Finite Group Databasebeta

← C2×C4×#99×C127 #44584063

Subgroups — C2×C4×#99×C127 #44584063

Order 32512.

2242 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C127C211yesmaximal, characteristic
2C22×#99×C127C212yesmaximal
3C2×C4×C22⋊C4×C127C212yesmaximal, complemented
4C4×#99×C127C218yesmaximal, direct product factor
5C2×#342×C127C212yesmaximal, complemented
6C23×D4×C127C2211yescharacteristic
7C24×C4×C127C411yescharacteristic, complemented
8C24×C4×C127C2211yescharacteristic
9C2×C4×D4×C127—22no—
10C2×C4×D4×C127—24no—
11C2×C4×D4×C127—22no—
12C2×C4×D4×C127C414yescomplemented
13C2×C4×D4×C127C2214yes—
14C22×C4⋊C4×C127C411yescharacteristic, complemented
15C22×C22⋊C4×C127C411yescharacteristic, complemented
16C22×C22⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2214yes—
18C22×C42×C127—22no—
19C22×C42×C127C411yescharacteristic, complemented
20C2×#99×C127C4116yesdirect product factor
21C2×#99×C127C2218yes—
22C4×C22⋊C4×C127—24no—
23C4×C22⋊C4×C127C2218yescomplemented
24C2×#95×C127—22no—
25#342×C127C2218yescomplemented
26C25×C127C2×C411yescharacteristic
27C25×C127C2311yescharacteristic
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—24no—
31C22×D4×C127C2×C418yes—
32C22×D4×C127C2×C412yes—
33C22×D4×C127C2312yes—
34C23×C4×C127—24no—
35C23×C4×C127—22no—
36C23×C4×C127—21no—
37C23×C4×C127—21no—
38C23×C4×C127—44no—
39C23×C4×C127—44no—
40C23×C4×C127C2×C411yescharacteristic
41C23×C4×C127C2×C411yescharacteristic
42C23×C4×C127C2×C411yescharacteristic
43C23×C4×C127C2×C411yescharacteristic
44C23×C4×C127C2×C411yescharacteristic
45C23×C4×C127C2×C412yes—
46C23×C4×C127C2×C414yescomplemented
47C23×C4×C127D412yes—
48C23×C4×C127D412yes—
49C23×C4×C127D414yes—
50C23×C4×C127C2312yes—
51C4×D4×C127—28no—
52C4×D4×C127—28no—
53C4×D4×C127—28no—
54C4×D4×C127—28no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127—22no—
57C2×C4⋊C4×C127C2×C414yescomplemented
58C2×C22⋊C4×C127—22no—
59C2×C22⋊C4×C127—22no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—28no—
62C2×C22⋊C4×C127—22no—
63C2×C22⋊C4×C127—22no—
64C2×C22⋊C4×C127C2×C412yes—
65C2×C22⋊C4×C127C2×C414yescomplemented
66C2×C22⋊C4×C127C2×C4116yescomplemented
67C2×C22⋊C4×C127C2312yes—
68C2×C22⋊C4×C127C2318yes—
69C2×C42×C127—21no—
70C2×C42×C127—22no—
71C2×C42×C127—21no—
72C2×C42×C127—28no—
73C2×C42×C127—48no—
74C2×C42×C127C2×C414yescomplemented
75#99×C127C2×C4132yesdirect product factor
76#95×C127—28no—
77C24×C127—21no—
78C24×C127—21no—
79C24×C127—22no—
80C24×C127—42no—
81C24×C127—42no—
82C24×C127—42no—
83C24×C127—42no—
84C24×C127C4214yescomplemented
85C24×C127C22×C411yescharacteristic
86C24×C127C22×C412yes—
87C24×C127C22×C411yescharacteristic
88C24×C127C22×C414yes—
89C24×C127C2×D412yes—
90C24×C127C2×D412yes—
91C24×C127#4114yes—
92C24×C127C2411yescharacteristic
93C2×D4×C127—24no—
94C2×D4×C127—24no—
95C2×D4×C127—216no—
96C2×D4×C127—216no—
97C2×D4×C127—24no—
98C2×D4×C127—24no—
99C2×D4×C127C4218yescomplemented
100C2×D4×C127C22×C418yes—
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103C22×C4×C127—22no—
104C22×C4×C127—28no—
105C22×C4×C127—24no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127—216no—
111C22×C4×C127—28no—
112C22×C4×C127—24no—
113C22×C4×C127—24no—
114C22×C4×C127—44no—
115C22×C4×C127—48no—
116C22×C4×C127—44no—
117C22×C4×C127—42no—
118C22×C4×C127—44no—
119C22×C4×C127—44no—
120C22×C4×C127—48no—
121C22×C4×C127—48no—
122C22×C4×C127—48no—
123C22×C4×C127—48no—
124C22×C4×C127—44no—
125C22×C4×C127—42no—
126C22×C4×C127—42no—
127C22×C4×C127C4214yescomplemented
128C22×C4×C127C22⋊C411yescharacteristic
129C22×C4×C127C22⋊C412yes—
130C22×C4×C127C22⋊C412yes—
131C22×C4×C127C22⋊C414yes—
132C22×C4×C127C22⋊C412yes—
133C22×C4×C127C22⋊C414yes—
134C22×C4×C127C22⋊C411yescharacteristic
135C22×C4×C127C22×C412yes—
136C22×C4×C127C22×C412yes—
137C22×C4×C127C22×C412yes—
138C22×C4×C127C22×C412yes—
139C22×C4×C127C22×C412yes—
140C22×C4×C127C2×D414yes—
141C22×C4×C127C2×D414yes—
142C4⋊C4×C127—28no—
143C22⋊C4×C127—24no—
144C22⋊C4×C127—216no—
145C22⋊C4×C127—24no—
146C22⋊C4×C127—28no—
147C22⋊C4×C127—28no—
148C22⋊C4×C127C22×C4116yescomplemented
149C42×C127—24no—
150C42×C127—24no—
151C42×C127—416no—
152C23×C127—24no—
153C23×C127—22no—
154C23×C127—24no—
155C23×C127—22no—
156C23×C127—22no—
157C23×C127—22no—
158C23×C127—24no—
159C23×C127—24no—
160C23×C127—41no—
161C23×C127—42no—
162C23×C127—41no—
163C23×C127—41no—
164C23×C127—48no—
165C23×C127—42no—
166C23×C127—48no—
167C23×C127—41no—
168C23×C127—42no—
169C23×C127—48no—
170C23×C127—42no—
171C23×C127—42no—
172C23×C127—42no—
173C23×C127—48no—
174C23×C127—42no—
175C23×C127—42no—
176C23×C127C2×C4212yes—
177C23×C127C2×C4212yes—
178C23×C127C2×C4214yescomplemented
179C23×C127C2×C22⋊C412yes—
180C23×C127C2×C22⋊C411yescharacteristic
181C23×C127C2×C22⋊C411yescharacteristic
182C23×C127#11712yes—
183C23×C127#11712yes—
184C23×C127C4×D418yes—
185C23×C127C4×D418yes—
186C23×C127C23×C411yescharacteristic
187C23×C127C23×C412yes—
188C23×C127C22×D412yes—
189C23×C127C2×#4112yes—
190D4×C127—216no—
191D4×C127—216no—
192C2×C4×C127—22no—
193C2×C4×C127—22no—
194C2×C4×C127—24no—
195C2×C4×C127—24no—
196C2×C4×C127—22no—
197C2×C4×C127—24no—
198C2×C4×C127—28no—
199C2×C4×C127—24no—
200C2×C4×C127—24no—
201C2×C4×C127—28no—
202C2×C4×C127—22no—
203C2×C4×C127—22no—
204C2×C4×C127—216no—
205C2×C4×C127—44no—
206C2×C4×C127—42no—
207C2×C4×C127—42no—
208C2×C4×C127—44no—
209C2×C4×C127—48no—
210C2×C4×C127—48no—
211C2×C4×C127—48no—
212C2×C4×C127—432no—
213C2×C4×C127—42no—
214C2×C4×C127—42no—
215C2×C4×C127—42no—
216C2×C4×C127—42no—
217C2×C4×C127—48no—
218C2×C4×C127—42no—
219C2×C4×C127—42no—
220C2×C4×C127#9914yesdirect product factor
221C2×C4×C127C2×C4214yescomplemented
222C2×C4×C127C2×C22⋊C412yes—
223C2×C4×C127C2×C22⋊C414yes—
224C2×C4×C127C2×C22⋊C412yes—
225C22×C127—22no—
226C22×C127—21no—
227C22×C127—21no—
228C22×C127—22no—
229C22×C127—22no—
230C22×C127—21no—
231C22×C127—21no—
232C22×C127—22no—
233C22×C127—216no—
234C22×C127—24no—
235C22×C127—24no—
236C22×C127—24no—
237C22×C127—42no—
238C22×C127—42no—
239C22×C127—48no—
240C22×C127—42no—
241C22×C127—42no—
242C22×C127—44no—
243C22×C127—44no—
244C22×C127—48no—
245C22×C127—42no—
246C22×C127—42no—
247C22×C127—48no—
248C22×C127C4×C22⋊C412yes—
249C22×C127C4×C22⋊C414yes—
250C22×C127C4×C22⋊C412yes—
251C22×C127C2×#9912yes—
252C22×C127#41012yes—
253C22×C127C22×C4211yescharacteristic
254C22×C127C22×C22⋊C411yescharacteristic
255C22×C127C2×#11711yescharacteristic
256C22×C127C2×C4×D414yes—
257C4×C127—22no—
258C4×C127—24no—
259C4×C127—24no—
260C4×C127—416no—
261C4×C127—42no—
262C4×C127—42no—
263C4×C127C2×#9914yesdirect product factor
264C2×C4×#99C12711yesmaximal, characteristic, direct product factor, Sylow
265C2×C127—24no—
266C2×C127—21no—
267C2×C127—22no—
268C2×C127—21no—
269C2×C127—24no—
270C2×C127—44no—
271C2×C127C4×#9914yesdirect product factor
272C2×C127C2×C4×C22⋊C411yescharacteristic
273C2×C127C22×#9911yescharacteristic
274C2×C127C2×#41011yescharacteristic
275C22×C4×D4C2×C12711yescharacteristic
276C22×#99C2×C12712yes—
277C2×C4×C22⋊C4C2×C12712yescomplemented
278C4×#99C2×C12718yesdirect product factor
279C2×#342C2×C12712yescomplemented
280C127C2×C4×#9911yescharacteristic, direct product factor, Sylow
281C23×D4C22×C12711yescharacteristic
282C24×C4C4×C12711yescharacteristic, complemented
283C24×C4C22×C12711yescharacteristic
284C2×C4×D4—22no—
285C2×C4×D4—22no—
286C2×C4×D4—24no—
287C2×C4×D4C4×C12714yescomplemented
288C2×C4×D4C22×C12714yes—
289C22×C4⋊C4C4×C12711yescharacteristic, complemented
290C22×C22⋊C4C4×C12711yescharacteristic, complemented
291C22×C22⋊C4C22×C12711yescharacteristic
292C22×C22⋊C4C22×C12714yes—
293C22×C42—22no—
294C22×C42C4×C12711yescharacteristic, complemented
295C2×#99C4×C127116yesdirect product factor
296C2×#99C22×C12718yes—
297C4×C22⋊C4—24no—
298C4×C22⋊C4C22×C12718yescomplemented
299C2×#95—22no—
300#342C22×C12718yescomplemented
301C25C2×C4×C12711yescharacteristic
302C25C23×C12711yescharacteristic
303C22×D4—22no—
304C22×D4—22no—
305C22×D4—24no—
306C22×D4C2×C4×C12718yes—
307C22×D4C2×C4×C12712yes—
308C22×D4C23×C12712yes—
309C23×C4—24no—
310C23×C4—22no—
311C23×C4—21no—
312C23×C4—21no—
313C23×C4—44no—
314C23×C4—44no—
315C23×C4C2×C4×C12711yescharacteristic
316C23×C4C2×C4×C12711yescharacteristic
317C23×C4C2×C4×C12711yescharacteristic
318C23×C4C2×C4×C12711yescharacteristic
319C23×C4C2×C4×C12711yescharacteristic
320C23×C4C2×C4×C12712yes—
321C23×C4C2×C4×C12714yescomplemented
322C23×C4D4×C12712yes—
323C23×C4D4×C12712yes—
324C23×C4D4×C12714yes—
325C23×C4C23×C12712yes—
326C4×D4—28no—
327C4×D4—28no—
328C4×D4—28no—
329C4×D4—28no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C2×C4×C12714yescomplemented
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×C4×C12712yes—
340C2×C22⋊C4C2×C4×C12714yescomplemented
341C2×C22⋊C4C2×C4×C127116yescomplemented
342C2×C22⋊C4C23×C12712yes—
343C2×C22⋊C4C23×C12718yes—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—28no—
348C2×C42—48no—
349C2×C42C2×C4×C12714yescomplemented
350#99C2×C4×C127132yesdirect product factor
351#95—28no—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C42×C12714yescomplemented
360C24C22×C4×C12711yescharacteristic
361C24C22×C4×C12712yes—
362C24C22×C4×C12711yescharacteristic
363C24C22×C4×C12714yes—
364C24C2×D4×C12712yes—
365C24C2×D4×C12712yes—
366C24#41×C12714yes—
367C24C24×C12711yescharacteristic
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4C42×C12718yescomplemented
375C2×D4C22×C4×C12718yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4C42×C12714yescomplemented
403C22×C4C22⋊C4×C12711yescharacteristic
404C22×C4C22⋊C4×C12712yes—
405C22×C4C22⋊C4×C12712yes—
406C22×C4C22⋊C4×C12714yes—
407C22×C4C22⋊C4×C12712yes—
408C22×C4C22⋊C4×C12714yes—
409C22×C4C22⋊C4×C12711yescharacteristic
410C22×C4C22×C4×C12712yes—
411C22×C4C22×C4×C12712yes—
412C22×C4C22×C4×C12712yes—
413C22×C4C22×C4×C12712yes—
414C22×C4C22×C4×C12712yes—
415C22×C4C2×D4×C12714yes—
416C22×C4C2×D4×C12714yes—
417C4⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—216no—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4C22×C4×C127116yescomplemented
424C42—24no—
425C42—24no—
426C42—416no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—41no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—48no—
440C23—42no—
441C23—48no—
442C23—41no—
443C23—42no—
444C23—48no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—42no—
450C23—42no—
451C23C2×C42×C12712yes—
452C23C2×C42×C12712yes—
453C23C2×C42×C12714yescomplemented
454C23C2×C22⋊C4×C12712yes—
455C23C2×C22⋊C4×C12711yescharacteristic
456C23C2×C22⋊C4×C12711yescharacteristic
457C23#117×C12712yes—
458C23#117×C12712yes—
459C23C4×D4×C12718yes—
460C23C4×D4×C12718yes—
461C23C23×C4×C12711yescharacteristic
462C23C23×C4×C12712yes—
463C23C22×D4×C12712yes—
464C23C2×#41×C12712yes—
465D4—216no—
466D4—216no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—216no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—432no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4#99×C12714yesdirect product factor
496C2×C4C2×C42×C12714yescomplemented
497C2×C4C2×C22⋊C4×C12712yes—
498C2×C4C2×C22⋊C4×C12714yes—
499C2×C4C2×C22⋊C4×C12712yes—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—216no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—48no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—44no—
519C22—48no—
520C22—42no—
521C22—42no—
522C22—48no—
523C22C4×C22⋊C4×C12712yes—
524C22C4×C22⋊C4×C12714yes—
525C22C4×C22⋊C4×C12712yes—
526C22C2×#99×C12712yes—
527C22#410×C12712yes—
528C22C22×C42×C12711yescharacteristic
529C22C22×C22⋊C4×C12711yescharacteristic
530C22C2×#117×C12711yescharacteristic
531C22C2×C4×D4×C12714yes—
532C4—22no—
533C4—24no—
534C4—24no—
535C4—416no—
536C4—42no—
537C4—42no—
538C4C2×#99×C12714yesdirect product factor
539C2—24no—
540C2—21no—
541C2—21no—
542C2—22no—
543C2—24no—
544C2—44no—
545C2C4×#99×C12714yesdirect product factor
546C2C2×C4×C22⋊C4×C12711yescharacteristic
547C2C22×#99×C12711yescharacteristic
548C2C2×#410×C12711yescharacteristic
subgroupG/H
centerC22×C4×C127C22⋊C4
commutatorC22C22×C42×C127
Frattini, cosocleC24C24×C127
socleC23×C127C2×C22⋊C4
prime radicalC23×D4×C127C22
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C4×C22⋊C4×C127C2semidirect
4. C4×#99×C127C2direct (N × H)
5. C2×#342×C127C2semidirect
7. C24×C4×C127C4semidirect
12. C2×C4×D4×C127C4semidirect
14. C22×C4⋊C4×C127C4semidirect
15. C22×C22⋊C4×C127C4semidirect
19. C22×C42×C127C4semidirect
20. C2×#99×C127C4direct (N × H)
23. C4×C22⋊C4×C127C22semidirect
25. #342×C127C22semidirect
46. C23×C4×C127C2×C4semidirect
57. C2×C4⋊C4×C127C2×C4semidirect
65. C2×C22⋊C4×C127C2×C4semidirect
66. C2×C22⋊C4×C127C2×C4semidirect
74. C2×C42×C127C2×C4semidirect
75. #99×C127C2×C4direct (N × H)
84. C24×C127C42semidirect
99. C2×D4×C127C42semidirect
127. C22×C4×C127C42semidirect
148. C22⋊C4×C127C22×C4semidirect
178. C23×C127C2×C42semidirect
220. C2×C4×C127#99direct (N × H)
221. C2×C4×C127C2×C42semidirect
263. C4×C127C2×#99direct (N × H)
264. C2×C4×#99C127direct (N × H)
271. C2×C127C4×#99direct (N × H)
277. C2×C4×C22⋊C4C2×C127semidirect
278. C4×#99C2×C127direct (N × H)
279. C2×#342C2×C127semidirect
280. C127C2×C4×#99direct (N × H)
282. C24×C4C4×C127semidirect
287. C2×C4×D4C4×C127semidirect
289. C22×C4⋊C4C4×C127semidirect
290. C22×C22⋊C4C4×C127semidirect
294. C22×C42C4×C127semidirect
295. C2×#99C4×C127direct (N × H)
298. C4×C22⋊C4C22×C127semidirect
300. #342C22×C127semidirect
321. C23×C4C2×C4×C127semidirect
332. C2×C4⋊C4C2×C4×C127semidirect
340. C2×C22⋊C4C2×C4×C127semidirect
341. C2×C22⋊C4C2×C4×C127semidirect
349. C2×C42C2×C4×C127semidirect
350. #99C2×C4×C127direct (N × H)
359. C24C42×C127semidirect
374. C2×D4C42×C127semidirect
402. C22×C4C42×C127semidirect
423. C22⋊C4C22×C4×C127semidirect
453. C23C2×C42×C127semidirect
495. C2×C4#99×C127direct (N × H)
496. C2×C4C2×C42×C127semidirect
538. C4C2×#99×C127direct (N × H)
545. C2C4×#99×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D4×C1272
C22×#99×C1272
C2×C4×C22⋊C4×C1272
C4×#99×C1272
C2×#342×C1272
C2×C4×#99127