Finite Group Databasebeta

← C22⋊C4×C4⋊C4×C127 #44577473

Subgroups — C22⋊C4×C4⋊C4×C127 #44577473

Order 32512.

2574 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C127C211yesmaximal, characteristic
2C2×#380×C127C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C127C212yesmaximal, complemented
4C2×C4×C22⋊C4×C127C212yesmaximal
5C2×C4×C22⋊C4×C127C211yesmaximal, characteristic
6C127×#1436C214yesmaximal, complemented
7C127×C4⋊#95C212yesmaximal, complemented
8C127×C22⋊#95C212yesmaximal
9C24×C4×C127C2212yes—
10C24×C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127—28no—
12C22×C4⋊C4×C127C418yescomplemented
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2212yes—
16C22×C4⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2211yescharacteristic
20C22×C42×C127C2214yes—
21C22×C42×C127C2212yes—
22#380×C127C418yescomplemented
23C4×C4⋊C4×C127—28no—
24C4×C22⋊C4×C127—28no—
25C4×C22⋊C4×C127C418yescomplemented
26C2×#95×C127C2214yes—
27C2×#95×C127C2214yes—
28C2×#95×C127C2214yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C25×C127C2311yescharacteristic
32C23×C4×C127—24no—
33C23×C4×C127—24no—
34C23×C4×C127—24no—
35C23×C4×C127—24no—
36C23×C4×C127—28no—
37C23×C4×C127—44no—
38C23×C4×C127C2×C418yes—
39C23×C4×C127C2×C414yes—
40C23×C4×C127C2×C414yes—
41C23×C4×C127C2×C414yes—
42C23×C4×C127C2314yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2312yes—
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2312yes—
47C23×C4×C127C2312yes—
48C23×C4×C127C2311yescharacteristic
49C2×C4⋊C4×C127—28no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—24no—
53C2×C4⋊C4×C127—232no—
54C2×C4⋊C4×C127C2×C414yes—
55C2×C4⋊C4×C127C2×C418yes—
56C2×C4⋊C4×C127C2×C414yes—
57C2×C4⋊C4×C127C2×C418yescomplemented
58C2×C4⋊C4×C127D418yes—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—22no—
61C2×C22⋊C4×C127—22no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127C2×C414yes—
64C2×C22⋊C4×C127C2×C418yes—
65C2×C22⋊C4×C127C2×C414yes—
66C2×C22⋊C4×C127D414yes—
67C2×C22⋊C4×C127Q814yes—
68C2×C42×C127—24no—
69C2×C42×C127—24no—
70C2×C42×C127—28no—
71C2×C42×C127—28no—
72C2×C42×C127—44no—
73C2×C42×C127C2×C418yescomplemented
74#95×C127—28no—
75#95×C127—28no—
76#95×C127—44no—
77#95×C127C2×C418yescomplemented
78#95×C127C2×C418yescomplemented
79C24×C127—22no—
80C24×C127—22no—
81C24×C127—22no—
82C24×C127—22no—
83C24×C127C22×C412yes—
84C24×C127C22×C412yes—
85C24×C127C22×C412yes—
86C24×C127C2×D412yes—
87C24×C127C2×Q812yes—
88C24×C127#4112yes—
89C24×C127#4112yes—
90C24×C127C2411yescharacteristic
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—24no—
99C22×C4×C127—22no—
100C22×C4×C127—28no—
101C22×C4×C127—28no—
102C22×C4×C127—216no—
103C22×C4×C127—216no—
104C22×C4×C127—216no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—24no—
114C22×C4×C127—24no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—24no—
118C22×C4×C127—24no—
119C22×C4×C127—24no—
120C22×C4×C127—22no—
121C22×C4×C127—216no—
122C22×C4×C127—48no—
123C22×C4×C127—416no—
124C22×C4×C127—416no—
125C22×C4×C127C42116yescomplemented
126C22×C4×C127C22×C414yes—
127C22×C4×C127C22×C418yes—
128C22×C4×C127C22×C414yes—
129C22×C4×C127C22×C412yes—
130C22×C4×C127C22×C412yes—
131C22×C4×C127C22×C412yes—
132C22×C4×C127C22×C414yes—
133C22×C4×C127C22×C414yes—
134C22×C4×C127C22×C412yes—
135C22×C4×C127C22×C412yes—
136C22×C4×C127C22×C412yes—
137C22×C4×C127C2×D418yes—
138C22×C4×C127C2×D414yes—
139C22×C4×C127C2×D414yes—
140C22×C4×C127C2×Q814yes—
141C22×C4×C127#4114yes—
142C4⋊C4×C127—216no—
143C4⋊C4×C127—28no—
144C4⋊C4×C127—216no—
145C4⋊C4×C127C22⋊C4116yesdirect product factor
146C22⋊C4×C127—28no—
147C22⋊C4×C127—216no—
148C22⋊C4×C127C4⋊C4116yesdirect product factor
149C42×C127—216no—
150C42×C127—416no—
151C23×C127—22no—
152C23×C127—22no—
153C23×C127—24no—
154C23×C127—24no—
155C23×C127—22no—
156C23×C127—22no—
157C23×C127—24no—
158C23×C127—24no—
159C23×C127—24no—
160C23×C127—24no—
161C23×C127—24no—
162C23×C127—24no—
163C23×C127—24no—
164C23×C127—24no—
165C23×C127—24no—
166C23×C127—24no—
167C23×C127C2×C4214yes—
168C23×C127C2×C4⋊C414yes—
169C23×C127#11714yes—
170C23×C127C4×D414yes—
171C23×C127C4×D414yes—
172C23×C127C4×Q814yes—
173C23×C127C4×Q814yes—
174C23×C127C23×C411yescharacteristic
175C23×C127C23×C411yescharacteristic
176C23×C127C23×C411yescharacteristic
177C23×C127C22×D412yes—
178C23×C127C22×D411yescharacteristic
179C23×C127C22×Q811yescharacteristic
180C23×C127C2×#4112yes—
181C23×C127C2×#4111yescharacteristic
182C23×C127C2×#4111yescharacteristic
183C23×C127H(5,𝔽2)12yes—
184C23×C1272−1+412yes—
185C2×C4×C127—22no—
186C2×C4×C127—22no—
187C2×C4×C127—24no—
188C2×C4×C127—24no—
189C2×C4×C127—24no—
190C2×C4×C127—24no—
191C2×C4×C127—24no—
192C2×C4×C127—24no—
193C2×C4×C127—24no—
194C2×C4×C127—28no—
195C2×C4×C127—28no—
196C2×C4×C127—24no—
197C2×C4×C127—24no—
198C2×C4×C127—24no—
199C2×C4×C127—232no—
200C2×C4×C127—24no—
201C2×C4×C127—24no—
202C2×C4×C127—24no—
203C2×C4×C127—28no—
204C2×C4×C127—28no—
205C2×C4×C127—24no—
206C2×C4×C127—24no—
207C2×C4×C127—24no—
208C2×C4×C127—28no—
209C2×C4×C127—432no—
210C2×C4×C127—44no—
211C2×C4×C127—44no—
212C2×C4×C127—42no—
213C2×C4×C127—44no—
214C2×C4×C127—44no—
215C2×C4×C127—42no—
216C2×C4×C127—44no—
217C2×C4×C127—42no—
218C2×C4×C127—42no—
219C2×C4×C127—44no—
220C2×C4×C127C2×C4214yes—
221C2×C4×C127C2×C4214yes—
222C2×C4×C127C2×C22⋊C418yes—
223C2×C4×C127C2×C22⋊C414yes—
224C2×C4×C127C2×C4⋊C418yescomplemented
225C2×C4×C127#11714yes—
226C2×C4×C127C4×D418yes—
227C2×C4×C127C4×D418yes—
228C22×C127—24no—
229C22×C127—24no—
230C22×C127—24no—
231C22×C127—24no—
232C22×C127—28no—
233C22×C127—28no—
234C22×C127—28no—
235C22×C127—28no—
236C22×C127—24no—
237C22×C127—24no—
238C22×C127C4×C4⋊C418yescomplemented
239C22×C127C22×C4211yescharacteristic
240C22×C127C22×C22⋊C411yescharacteristic
241C22×C127C22×C4⋊C411yescharacteristic
242C22×C127C2×#11711yescharacteristic
243C22×C127C2×#11711yescharacteristic
244C22×C127C2×C4×D411yescharacteristic
245C22×C127C2×C4×D412yes—
246C22×C127C2×C4×D412yes—
247C22×C127C2×C4×D411yescharacteristic
248C22×C127C2×C4×Q811yescharacteristic
249C22×C127C2×C4×Q811yescharacteristic
250C22×C127#51811yescharacteristic
251C22×C127#51911yescharacteristic
252C22×C127#52012yes—
253C22×C127#52012yes—
254C22×C127D4212yes—
255C22×C127#54612yes—
256C22×C127#54712yes—
257C22×C127#54812yes—
258C22×C127D4×Q812yes—
259C22×C127#55212yes—
260C22×C127#55312yes—
261C22×C127#55412yes—
262C4×C127—28no—
263C4×C127—28no—
264C4×C127—24no—
265C4×C127—44no—
266C4×C127—44no—
267C4×C127—48no—
268C4×C127C4×C22⋊C418yescomplemented
269C22⋊C4×C4⋊C4C12711yesmaximal, characteristic, direct product factor, Sylow
270C2×C127—28no—
271C2×C127C2×C4×C22⋊C411yescharacteristic
272C2×C127C2×C4×C4⋊C411yescharacteristic
273C2×C127#227411yescharacteristic
274C2×C127D4×C22⋊C411yescharacteristic
275C2×C127Q8×C22⋊C411yescharacteristic
276C2×C127#234411yescharacteristic
277C2×C127#234511yescharacteristic
278C2×C127D4×C4⋊C412yes—
279C2×C127#235212yes—
280C2×C127#236012yes—
281C2×C127#236112yes—
282C23×C4⋊C4C2×C12711yescharacteristic
283C2×#380C2×C12711yescharacteristic
284C2×C4×C4⋊C4C2×C12712yescomplemented
285C2×C4×C22⋊C4C2×C12712yes—
286C2×C4×C22⋊C4C2×C12711yescharacteristic
287#1436C2×C12714yescomplemented
288C4⋊#95C2×C12712yescomplemented
289C22⋊#95C2×C12712yes—
290C127C22⋊C4×C4⋊C411yescharacteristic, direct product factor, Sylow
291C24×C4C22×C12712yes—
292C24×C4C22×C12711yescharacteristic
293C22×C4⋊C4—28no—
294C22×C4⋊C4C4×C12718yescomplemented
295C22×C4⋊C4C22×C12711yescharacteristic
296C22×C4⋊C4C22×C12712yes—
297C22×C4⋊C4C22×C12712yes—
298C22×C4⋊C4C22×C12711yescharacteristic
299C22×C22⋊C4C22×C12711yescharacteristic
300C22×C22⋊C4C22×C12712yes—
301C22×C22⋊C4C22×C12711yescharacteristic
302C22×C42C22×C12714yes—
303C22×C42C22×C12712yes—
304#380C4×C12718yescomplemented
305C4×C4⋊C4—28no—
306C4×C22⋊C4—28no—
307C4×C22⋊C4C4×C12718yescomplemented
308C2×#95C22×C12714yes—
309C2×#95C22×C12714yes—
310C2×#95C22×C12714yes—
311C2×#95C22×C12712yes—
312C2×#95C22×C12712yes—
313C25C23×C12711yescharacteristic
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—24no—
317C23×C4—24no—
318C23×C4—28no—
319C23×C4—44no—
320C23×C4C2×C4×C12718yes—
321C23×C4C2×C4×C12714yes—
322C23×C4C2×C4×C12714yes—
323C23×C4C2×C4×C12714yes—
324C23×C4C23×C12714yes—
325C23×C4C23×C12712yes—
326C23×C4C23×C12712yes—
327C23×C4C23×C12711yescharacteristic
328C23×C4C23×C12712yes—
329C23×C4C23×C12712yes—
330C23×C4C23×C12711yescharacteristic
331C2×C4⋊C4—28no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—232no—
336C2×C4⋊C4C2×C4×C12714yes—
337C2×C4⋊C4C2×C4×C12718yes—
338C2×C4⋊C4C2×C4×C12714yes—
339C2×C4⋊C4C2×C4×C12718yescomplemented
340C2×C4⋊C4D4×C12718yes—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4C2×C4×C12714yes—
346C2×C22⋊C4C2×C4×C12718yes—
347C2×C22⋊C4C2×C4×C12714yes—
348C2×C22⋊C4D4×C12714yes—
349C2×C22⋊C4Q8×C12714yes—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—28no—
353C2×C42—28no—
354C2×C42—44no—
355C2×C42C2×C4×C12718yescomplemented
356#95—28no—
357#95—28no—
358#95—44no—
359#95C2×C4×C12718yescomplemented
360#95C2×C4×C12718yescomplemented
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C22×C4×C12712yes—
366C24C22×C4×C12712yes—
367C24C22×C4×C12712yes—
368C24C2×D4×C12712yes—
369C24C2×Q8×C12712yes—
370C24#41×C12712yes—
371C24#41×C12712yes—
372C24C24×C12711yescharacteristic
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4C42×C127116yescomplemented
408C22×C4C22×C4×C12714yes—
409C22×C4C22×C4×C12718yes—
410C22×C4C22×C4×C12714yes—
411C22×C4C22×C4×C12712yes—
412C22×C4C22×C4×C12712yes—
413C22×C4C22×C4×C12712yes—
414C22×C4C22×C4×C12714yes—
415C22×C4C22×C4×C12714yes—
416C22×C4C22×C4×C12712yes—
417C22×C4C22×C4×C12712yes—
418C22×C4C22×C4×C12712yes—
419C22×C4C2×D4×C12718yes—
420C22×C4C2×D4×C12714yes—
421C22×C4C2×D4×C12714yes—
422C22×C4C2×Q8×C12714yes—
423C22×C4#41×C12714yes—
424C4⋊C4—216no—
425C4⋊C4—28no—
426C4⋊C4—216no—
427C4⋊C4C22⋊C4×C127116yesdirect product factor
428C22⋊C4—28no—
429C22⋊C4—216no—
430C22⋊C4C4⋊C4×C127116yesdirect product factor
431C42—216no—
432C42—416no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23C2×C42×C12714yes—
450C23C2×C4⋊C4×C12714yes—
451C23#117×C12714yes—
452C23C4×D4×C12714yes—
453C23C4×D4×C12714yes—
454C23C4×Q8×C12714yes—
455C23C4×Q8×C12714yes—
456C23C23×C4×C12711yescharacteristic
457C23C23×C4×C12711yescharacteristic
458C23C23×C4×C12711yescharacteristic
459C23C22×D4×C12712yes—
460C23C22×D4×C12711yescharacteristic
461C23C22×Q8×C12711yescharacteristic
462C23C2×#41×C12712yes—
463C23C2×#41×C12711yescharacteristic
464C23C2×#41×C12711yescharacteristic
465C23H(5,𝔽2)×C12712yes—
466C232−1+4×C12712yes—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—232no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—432no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4C2×C42×C12714yes—
503C2×C4C2×C42×C12714yes—
504C2×C4C2×C22⋊C4×C12718yes—
505C2×C4C2×C22⋊C4×C12714yes—
506C2×C4C2×C4⋊C4×C12718yescomplemented
507C2×C4#117×C12714yes—
508C2×C4C4×D4×C12718yes—
509C2×C4C4×D4×C12718yes—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—28no—
515C22—28no—
516C22—28no—
517C22—28no—
518C22—24no—
519C22—24no—
520C22C4×C4⋊C4×C12718yescomplemented
521C22C22×C42×C12711yescharacteristic
522C22C22×C22⋊C4×C12711yescharacteristic
523C22C22×C4⋊C4×C12711yescharacteristic
524C22C2×#117×C12711yescharacteristic
525C22C2×#117×C12711yescharacteristic
526C22C2×C4×D4×C12711yescharacteristic
527C22C2×C4×D4×C12712yes—
528C22C2×C4×D4×C12712yes—
529C22C2×C4×D4×C12711yescharacteristic
530C22C2×C4×Q8×C12711yescharacteristic
531C22C2×C4×Q8×C12711yescharacteristic
532C22#518×C12711yescharacteristic
533C22#519×C12711yescharacteristic
534C22#520×C12712yes—
535C22#520×C12712yes—
536C22D42×C12712yes—
537C22#546×C12712yes—
538C22#547×C12712yes—
539C22#548×C12712yes—
540C22D4×Q8×C12712yes—
541C22#552×C12712yes—
542C22#553×C12712yes—
543C22#554×C12712yes—
544C4—28no—
545C4—28no—
546C4—24no—
547C4—44no—
548C4—44no—
549C4—48no—
550C4C4×C22⋊C4×C12718yescomplemented
551C2—28no—
552C2C2×C4×C22⋊C4×C12711yescharacteristic
553C2C2×C4×C4⋊C4×C12711yescharacteristic
554C2C127×#227411yescharacteristic
555C2D4×C22⋊C4×C12711yescharacteristic
556C2Q8×C22⋊C4×C12711yescharacteristic
557C2C127×#234411yescharacteristic
558C2C127×#234511yescharacteristic
559C2D4×C4⋊C4×C12712yes—
560C2C127×#235212yes—
561C2C127×#236012yes—
562C2C127×#236112yes—
subgroupG/H
center, socleC24×C127C24
commutatorC22C22×C42×C127
Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C4×C4⋊C4×C127C2semidirect
6. C127×#1436C2semidirect
7. C127×C4⋊#95C2semidirect
12. C22×C4⋊C4×C127C4semidirect
22. #380×C127C4semidirect
25. C4×C22⋊C4×C127C4semidirect
57. C2×C4⋊C4×C127C2×C4semidirect
73. C2×C42×C127C2×C4semidirect
77. #95×C127C2×C4semidirect
78. #95×C127C2×C4semidirect
125. C22×C4×C127C42semidirect
145. C4⋊C4×C127C22⋊C4direct (N × H)
148. C22⋊C4×C127C4⋊C4direct (N × H)
224. C2×C4×C127C2×C4⋊C4semidirect
238. C22×C127C4×C4⋊C4semidirect
268. C4×C127C4×C22⋊C4semidirect
269. C22⋊C4×C4⋊C4C127direct (N × H)
284. C2×C4×C4⋊C4C2×C127semidirect
287. #1436C2×C127semidirect
288. C4⋊#95C2×C127semidirect
290. C127C22⋊C4×C4⋊C4direct (N × H)
294. C22×C4⋊C4C4×C127semidirect
304. #380C4×C127semidirect
307. C4×C22⋊C4C4×C127semidirect
339. C2×C4⋊C4C2×C4×C127semidirect
355. C2×C42C2×C4×C127semidirect
359. #95C2×C4×C127semidirect
360. #95C2×C4×C127semidirect
407. C22×C4C42×C127semidirect
427. C4⋊C4C22⋊C4×C127direct (N × H)
430. C22⋊C4C4⋊C4×C127direct (N × H)
506. C2×C4C2×C4⋊C4×C127semidirect
520. C22C4×C4⋊C4×C127semidirect
550. C4C4×C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C1272
C2×#380×C1272
C2×C4×C4⋊C4×C1272
C2×C4×C22⋊C4×C1272
C127×#14362
C127×C4⋊#952
C127×C22⋊#952
C22⋊C4×C4⋊C4127