Finite Group Databasebeta

← #362115

Subgroups — #362115

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#21146C211yesmaximal, characteristic, complemented
3C2×#1806C212yesmaximal
4C2×#1806C211yesmaximal, characteristic
5C2×#1759C211yesmaximal, characteristic
6#13848C211yesmaximal, characteristic, complemented
7#11226C212yesmaximal, complemented
8#11209C212yesmaximal, complemented
9#10851C211yesmaximal, characteristic, complemented
10#10840C211yesmaximal, characteristic, complemented
11#10790C212yesmaximal
12C2×C4×#41C2212yes—
13C2×C4×#41C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2212yes—
16C22×#104C2212yes—
17C22×#104C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2351C411yescharacteristic, complemented
21#2351C411yescharacteristic, complemented
22Q8×C4⋊C4C412yescomplemented
23#2349—21no—
24#2349—21no—
25#2349C411yescharacteristic, complemented
26#2349C411yescharacteristic, complemented
27D4×C4⋊C4—22no—
28D4×C4⋊C4C412yescomplemented
29#2296—22no—
30#2296C2212yes—
31#2285—21no—
32#2285C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38#1940—21no—
39#1940—21no—
40#1939—22no—
41#1806—28no—
42#1806—24no—
43#1759—24no—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1728C2212yes—
52#1444—21no—
53#1442—21no—
54#1441—22no—
55C4×#95C2212yes—
56C22×#41C2311yescharacteristic
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—24no—
63C4×#41—24no—
64C4×#41—42no—
65C4×#41—42no—
66C4×#41D414yes—
67C4×#41Q814yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×#117—21no—
81C2×#117—22no—
82C2×#117C2312yes—
83C2×#117C2311yescharacteristic
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D412yes—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#423—24no—
96#423—22no—
97#423—22no—
98#423—42no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—24no—
108C2×#104—24no—
109C2×#104—24no—
110C2×#104D414yes—
111C2×#104Q814yes—
112C4×M16—22no—
113C4×M16—22no—
114C2×#97—21no—
115C2×C4×C8—21no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389C2×C412yes—
121#389C2×C412yes—
122#389C2×C412yes—
123#389C2×C412yes—
124#385—22no—
125#385—22no—
126#385—22no—
127#385—22no—
128#385—22no—
129#385—22no—
130#384—22no—
131#384C2×C412yes—
132#382—21no—
133#382—21no—
134#382C2×C411yescharacteristic
135#382C2×C411yescharacteristic
136#380—21no—
137#380—21no—
138#380—22no—
139#380—21no—
140#380—21no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—42no—
152C4×C4⋊C4C2×C412yes—
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4D412yes—
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158#376—22no—
159#376—22no—
160C2×#95—21no—
161C2×#95—21no—
162C2×#95—22no—
163C2×#95—22no—
164C2×#95—21no—
165C2×#95—21no—
166C2×#95—22no—
167C2×#95—21no—
168C2×#95D411yescharacteristic
169C2×#95D411yescharacteristic
170#339—24no—
171#339—24no—
172#339—22no—
173#339—22no—
174#339—24no—
175#339—42no—
176#339—42no—
177#336—22no—
178#336—22no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41C2×D412yes—
186C2×#41C2×D412yes—
187C2×#41C2×Q812yes—
188C2×#41C2×Q812yes—
189C2×#41#4112yes—
190C2×#41#4112yes—
191C22×Q8C22×C411yescharacteristic
192C22×D4—21no—
193C22×D4C22×C411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16C2×D412yes—
224C2×M16C2×Q812yes—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—42no—
233C4×Q8—42no—
234C4×Q8C4⋊C414yescomplemented
235C4×Q8C4⋊C414yescomplemented
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4C4⋊C414yescomplemented
251C4×D4C4⋊C414yescomplemented
252#117—21no—
253#117—21no—
254#117—24no—
255#117—24no—
256#117—22no—
257#117—22no—
258#117—22no—
259#117—22no—
260#117—44no—
261#117—44no—
262#117C2×D412yes—
263#117C2×Q812yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22⋊C412yes—
282C2×C4⋊C4C22⋊C412yes—
283C2×C4⋊C4C22⋊C412yes—
284C2×C4⋊C4C22⋊C412yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C22×C412yes—
326C2×C42C22×C412yes—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×Q812yes—
336C2×C42C2×Q812yes—
337C2×C42#4112yes—
338C2×C42#4112yes—
339C4⋊C8—44no—
340C4⋊C8—44no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343#104—28no—
344#104—28no—
345#104—48no—
346#97—42no—
347C4×C8—42no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—21no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—42no—
360#95—42no—
361#95—42no—
362#95C22⋊C411yescharacteristic
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C412yes—
367#95C22⋊C412yes—
368C24—21no—
369C24C22×D411yescharacteristic
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41#12114yes—
383#41C22⋊Q814yes—
384#41#12514yes—
385#41C4⋊Q814yes—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C2×C4⋊C412yes—
390C2×Q8C4×D412yes—
391C2×Q8C4×Q812yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C2×C4⋊C412yes—
402C2×D4C4×D412yes—
403C2×D4C4×Q812yes—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4C2×C22⋊C412yes—
453C22×C4C2×C22⋊C412yes—
454C22×C4C2×C22⋊C412yes—
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C412yes—
460C22×C4#12112yes—
461C22×C4#12112yes—
462C22×C4C22⋊Q812yes—
463C22×C4C22⋊Q812yes—
464C22×C4C23×C411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×Q811yescharacteristic
469C22×C4C22×Q811yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4H(5,𝔽2)11yescharacteristic
475C22×C4H(5,𝔽2)11yescharacteristic
476C22×C42−1+411yescharacteristic
477C22×C42−1+411yescharacteristic
478M16—22no—
479M16—22no—
480M16—22no—
481M16—22no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—41no—
486M16—42no—
487M16—42no—
488M16—42no—
489M16—42no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—416no—
517C4⋊C4—44no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4C2×C4⋊C412yes—
522C4⋊C4C2×C4⋊C412yes—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—44no—
549C42—41no—
550C42—41no—
551C42—42no—
552C42—42no—
553C42—41no—
554C42—41no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—42no—
559C42C2×C4⋊C412yes—
560C42C2×C4⋊C412yes—
561C42#11712yes—
562C42#11712yes—
563C42#12112yes—
564C42#12112yes—
565C42C22⋊Q812yes—
566C42C22⋊Q812yes—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—42no—
573C23—42no—
574C23C22×C22⋊C411yescharacteristic
575C23C2×#12111yescharacteristic
576C23C2×C22⋊Q811yescharacteristic
577C23D4211yescharacteristic
578C23#54611yescharacteristic
579C23#54811yescharacteristic
580C23D4×Q811yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8#38914yes—
585D4—24no—
586D4—24no—
587D4—24no—
588D4—24no—
589D4—22no—
590D4—22no—
591D4—22no—
592D4#38914yes—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—88no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—88no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—81no—
649C2×C4—82no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—82no—
654C2×C4#38012yes—
655C2×C4#38012yes—
656C2×C4#38012yes—
657C2×C4#38012yes—
658C2×C4#42114yes—
659C2×C4C22×C4⋊C411yescharacteristic
660C2×C4C2×#11711yescharacteristic
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C2×C4×Q811yescharacteristic
663C2×C4#51811yescharacteristic
664C2×C4#51911yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4C2×C22⋊Q811yescharacteristic
668C2×C4C2×C22⋊Q811yescharacteristic
669C2×C4C2×#12511yescharacteristic
670C2×C4C2×C4⋊Q811yescharacteristic
671C2×C4#53711yescharacteristic
672C2×C4#53711yescharacteristic
673C2×C4#54411yescharacteristic
674C2×C4#54411yescharacteristic
675C2×C4D4211yescharacteristic
676C2×C4#54611yescharacteristic
677C2×C4#54711yescharacteristic
678C2×C4#54711yescharacteristic
679C2×C4#54811yescharacteristic
680C2×C4D4×Q811yescharacteristic
681C2×C4#55211yescharacteristic
682C2×C4#55211yescharacteristic
683C2×C4#55311yescharacteristic
684C2×C4#55311yescharacteristic
685C2×C4#55411yescharacteristic
686C2×C4#55411yescharacteristic
687C8—42no—
688C8—42no—
689C8—82no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—24no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22C2×#38011yescharacteristic
704C22D4×C22⋊C411yescharacteristic
705C22Q8×C22⋊C411yescharacteristic
706C22#241611yescharacteristic
707C22#241711yescharacteristic
708C22#244011yescharacteristic
709C22#244211yescharacteristic
710C22C2×#42112yes—
711C22#290112yes—
712C4—24no—
713C4—24no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—84no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4C2×#38911yescharacteristic
729C4#231611yescharacteristic
730C4#234411yescharacteristic
731C4#234511yescharacteristic
732C4#241611yescharacteristic
733C4#241711yescharacteristic
734C4#244011yescharacteristic
735C4#244211yescharacteristic
736C2—21no—
737C2—21no—
738C2—21no—
739C2—21no—
740C2—44no—
741C2#1390711yescharacteristic
742C2#2065411yescharacteristic
743C2#2065511yescharacteristic
744C2#2243412yes—
745C2#2252912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×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
2. #21146C2semidirect
6. #13848C2semidirect
7. #11226C2semidirect
8. #11209C2semidirect
9. #10851C2semidirect
10. #10840C2semidirect
20. #2351C4semidirect
21. #2351C4semidirect
22. Q8×C4⋊C4C4semidirect
25. #2349C4semidirect
26. #2349C4semidirect
28. D4×C4⋊C4C4semidirect
234. C4×Q8C4⋊C4semidirect
235. C4×Q8C4⋊C4semidirect
250. C4×D4C4⋊C4semidirect
251. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#211462
C2×#18062
C2×#17592
#138482
#112262
#112092
#108512
#108402
#107902