Finite Group Databasebeta

← C4⋊C4×#99 #356182

Subgroups — C4⋊C4×#99 #356182

Order 512.

2150 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C212yesmaximal
4C2×C4×#99C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C212yesmaximal, complemented
6#10912C214yesmaximal, complemented
7#10854C212yesmaximal, complemented
8#10761C212yesmaximal
9C22×C4×D4C2212yes—
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C411yescharacteristic, complemented
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2211yescharacteristic
14C22×#99C2212yes—
15C22×#99C2211yescharacteristic
16D4×C4⋊C4—22no—
17D4×C4⋊C4—22no—
18D4×C4⋊C4—24no—
19C2×#389C412yescomplemented
20C2×#385C412yescomplemented
21C2×#385C2212yes—
22C2×#384C411yescharacteristic, complemented
23C2×#380C411yescharacteristic, complemented
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—22no—
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2212yes—
30#1799C418yescomplemented
31C4×#99—28no—
32C4×#99C418yescomplemented
33C2×#342C2214yes—
34C2×#342C2214yes—
35C2×#342C2212yes—
36C2×#342C2212yes—
37#1436—24no—
38C4⋊#95—22no—
39C22⋊#95C2214yes—
40C23×D4C2311yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C412yes—
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—44no—
53C2×C4×D4C2×C414yes—
54C2×C4×D4C2×C414yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4D414yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C22⋊C4C2312yes—
80C22×C42—22no—
81C22×C42—24no—
82C22×C42C2×C412yes—
83C22×C42C2×C411yescharacteristic
84C2×#99—24no—
85C2×#99—22no—
86C2×#99—22no—
87C2×#99—24no—
88C2×#99C2×C418yes—
89C2×#99C2×C414yes—
90C2×#99C2×C414yes—
91C2×#99D414yes—
92C2×#99Q814yes—
93#389—24no—
94#389—24no—
95#385—24no—
96#385—24no—
97#385—24no—
98#385—24no—
99#384—22no—
100#384—22no—
101#380—22no—
102#380—22no—
103#380—24no—
104#380—22no—
105#380—22no—
106#380C2×C418yescomplemented
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—48no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—28no—
113C4×C22⋊C4—44no—
114C4×C22⋊C4C2×C418yescomplemented
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—24no—
118C2×#95—24no—
119C2×#95—24no—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123#342—28no—
124#342—44no—
125#342C2×C418yescomplemented
126#342C2×C418yescomplemented
127C25C22×C411yescharacteristic
128C25C2411yescharacteristic
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C22×C412yes—
134C22×D4C22×C412yes—
135C22×D4C2×D412yes—
136C22×D4C2×Q812yes—
137C22×D4#4112yes—
138C22×D4#4112yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4C4214yescomplemented
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C2×D414yes—
173C23×C4C2×D412yes—
174C23×C4C2×D414yes—
175C23×C4C2×D412yes—
176C23×C4#4114yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C4214yescomplemented
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C414yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4C4214yescomplemented
246C2×C22⋊C4C22×C414yes—
247C2×C22⋊C4C22×C418yes—
248C2×C22⋊C4C22×C412yes—
249C2×C22⋊C4C22×C412yes—
250C2×C22⋊C4C22×C414yes—
251C2×C22⋊C4C2×D414yes—
252C2×C22⋊C4C2×Q814yes—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—42no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—48no—
264C2×C42—48no—
265C2×C42—84no—
266C2×C42C4214yescomplemented
267#99—28no—
268#99—216no—
269#99C4⋊C4116yesdirect product factor
270#95—22no—
271#95—22no—
272#95—24no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—48no—
277#95—48no—
278#95—84no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24C2×C4212yes—
292C24C4×D412yes—
293C24C4×Q812yes—
294C24C23×C411yescharacteristic
295C24C23×C411yescharacteristic
296C24C23×C411yescharacteristic
297C24C22×D412yes—
298C24C22×D411yescharacteristic
299C24C22×Q811yescharacteristic
300C24C2×#4112yes—
301C24C2×#4111yescharacteristic
302C24C2×#4111yescharacteristic
303C24H(5,𝔽2)12yes—
304C242−1+412yes—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4C2×C4⋊C414yes—
318C2×D4#11714yes—
319C2×D4C4×D414yes—
320C2×D4C4×Q814yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—44no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4C2×C4212yes—
411C22×C4C2×C4212yes—
412C22×C4C2×C4212yes—
413C22×C4C2×C4212yes—
414C22×C4C2×C4212yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C414yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4#11712yes—
422C22×C4#11712yes—
423C22×C4C4×D414yes—
424C22×C4C4×D414yes—
425C22×C4C4×D412yes—
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4C4×Q812yes—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—416no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—84no—
445C4⋊C4#9914yesdirect product factor
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—28no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4C2×C4⋊C418yescomplemented
464C42—24no—
465C42—24no—
466C42—44no—
467C42—44no—
468C42—416no—
469C42—816no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C4×C4⋊C414yescomplemented
519C23C22×C4211yescharacteristic
520C23C22×C22⋊C411yescharacteristic
521C23C22×C4⋊C411yescharacteristic
522C23C2×#11711yescharacteristic
523C23C2×#11711yescharacteristic
524C23C2×C4×D411yescharacteristic
525C23C2×C4×D412yes—
526C23C2×C4×D412yes—
527C23C2×C4×D411yescharacteristic
528C23C2×C4×Q811yescharacteristic
529C23C2×C4×Q811yescharacteristic
530C23#51811yescharacteristic
531C23#51911yescharacteristic
532C23#52012yes—
533C23#52012yes—
534C23D4212yes—
535C23#54612yes—
536C23#54712yes—
537C23#54812yes—
538C23D4×Q812yes—
539C23#55212yes—
540C23#55312yes—
541C23#55412yes—
542D4—28no—
543D4—28no—
544D4—28no—
545D4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—88no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—84no—
614C2×C4—82no—
615C2×C4—84no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4—82no—
620C2×C4—84no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4C4×C22⋊C412yes—
626C2×C4C4×C22⋊C414yes—
627C2×C4C4×C22⋊C412yes—
628C2×C4C4×C4⋊C414yescomplemented
629C2×C4C2×#9914yes—
630C2×C4C2×#9912yes—
631C2×C4#41012yes—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—22no—
646C22—21no—
647C22—21no—
648C22—22no—
649C22—22no—
650C22—21no—
651C22—21no—
652C22—22no—
653C22—22no—
654C22—21no—
655C22—21no—
656C22—24no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22C2×C4×C22⋊C411yescharacteristic
678C22C2×C4×C4⋊C411yescharacteristic
679C22#227411yescharacteristic
680C22D4×C22⋊C411yescharacteristic
681C22Q8×C22⋊C411yescharacteristic
682C22#234411yescharacteristic
683C22#234511yescharacteristic
684C22D4×C4⋊C412yes—
685C22#235212yes—
686C22#236012yes—
687C22#236112yes—
688C22C22×#9911yescharacteristic
689C22C2×#41011yescharacteristic
690C22#288311yescharacteristic
691C22#288411yescharacteristic
692C4—24no—
693C4—22no—
694C4—24no—
695C4—24no—
696C4—48no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—88no—
702C4—84no—
703C4—82no—
704C4—82no—
705C4C4×#9914yescomplemented
706C2—22no—
707C2—22no—
708C2—22no—
709C2—21no—
710C2—21no—
711C2—21no—
712C2—21no—
713C2—22no—
714C2—44no—
715C2C22⋊C4×C4⋊C411yescharacteristic
716C2C2×C4×#9911yescharacteristic
717C2#2039311yescharacteristic
718C2D4×#9911yescharacteristic
719C2#2240611yescharacteristic
720C2Q8×#9911yescharacteristic
721C2#2251311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22⋊C4×C4⋊C4C2semidirect
6. #10912C2semidirect
7. #10854C2semidirect
11. C23×C4⋊C4C4semidirect
19. C2×#389C4semidirect
20. C2×#385C4semidirect
22. C2×#384C4semidirect
23. C2×#380C4semidirect
27. C2×C4×C4⋊C4C4semidirect
30. #1799C4semidirect
32. C4×#99C4semidirect
106. #380C2×C4semidirect
114. C4×C22⋊C4C2×C4semidirect
125. #342C2×C4semidirect
126. #342C2×C4semidirect
160. C23×C4C42semidirect
213. C2×C4⋊C4C42semidirect
245. C2×C22⋊C4C42semidirect
266. C2×C42C42semidirect
269. #99C4⋊C4direct (N × H)
445. C4⋊C4#99direct (N × H)
463. C22⋊C4C2×C4⋊C4semidirect
518. C23C4×C4⋊C4semidirect
628. C2×C4C4×C4⋊C4semidirect
705. C4C4×#99semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#17992
C2×C4×#992
C22⋊C4×C4⋊C42
#109122
#108542
#107612