Finite Group Databasebeta

← #223002

Subgroups — #223002

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34898C211yesmaximal, characteristic
2#22751C211yesmaximal, characteristic
3#21264C211yesmaximal, characteristic
4#9871C211yesmaximal, characteristic, complemented
5#9866C211yesmaximal, characteristic, complemented
6#9073C211yesmaximal, characteristic, complemented
7#9073C211yesmaximal, characteristic, complemented
8#8942C211yesmaximal, characteristic, complemented
9#8941C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic, complemented
11#8710C211yesmaximal, characteristic, complemented
12#8566C211yesmaximal, characteristic, complemented
13C4×Q8⋊C8C211yesmaximal, characteristic, complemented
14#7900C211yesmaximal, characteristic, complemented
15#7899C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20#2577—21no—
21#2575—21no—
22#2570C411yescharacteristic
23#2565C411yescharacteristic
24#2562C411yescharacteristic
25#2536C411yescharacteristic
26#2524—21no—
27#2496C411yescharacteristic
28#2495C411yescharacteristic
29#2468—21no—
30#2423C411yescharacteristic
31#2421C411yescharacteristic
32#2402—21no—
33#2395—21no—
34C4×#125C2211yescharacteristic
35C4×#126—21no—
36C4×C22⋊Q8C2211yescharacteristic
37C4×C22⋊Q8C2211yescharacteristic
38#2298—21no—
39#2285C2211yescharacteristic
40#2283C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42C42×D4C2211yescharacteristic
43#1954—21no—
44#1939—21no—
45#1923—21no—
46#1923—21no—
47#1923—21no—
48#1917—21no—
49#1870—21no—
50#1831C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55#1753—21no—
56C4×C22⋊C8C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58C42×C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C22×M16—21no—
76C2×#126—21no—
77C2×#125C2×C411yescharacteristic
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—22no—
90#432—41no—
91#431—22no—
92#431—22no—
93#431—41no—
94#431—41no—
95#423—22no—
96#423—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#400—21no—
121#400—21no—
122#400—21no—
123#400—21no—
124#400—21no—
125#399—21no—
126#398—21no—
127#398—21no—
128#398—21no—
129#398—21no—
130#398—21no—
131#397—21no—
132#397—21no—
133#397—21no—
134#394—21no—
135#394—21no—
136#391C2×C411yescharacteristic
137#391C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#385—21no—
161#385—21no—
162#384C2×C411yescharacteristic
163#384C2×C411yescharacteristic
164#383—21no—
165#383—21no—
166#382—21no—
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169#382C2×C411yescharacteristic
170#381—21no—
171#380—21no—
172#380C2×C411yescharacteristic
173#380C2×C411yescharacteristic
174#380C2×C411yescharacteristic
175#380C2×C411yescharacteristic
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4C2×C411yescharacteristic
188C4×C4⋊C4C2×C411yescharacteristic
189C4×C4⋊C4D411yescharacteristic
190C4×C4⋊C4D411yescharacteristic
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2311yescharacteristic
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4C2×C411yescharacteristic
198C4×C22⋊C4C2×C411yescharacteristic
199C4×C22⋊C4D411yescharacteristic
200C4×C22⋊C4D411yescharacteristic
201C4×C22⋊C4D411yescharacteristic
202C4×C22⋊C4D411yescharacteristic
203#376—21no—
204C43—21no—
205C43C2311yescharacteristic
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214Q8⋊C8—22no—
215Q8⋊C8—22no—
216Q8⋊C8—41no—
217Q8⋊C8—41no—
218C22×Q8C22×C411yescharacteristic
219C22×D4C22⋊C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C22×C411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C2×M16—42no—
227C2×M16—42no—
228C2×M16—81no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237#126—44no—
238#125—22no—
239#125—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—41no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—42no—
255C4×D4—41no—
256C4×D4—41no—
257#117—42no—
258#117—42no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
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—21no—
313C2×C42—21no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2411yescharacteristic
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8#4112yes—
339C22⋊C8—42no—
340C22⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—81no—
346#97—42no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—42no—
355C4×C8—42no—
356C4×C8—81no—
357C4×C8#4112yes—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389C24C2×C22⋊C411yescharacteristic
390C24C2×C22⋊C411yescharacteristic
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8#13112yes—
397C2×D4—22no—
398C2×D4—41no—
399C2×D4—41no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C23×C411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453M16—84no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—22no—
544C42—41no—
545C42—42no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—42no—
555C42—42no—
556C42—81no—
557C42#13112yes—
558C42#13112yes—
559C42C2×#4111yescharacteristic
560C42C2×#4111yescharacteristic
561C42H(5,𝔽2)11yescharacteristic
562C422−1+411yescharacteristic
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23C22×C22⋊C411yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—41no—
581Q8—41no—
582D4—44no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4#40812yes—
657C2×C4#40812yes—
658C2×C4#42112yes—
659C2×C4C4×#4111yescharacteristic
660C2×C4#52011yescharacteristic
661C2×C4#54611yescharacteristic
662C2×C4#54611yescharacteristic
663C2×C4#54711yescharacteristic
664C2×C4#54711yescharacteristic
665C2×C4C2×#13111yescharacteristic
666C2×C4C2×#13111yescharacteristic
667C2×C4C2×#13111yescharacteristic
668C2×C4#56811yescharacteristic
669C2×C4#57511yescharacteristic
670C2×C4#57511yescharacteristic
671C2×C4#57711yescharacteristic
672C2×C4#57811yescharacteristic
673C8—42no—
674C8—42no—
675C8—42no—
676C8—81no—
677C8—81no—
678C8—81no—
679C8—81no—
680C8—81no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22#233911yescharacteristic
696C22C2×#40811yescharacteristic
697C22#288011yescharacteristic
698C22#289611yescharacteristic
699C22#289811yescharacteristic
700C22C2×#42111yescharacteristic
701C22#290111yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—21no—
706C4—21no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—42no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4#296611yescharacteristic
723C4#296911yescharacteristic
724C4#299311yescharacteristic
725C4#299411yescharacteristic
726C4#332311yescharacteristic
727C4#332711yescharacteristic
728C4#335111yescharacteristic
729C4#335211yescharacteristic
730C2—81no—
731C2—81no—
732C2#1747711yescharacteristic
733C2#1748411yescharacteristic
734C2#1751611yescharacteristic
735C2#1752411yescharacteristic
736C2#2247411yescharacteristic
737C2#2249911yescharacteristic
738C2#2250711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9871C2semidirect
5. #9866C2semidirect
6. #9073C2semidirect
7. #9073C2semidirect
8. #8942C2semidirect
9. #8941C2semidirect
10. #8711C2semidirect
11. #8710C2semidirect
12. #8566C2semidirect
13. C4×Q8⋊C8C2semidirect
14. #7900C2semidirect
15. #7899C2semidirect

Maximal subgroups

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

subgroupindex
#348982
#227512
#212642
#98712
#98662
#90732
#89422
#89412
#87112
#87102
#85662
C4×Q8⋊C82
#79002
#78992