Finite Group Databasebeta

← C4⋊C4×#104 #356205

Subgroups — C4⋊C4×#104 #356205

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2C2×#1806C211yesmaximal, characteristic
3C4⋊C4×M16C211yesmaximal, characteristic, complemented
4C2×C4×#104C212yesmaximal
5C2×C4×#104C211yesmaximal, characteristic
6C42×C4⋊C4C211yesmaximal, characteristic, complemented
7#10886C212yesmaximal, complemented
8#10878C212yesmaximal, complemented
9#10851C211yesmaximal, characteristic, complemented
10#10839C211yesmaximal, characteristic, complemented
11#10790C212yesmaximal
12C2×C4×#41C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×#104C2211yescharacteristic
16C22×#104C2212yes—
17C22×#104C2211yescharacteristic
18C2×C4×M16C2212yes—
19C2×C4×M16C2211yescharacteristic
20#2351C412yescomplemented
21Q8×C4⋊C4C412yescomplemented
22#2349—22no—
23#2349C412yescomplemented
24D4×C4⋊C4—22no—
25D4×C4⋊C4C412yescomplemented
26#2296—22no—
27#2296C2212yes—
28#2285—21no—
29#2285C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—22no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C43C2212yes—
35C2×C43C2211yescharacteristic
36#1806C418yescomplemented
37#1771—22no—
38#1770—21no—
39C8×C4⋊C4—21no—
40C4×#104—28no—
41C4×#104C418yescomplemented
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49#1728C2212yes—
50C4×#95—22no—
51C4×#95C2212yes—
52C22×#41C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—24no—
59C4×#41—24no—
60C4×#41—42no—
61C4×#41—42no—
62C4×#41C2×C414yes—
63C4×#41C2×C414yes—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—24no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×#117—21no—
74C2×#117—22no—
75C2×#117C2312yes—
76C2×#117C2311yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4D412yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42—24no—
84C22×C42—22no—
85C22×C42C2311yescharacteristic
86C22×C42C2312yes—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#423—22no—
91#423C2×C414yescomplemented
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×#104—24no—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—24no—
98C2×#104C2×C418yes—
99C2×#104C2×C414yes—
100C2×#104C2×C414yes—
101C2×#104D414yes—
102C2×#104Q814yes—
103C4×M16—22no—
104C4×M16—24no—
105C4×M16—42no—
106C4×M16C2×C414yescomplemented
107C2×#97—21no—
108C2×#97—22no—
109C2×C4×C8—21no—
110C2×C4×C8—22no—
111#389—24no—
112#389—24no—
113#389C2×C414yes—
114#389C2×C414yes—
115#385—24no—
116#385—24no—
117#385—24no—
118#384—22no—
119#384C2×C412yes—
120#382—22no—
121#382C2×C412yes—
122#380—21no—
123#380—21no—
124#380—22no—
125#380—22no—
126C4×C4⋊C4—216no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2×C414yescomplemented
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D412yes—
140C2×#95—21no—
141C2×#95—24no—
142C2×#95—22no—
143C2×#95D412yes—
144C43—22no—
145C43—24no—
146C43—42no—
147C43C2×C414yescomplemented
148#339—24no—
149#339—24no—
150#339—42no—
151#339—42no—
152#339C2×C418yescomplemented
153#339C2×C414yescomplemented
154#339C2×C414yescomplemented
155#336—22no—
156#336—22no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41C22×C412yes—
164C2×#41C22×C412yes—
165C2×#41C2×D412yes—
166C2×#41C2×Q812yes—
167C2×#41#4112yes—
168C2×#41#4112yes—
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—24no—
179C23×C4C2×D414yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16C22×C412yes—
197C2×M16C22×C414yes—
198C2×M16C22×C412yes—
199C2×M16C2×D412yes—
200C2×M16C2×Q812yes—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—28no—
207C4×Q8—44no—
208C4×Q8C4214yescomplemented
209C4×Q8C4214yescomplemented
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—28no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—44no—
218C4×D4—48no—
219C4×D4—48no—
220C4×D4C4214yescomplemented
221C4×D4C4214yescomplemented
222#117—21no—
223#117—21no—
224#117—24no—
225#117—24no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—44no—
231#117—44no—
232#117C22×C412yes—
233#117C22×C412yes—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22⋊C414yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C42—28no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—216no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—48no—
287C2×C42—44no—
288C2×C42—48no—
289C2×C42C22×C412yes—
290C2×C42C22×C414yes—
291C2×C42C22×C414yes—
292C2×C42C22×C412yes—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×Q812yes—
303C2×C42#4112yes—
304C2×C42#4112yes—
305C4⋊C8—24no—
306C4⋊C8—24no—
307#104—28no—
308#104—216no—
309#104C4⋊C4116yesdirect product factor
310#97—24no—
311#97—44no—
312C4×C8—24no—
313C4×C8—44no—
314#95—216no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95C22⋊C412yes—
321#95C22⋊C412yes—
322#95C22⋊C414yes—
323C24—21no—
324C24C22×D411yescharacteristic
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41C2×C4⋊C414yes—
338#41#11714yes—
339#41C4×D414yes—
340#41C4×Q814yes—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8C2×C4212yes—
345C2×Q8C4×D412yes—
346C2×Q8C4×Q812yes—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4C2×C4212yes—
357C2×D4C4×D412yes—
358C2×D4C4×Q812yes—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—82no—
396C22×C4—84no—
397C22×C4C2×C22⋊C414yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C414yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4C4×D412yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C42−1+411yescharacteristic
420C22×C42−1+411yescharacteristic
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—24no—
426M16—22no—
427M16—24no—
428M16—42no—
429M16—41no—
430M16—41no—
431M16—42no—
432M16C2×C4⋊C414yescomplemented
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—42no—
442C4⋊C4—216no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4#10418yesdirect product factor
457C4⋊C4C2×C4212yes—
458C4⋊C4C2×C4212yes—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—48no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C42—216no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—24no—
482C42—22no—
483C42—24no—
484C42—48no—
485C42—432no—
486C42—48no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42C2×C4212yes—
494C42C2×C4212yes—
495C42C2×C4⋊C414yescomplemented
496C42#11712yes—
497C42#11712yes—
498C42C4×D412yes—
499C42C4×D412yes—
500C42C4×D412yes—
501C42C4×D412yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—42no—
508C23—42no—
509C23C22×C22⋊C411yescharacteristic
510C23C2×C4×D411yescharacteristic
511C23C2×C4×D411yescharacteristic
512C23D4211yescharacteristic
513C23#54611yescharacteristic
514C23#54811yescharacteristic
515C23D4×Q811yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8C4×C4⋊C414yescomplemented
520D4—24no—
521D4—24no—
522D4—24no—
523D4—24no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4C4×C4⋊C414yescomplemented
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—84no—
573C2×C4—816no—
574C2×C4C4×C22⋊C412yes—
575C2×C4C4×C22⋊C412yes—
576C2×C4C4×C22⋊C412yes—
577C2×C4C4×C22⋊C412yes—
578C2×C4C2×#10418yes—
579C2×C4C2×#10414yes—
580C2×C4C22×C4211yescharacteristic
581C2×C4C22×C4⋊C411yescharacteristic
582C2×C4C2×#11711yescharacteristic
583C2×C4C2×#11711yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×Q811yescharacteristic
589C2×C4C2×C4×Q811yescharacteristic
590C2×C4#51811yescharacteristic
591C2×C4#51911yescharacteristic
592C2×C4#52011yescharacteristic
593C2×C4#52011yescharacteristic
594C2×C4#52011yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4D4211yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4#54811yescharacteristic
601C2×C4D4×Q811yescharacteristic
602C2×C4#55211yescharacteristic
603C2×C4#55211yescharacteristic
604C2×C4#55311yescharacteristic
605C2×C4#55311yescharacteristic
606C2×C4#55411yescharacteristic
607C2×C4#55411yescharacteristic
608C8—24no—
609C8—42no—
610C8—44no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—24no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22C2×C4×C22⋊C411yescharacteristic
625C22D4×C22⋊C411yescharacteristic
626C22Q8×C22⋊C411yescharacteristic
627C22D4×C4⋊C411yescharacteristic
628C22#235211yescharacteristic
629C22#236011yescharacteristic
630C22#236111yescharacteristic
631C22C22×#10412yes—
632C22#290112yes—
633C4—28no—
634C4—28no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—48no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—84no—
645C4C4×#10414yescomplemented
646C4C2×C4×C4⋊C411yescharacteristic
647C4#227411yescharacteristic
648C4#234411yescharacteristic
649C4#234511yescharacteristic
650C4D4×C4⋊C411yescharacteristic
651C4#235211yescharacteristic
652C4#236011yescharacteristic
653C4#236111yescharacteristic
654C2—21no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2—44no—
659C2C22⋊C4×C4⋊C411yescharacteristic
660C2C2×C4×#10411yescharacteristic
661C2#2040511yescharacteristic
662C2D4×#10412yes—
663C2Q8×#10412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×M16C2semidirect
6. C42×C4⋊C4C2semidirect
7. #10886C2semidirect
8. #10878C2semidirect
9. #10851C2semidirect
10. #10839C2semidirect
20. #2351C4semidirect
21. Q8×C4⋊C4C4semidirect
23. #2349C4semidirect
25. D4×C4⋊C4C4semidirect
36. #1806C4semidirect
41. C4×#104C4semidirect
91. #423C2×C4semidirect
106. C4×M16C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect
147. C43C2×C4semidirect
152. #339C2×C4semidirect
153. #339C2×C4semidirect
154. #339C2×C4semidirect
208. C4×Q8C42semidirect
209. C4×Q8C42semidirect
220. C4×D4C42semidirect
221. C4×D4C42semidirect
309. #104C4⋊C4direct (N × H)
432. M16C2×C4⋊C4semidirect
456. C4⋊C4#104direct (N × H)
495. C42C2×C4⋊C4semidirect
519. Q8C4×C4⋊C4semidirect
527. D4C4×C4⋊C4semidirect
645. C4C4×#104semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
C2×#18062
C4⋊C4×M162
C2×C4×#1042
C42×C4⋊C42
#108862
#108782
#108512
#108392
#107902