Finite Group Databasebeta

← #30642046

Subgroups — #30642046

Order 16256.

1056 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523247C211yesmaximal, characteristic, complemented
2C2×#16861178C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic127C211yesmaximal, characteristic, complemented
4#22549788C212yesmaximal, complemented
5#22549784C212yesmaximal, complemented
6#22549756C211yesmaximal, characteristic, complemented
7#22549756C212yesmaximal, complemented
8#22549753C211yesmaximal, characteristic, complemented
9#22549752C212yesmaximal, complemented
10#22549737C211yesmaximal, characteristic, complemented
11C2×#120×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—24no—
13C2×#14523247—24no—
14C2×#14523247—24no—
15C22×Dic254C2211yescharacteristic, complemented
16C22×C22⋊D127C2212yes—
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2211yescharacteristic, complemented
19C23×Dic127C2211yescharacteristic
20C23×Dic127C2211yescharacteristic
21C23×Dic127C2211yescharacteristic, complemented
22C22×C4×D127C2211yescharacteristic, complemented
23#16861178—24no—
24C2×C22⋊Dic127—24no—
25C2×C22⋊Dic127C2212yes—
26C2×C22⋊Dic127C2212yes—
27C2×C22⋊Dic127C2212yes—
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2212yescomplemented
30C2×(C2×D127)⋊C4C2212yescomplemented
31C127⋊#95C2212yescomplemented
32C127⋊#95C2212yescomplemented
33C127⋊#95C2212yescomplemented
34C127⋊#95C2212yescomplemented
35C25×C127C2211yescharacteristic
36C22×D4×C127C2212yes—
37C22×D4×C127C2211yescharacteristic
38#120×C127—24no—
39C2×C22⋊C4×C127C2212yes—
40C2×C22⋊C4×C127C2211yescharacteristic
41C23×D127C2311yescharacteristic
42#14523247—232no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254—22no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—24no—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52C2×C22⋊D127—24no—
53C2×C22⋊D127—42no—
54C2×C22⋊D127D414yescomplemented
55C22×Dic127—22no—
56C22×Dic127—22no—
57C22×Dic127—22no—
58C22×Dic127—22no—
59C22×Dic127—22no—
60C22×Dic127—22no—
61C22×Dic127—42no—
62C22×Dic127D414yescomplemented
63C22×Dic127C2312yes—
64C22×Dic127C2312yes—
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C2×C4×D127—22no—
69C2×C4×D127—22no—
70C2×C4×D127—22no—
71C22⋊Dic127—24no—
72C22⋊Dic127—24no—
73C22⋊Dic127—24no—
74C22⋊Dic127—22no—
75C22⋊Dic127—42no—
76(C2×D127)⋊C4—42no—
77C24×C127—22no—
78C24×C127—24no—
79C24×C127—22no—
80C24×C127—22no—
81C24×C127—42no—
82C24×C127C2311yescharacteristic
83C24×C127C2312yes—
84C24×C127C2311yescharacteristic
85C2×D4×C127—22no—
86C2×D4×C127—22no—
87C2×D4×C127—24no—
88C2×D4×C127—24no—
89C2×D4×C127—42no—
90C2×D4×C127D414yescomplemented
91C22×C4×C127C2312yes—
92C22×C4×C127C2311yescharacteristic
93C22⋊C4×C127—22no—
94C22⋊C4×C127—24no—
95C22×D127—21no—
96C22×D127—21no—
97C22×D127—22no—
98C22×D127—22no—
99C22×D127C2×D412yes—
100Dic254—28no—
101C22⋊D127—28no—
102C22⋊D127—28no—
103C22⋊D127—48no—
104C2×Dic127—28no—
105C2×Dic127—28no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—22no—
113C2×Dic127—22no—
114C2×Dic127—22no—
115C2×Dic127—22no—
116C2×Dic127—22no—
117C2×Dic127—22no—
118C2×Dic127—44no—
119C2×Dic127C2×D412yes—
120C2×Dic127C2×D412yes—
121C2×Dic127C2×D412yes—
122C4×D127—28no—
123C23×C127—22no—
124C23×C127—22no—
125C23×C127—28no—
126C23×C127—22no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—22no—
130C23×C127—22no—
131C23×C127—22no—
132C23×C127—22no—
133C23×C127—22no—
134C23×C127—22no—
135C23×C127—22no—
136C23×C127—48no—
137C23×C127—48no—
138C23×C127—44no—
139C23×C127—44no—
140C23×C127C2×D412yes—
141C23×C127C2×D412yes—
142C23×C127C2×D412yes—
143C23×C127C2×D412yes—
144C23×C127#4114yescomplemented
145C23×C127C2411yescharacteristic
146D4×C127—28no—
147D4×C127—48no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—22no—
151C2×C4×C127—22no—
152C2×C4×C127—22no—
153C2×C4×C127C2×D412yes—
154C22×C127—22no—
155C22×C127—22no—
156C22×C127—24no—
157C22×C127—24no—
158C22×C127—22no—
159C22×C127—24no—
160C22×C127—24no—
161C22×C127—24no—
162C22×C127—24no—
163C22×C127—416no—
164C22×C127—42no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—42no—
168C22×C127—42no—
169C22×C127—42no—
170C22×C127—42no—
171C22×C127#12014yes—
172C22×C127C22×D411yescharacteristic
173C22×C127C22×D411yescharacteristic
174C22×C127C22×D411yescharacteristic
175C22×C127C2×#4112yes—
176C22×C127H(5,𝔽2)12yes—
177C2×D127—22no—
178C2×D127—24no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—42no—
182C4×C127—22no—
183C4×C127—42no—
184Dic127—22no—
185Dic127—22no—
186Dic127—22no—
187Dic127—42no—
188Dic127—42no—
189C2×C127—24no—
190C2×C127—42no—
191C2×C127—44no—
192C2×C127C2×#12011yescharacteristic
193C2×C127#52511yescharacteristic
194C2×C127#53511yescharacteristic
195C2×C127#54612yes—
196C2×C127#54612yes—
197D127—42no—
198#2404—1271nomaximal, Sylow
199C127#240411yescharacteristic, complemented, Sylow
200C22×#41—1271no—
201C2×#120—1271no—
202C2×#120D12711yescharacteristic, complemented
203C22×C22⋊C4—1271no—
204#393—1272no—
205#392—1272no—
206#386—1272no—
207#386—1271no—
208#386—1272no—
209#386—1271no—
210#381—1271no—
211C25C2×D12711yescharacteristic
212C2×#41—2544no—
213C2×#41—2544no—
214C2×#41—2544no—
215C22×Q8—1271no—
216C22×D4—1271no—
217C22×D4—1272no—
218C22×D4—1271no—
219C22×D4C2×D12712yes—
220C22×D4C2×D12711yescharacteristic
221C23×C4—1271no—
222C23×C4—1271no—
223C23×C4—1271no—
224C23×C4—1271no—
225#120—24no—
226#120—2544no—
227C2×C22⋊C4—1272no—
228C2×C22⋊C4—1272no—
229C2×C22⋊C4—1272no—
230C2×C22⋊C4—1272no—
231C2×C22⋊C4—1271no—
232C2×C22⋊C4—1272no—
233C2×C22⋊C4—2544no—
234C2×C22⋊C4C2×D12711yescharacteristic
235C2×C22⋊C4C2×D12712yes—
236#95—1272no—
237#95—1272no—
238#95—1272no—
239#95—1272no—
240C24—22no—
241C24—24no—
242C24—22no—
243C24—22no—
244C24—42no—
245C24—1271no—
246C24C22×D12711yescharacteristic
247C24C22×D12712yes—
248C24C22×D12711yescharacteristic
249#41—25432no—
250C2×Q8—2542no—
251C2×Q8—2542no—
252C2×Q8—2542no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—42no—
258C2×D4—1274no—
259C2×D4—5082no—
260C2×D4—2542no—
261C2×D4—2544no—
262C2×D4—2544no—
263C2×D4—2542no—
264C2×D4—2542no—
265C2×D4—2542no—
266C2×D4—2542no—
267C2×D4C22⋊D12714yescomplemented
268C22×C4—1274no—
269C22×C4—1271no—
270C22×C4—1271no—
271C22×C4—1271no—
272C22×C4—1272no—
273C22×C4—1272no—
274C22×C4—5082no—
275C22×C4—2542no—
276C22×C4—2542no—
277C22×C4—2542no—
278C22×C4—2542no—
279C22×C4—2542no—
280C22×C4—2542no—
281C22×C4—2542no—
282C22×C4—2542no—
283C22×C4—2542no—
284C22×C4C22×D12712yes—
285C22×C4C22×D12711yescharacteristic
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—5082no—
289C22⋊C4—5082no—
290C22⋊C4—2544no—
291C22⋊C4—2544no—
292C22⋊C4—2544no—
293C22⋊C4—2542no—
294C23—22no—
295C23—22no—
296C23—28no—
297C23—22no—
298C23—22no—
299C23—21no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—48no—
308C23—48no—
309C23—44no—
310C23—44no—
311C23—1272no—
312C23—2541no—
313C23—2541no—
314C23—2542no—
315C23—2542no—
316C23D4×D12712yes—
317C23D4×D12712yes—
318C23C2×C22⋊D12712yes—
319C23C2×C22⋊D12712yes—
320C23#1452324714yescomplemented
321C23C23×D12711yescharacteristic
322Q8—2548no—
323D4—28no—
324D4—48no—
325D4—5088no—
326D4—2548no—
327D4—2548no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—1272no—
334C2×C4—1272no—
335C2×C4—1272no—
336C2×C4—5084no—
337C2×C4—2548no—
338C2×C4—2542no—
339C2×C4—2542no—
340C2×C4—2542no—
341C2×C4—2542no—
342C2×C4—2542no—
343C2×C4—2542no—
344C2×C4—2548no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2548no—
352C2×C4C2×C22⋊D12712yes—
353C22—22no—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—22no—
358C22—24no—
359C22—24no—
360C22—24no—
361C22—24no—
362C22—416no—
363C22—42no—
364C22—41no—
365C22—42no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—5081no—
371C22—5081no—
372C22—5082no—
373C22—2542no—
374C22—2544no—
375C22#1686117714yes—
376C22C2×D4×D12711yescharacteristic
377C22C2×D4×D12711yescharacteristic
378C22C22×C22⋊D12711yescharacteristic
379C22C2×#1452324712yes—
380C22#1686125412yes—
381C4—22no—
382C4—42no—
383C4—5082no—
384C4—5082no—
385C4—2542no—
386C4—2542no—
387C4—2542no—
388C2—24no—
389C2—44no—
390C2—42no—
391C2—5082no—
392C2C2×#1686117711yescharacteristic
393C2#2255038211yescharacteristic
394C2#2255044111yescharacteristic
395C2#2255051312yes—
396C2#2255052012yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#120×C127C2
hypercenterC2×#120D127
hypocenterC127#2404
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C2×#16861178C2semidirect
3. C22×C22⋊Dic127C2semidirect
4. #22549788C2semidirect
5. #22549784C2semidirect
6. #22549756C2semidirect
7. #22549756C2semidirect
8. #22549753C2semidirect
9. #22549752C2semidirect
10. #22549737C2semidirect
11. C2×#120×C127C2semidirect
15. C22×Dic254C22semidirect
18. C22×C22⋊D127C22semidirect
21. C23×Dic127C22semidirect
22. C22×C4×D127C22semidirect
29. C2×C22⋊Dic127C22semidirect
30. C2×(C2×D127)⋊C4C22semidirect
31. C127⋊#95C22semidirect
32. C127⋊#95C22semidirect
33. C127⋊#95C22semidirect
34. C127⋊#95C22semidirect
54. C2×C22⋊D127D4semidirect
62. C22×Dic127D4semidirect
90. C2×D4×C127D4semidirect
144. C23×C127#41semidirect
199. C127#2404semidirect
202. C2×#120D127semidirect
267. C2×D4C22⋊D127semidirect
320. C23#14523247semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611782
C22×C22⋊Dic1272
#225497882
#225497842
#225497562
#225497532
#225497522
#225497372
C2×#120×C1272
#2404127