2542no—
303C22C22⋊C4×D12711yescharacteristic
304C22#1686117711yescharacteristic
305C22#1686117911yescharacteristic
306C22#1686122811yescharacteristic
307C22#1686123011yescharacteristic
308C22#1686123311yescharacteristic
309C22#1686123811yescharacteristic
310C4—81no—
311C4—10161no—
312C4—2542no—
313C4—2544no—
314C4C22⋊C4×D12711yescharacteristic
315C4C2×(C2×D127)⋊C411yescharacteristic
316C4#1686117711yescharacteristic
317C4#1686117911yescharacteristic
318C2—44no—
319C2—5082no—
320C2#2254973111yescharacteristic
321C2#2254986911yescharacteristic
322C2#2254987411yescharacteristic
323C2#2255001811yescharacteristic
324C2#2255002211yescharacteristic
325C2#2255003511yescharacteristic
326C2#2255003911yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523247C2
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1788
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861134C2semidirect
3. C2×#16861132C2semidirect
4. C2×#16861130C2semidirect
5. C2×(C2×D127)⋊C8C2semidirect
6. #22549733C2semidirect
7. C2×#102×C127C2semidirect
13. C2×#14523247C4semidirect
86. Dic254C22⋊C4semidirect
114. D4×C127C22⋊C4semidirect
164. C127#1788semidirect
169. C2×#102D127semidirect
263. D4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611342
C2×#168611322
C2×#168611302
C2×(C2×D127)⋊C82
#225497332
C2×#102×C1272
#1788127