Finite Group Databasebeta

← #30642075

Subgroups — #30642075

Order 16256.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523248C211yesmaximal, characteristic, complemented
2C2×#16861188C211yesmaximal, characteristic, complemented
3C2×#16861183C211yesmaximal, characteristic, complemented
4#22549791C212yesmaximal, complemented
5#22549784C212yesmaximal, complemented
6#22549782C211yesmaximal, characteristic, complemented
7#22549778C211yesmaximal, characteristic, complemented
8#22549752C212yesmaximal, complemented
9#22549747C212yesmaximal, complemented
10#22549736C211yesmaximal, characteristic, complemented
11C2×C4×D4×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—24no—
13C2×#14523248—24no—
14C2×#14523248—24no—
15C22×Dic254C2211yescharacteristic
16C22×D508C2211yescharacteristic
17C22×C22⋊D127C2212yes—
18C22×C22⋊D127C2212yes—
19C22×C4×D127C2212yes—
20#16861188—24no—
21#16861183—24no—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×C4⋊Dic127C2211yescharacteristic
24C2×Dic127⋊C4C2212yes—
25C2×C22⋊Dic127C2212yes—
26C2×C22⋊Dic127C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2212yes—
29C2×(C2×D127)⋊C4C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C127⋊#95C2212yes—
33C22×D4×C127C2211yescharacteristic
34C23×C4×C127C2211yescharacteristic
35C23×C4×C127C2211yescharacteristic
36C4×D4×C127—24no—
37C2×C4⋊C4×C127C2211yescharacteristic
38C2×C22⋊C4×C127C2211yescharacteristic
39C2×C22⋊C4×C127C2211yescharacteristic
40C2×C42×C127C2211yescharacteristic
41C23×D127C2312yes—
42#14523248—232no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254D414yescomplemented
46C2×D508—22no—
47C2×D508—22no—
48C2×D508D414yescomplemented
49C2×C22⋊D127—24no—
50C2×C22⋊D127—24no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127—24no—
53C2×C22⋊D127—24no—
54C2×C22⋊D127—42no—
55C22×Dic127C2312yes—
56C22×Dic127C2312yes—
57C22×Dic127C2312yes—
58C2×C4×D127—24no—
59C2×C4×D127—24no—
60C2×C4×D127—24no—
61C4⋊Dic127—22no—
62C4⋊Dic127—22no—
63Dic127⋊C4—24no—
64C22⋊Dic127—24no—
65C22⋊Dic127—42no—
66(C2×D127)⋊C4—24no—
67(C2×D127)⋊C4—24no—
68(C2×D127)⋊C4—42no—
69C24×C127C2311yescharacteristic
70C24×C127C2311yescharacteristic
71C2×D4×C127—22no—
72C2×D4×C127—22no—
73C2×D4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—22no—
79C22×C4×C127D414yescomplemented
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C4⋊C4×C127—22no—
86C22⋊C4×C127—22no—
87C22⋊C4×C127—22no—
88C42×C127—22no—
89C22×D127—22no—
90C22×D127—22no—
91C22×D127—22no—
92C22×D127—22no—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127C2×D414yes—
96Dic254—28no—
97D508—28no—
98C22⋊D127—216no—
99C22⋊D127—48no—
100C2×Dic127—22no—
101C2×Dic127—22no—
102C2×Dic127—22no—
103C2×Dic127—22no—
104C2×Dic127—22no—
105C2×Dic127—22no—
106C2×Dic127—22no—
107C2×Dic127—22no—
108C2×Dic127C2×D414yes—
109C4×D127—216no—
110C23×C127—21no—
111C23×C127—21no—
112C23×C127—21no—
113C23×C127—21no—
114C23×C127—21no—
115C23×C127—21no—
116C23×C127—21no—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127C2×D412yes—
123C23×C127#4112yescomplemented
124C23×C127C2411yescharacteristic
125D4×C127—28no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—28no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—44no—
138C2×C4×C127C2×D412yes—
139C2×C4×C127C2×D412yes—
140C2×C4×C127C2×D412yes—
141C2×C4×C127C2×D412yes—
142C2×C4×C127#4112yescomplemented
143C22×C127—22no—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—22no—
147C22×C127—22no—
148C22×C127—22no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C22×C127C22×D411yescharacteristic
159C22×C127C22×D411yescharacteristic
160C22×C127C2×#4111yescharacteristic
161C22×C127C2×#4111yescharacteristic
162C22×C127H(5,𝔽2)11yescharacteristic
163C22×C1272−1+411yescharacteristic
164C2×D127—24no—
165C2×D127—24no—
166C2×D127—24no—
167C2×D127—42no—
168C2×D127—42no—
169C2×D127—42no—
170C2×D127—42no—
171C4×C127—22no—
172C4×C127—22no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127#12014yes—
176Dic127—24no—
177Dic127—42no—
178Dic127—42no—
179C2×C127—42no—
180C2×C127—42no—
181C2×C127C2×#12011yescharacteristic
182C2×C127#52511yescharacteristic
183C2×C127#53711yescharacteristic
184C2×C127#54611yescharacteristic
185C2×C127#54711yescharacteristic
186C2×C127#54811yescharacteristic
187C2×C127#55011yescharacteristic
188D127—44no—
189#2407—1271nomaximal, Sylow
190C127#240711yescharacteristic, complemented, Sylow
191C22×#41—1271no—
192C2×C22⋊Q8—1271no—
193C2×#121—1271no—
194C2×C4×D4D12711yescharacteristic, complemented
195#394—1272no—
196#392—1272no—
197#391—1271no—
198#390—1271no—
199#386—1272no—
200#385—1272no—
201#380—1271no—
202C2×#41—2544no—
203C2×#41—2544no—
204C2×#41—2544no—
205C22×Q8—1271no—
206C22×D4—1272no—
207C22×D4—1272no—
208C22×D4—1271no—
209C22×D4C2×D12711yescharacteristic
210C23×C4—1272no—
211C23×C4C2×D12711yescharacteristic
212C23×C4C2×D12711yescharacteristic
213C22⋊Q8—2544no—
214#121—2544no—
215C4×D4—24no—
216C2×C4⋊C4—1271no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4—1272no—
219C2×C4⋊C4C2×D12711yescharacteristic
220C2×C22⋊C4—1272no—
221C2×C22⋊C4—1272no—
222C2×C22⋊C4—1272no—
223C2×C22⋊C4—1272no—
224C2×C22⋊C4—1272no—
225C2×C22⋊C4C2×D12711yescharacteristic
226C2×C22⋊C4C2×D12711yescharacteristic
227C2×C42C2×D12711yescharacteristic
228#95—1272no—
229#95—1272no—
230#95—1272no—
231C24—1272no—
232C24C22×D12711yescharacteristic
233C24C22×D12711yescharacteristic
234#41—25432no—
235C2×Q8—1274no—
236C2×Q8—2542no—
237C2×Q8—2542no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—1274no—
242C2×D4—5082no—
243C2×D4—2544no—
244C2×D4—2544no—
245C2×D4—2544no—
246C2×D4—2544no—
247C2×D4—2544no—
248C2×D4—2542no—
249C2×D4—2542no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—1272no—
256C22×C4—1272no—
257C22×C4—1272no—
258C22×C4—2544no—
259C22×C4—2544no—
260C22×C4—2544no—
261C22×C4C22⋊D12714yescomplemented
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264C22×C4C22×D12711yescharacteristic
265C22×C4C22×D12711yescharacteristic
266C22×C4C22×D12711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—2544no—
269C4⋊C4—2542no—
270C4⋊C4—2542no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—5082no—
274C22⋊C4—5082no—
275C22⋊C4—2544no—
276C22⋊C4—2544no—
277C22⋊C4—2544no—
278C42—22no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—1274no—
292C23—2542no—
293C23—2542no—
294C23—2542no—
295C23—2542no—
296C23—2542no—
297C23—2542no—
298C23C2×C22⋊D12712yes—
299C23#1452324812yescomplemented
300C23C23×D12711yescharacteristic
301Q8—2548no—
302D4—28no—
303D4—5088no—
304D4—25416no—
305D4—2548no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—28no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—1274no—
319C2×C4—25416no—
320C2×C4—2542no—
321C2×C4—2542no—
322C2×C4—2542no—
323C2×C4—2542no—
324C2×C4—2542no—
325C2×C4—2542no—
326C2×C4—2542no—
327C2×C4—2542no—
328C2×C4D4×D12712yes—
329C2×C4D4×D12712yes—
330C2×C4C2×C22⋊D12712yes—
331C2×C4C2×C22⋊D12712yes—
332C2×C4#1452324812yescomplemented
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—5082no—
348C22—5082no—
349C22—5082no—
350C22—5082no—
351C22—2544no—
352C22—2544no—
353C22—2544no—
354C22C2×D4×D12711yescharacteristic
355C22C2×D4×D12711yescharacteristic
356C22C22×C22⋊D12711yescharacteristic
357C22C2×#1452324811yescharacteristic
358C22C2×#1452324811yescharacteristic
359C22#1686125511yescharacteristic
360C22#1686125611yescharacteristic
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—5082no—
366C4—5082no—
367C4—2544no—
368C4#1686117714yes—
369C2—42no—
370C2—42no—
371C2—5084no—
372C2C2×#1686117711yescharacteristic
373C2#2255038711yescharacteristic
374C2#2255045411yescharacteristic
375C2#2255052211yescharacteristic
376C2#2255053111yescharacteristic
377C2#2255053511yescharacteristic
378C2#2255054711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×C4×D4×C127C2
hypercenterC2×C4×D4D127
hypocenterC127#2407
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14523248C2semidirect
2. C2×#16861188C2semidirect
3. C2×#16861183C2semidirect
4. #22549791C2semidirect
5. #22549784C2semidirect
6. #22549782C2semidirect
7. #22549778C2semidirect
8. #22549752C2semidirect
9. #22549747C2semidirect
10. #22549736C2semidirect
11. C2×C4×D4×C127C2semidirect
45. C2×Dic254D4semidirect
48. C2×D508D4semidirect
79. C22×C4×C127D4semidirect
123. C23×C127#41semidirect
142. C2×C4×C127#41semidirect
190. C127#2407semidirect
194. C2×C4×D4D127semidirect
261. C22×C4C22⋊D127semidirect
299. C23#14523248semidirect
332. C2×C4#14523248semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611882
C2×#168611832
#225497912
#225497842
#225497822
#225497782
#225497522
#225497472
#225497362
C2×C4×D4×C1272
#2407127