Finite Group Databasebeta

← C2×#59071 #7667016

Subgroups — C2×#59071 #7667016

Order 512.

7406 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C22×#533C211yesmaximal, characteristic
3#59071C2132yesmaximal, direct product factor
4C2×#2719C214yesmaximal, complemented
5C2×#2568C211yesmaximal, characteristic, complemented
6C2×#2546C211yesmaximal, characteristic, complemented
7C2×#2489C212yesmaximal
8C2×#2475C214yesmaximal
9C2×#2453C212yesmaximal, complemented
10C2×#2421C212yesmaximal
11C2×#2446C212yesmaximal, complemented
12C2×#2429C212yesmaximal
13C2×#2429C212yesmaximal
14C2×#2365C211yesmaximal, characteristic, complemented
15C2×#2365C212yesmaximal, complemented
16C2×#2351C212yesmaximal, complemented
17C2×C4×C22⋊Q8C211yesmaximal, characteristic
18C2×C4×C22⋊Q8C211yesmaximal, characteristic
19C2×#557—216no—
20C2×#557C22116yescomplemented
21C2×#533—216no—
22C2×#533C22116yes—
23C22×C4⋊Q8C2212yes—
24C22×C4⋊Q8C2211yescharacteristic
25C22×C4⋊Q8C2212yes—
26C22×C4⋊Q8C2211yescharacteristic
27C22×#125C2212yes—
28C22×#125C2214yes—
29C22×#125C2212yes—
30C22×C22⋊Q8C2211yescharacteristic
31C22×C22⋊Q8C2211yescharacteristic
32C22×C22⋊Q8C2212yes—
33C22×C22⋊Q8C2212yes—
34C22×C4×Q8C2212yes—
35C22×C4×Q8C2212yes—
36C22×C4×Q8C2212yes—
37C22×C4×Q8C2212yes—
38C22×C4×Q8C2212yes—
39C22×C4×Q8C2211yescharacteristic
40C22×C4×Q8C2211yescharacteristic
41C22×#117C2212yes—
42C23×C42C2211yescharacteristic
43#2719C22164yescomplemented
44#2568C22116yescomplemented
45#2546C22116yescomplemented
46#2489C22132yes—
47#2475C22164yes—
48#2453C22132yescomplemented
49#2446C22132yescomplemented
50#2429C22132yes—
51#2429C22132yes—
52#2421C22132yes—
53C2×#400C2214yes—
54C2×#400C2214yes—
55C2×#400C2214yes—
56C2×#398C2214yes—
57C2×#398C2214yes—
58C2×#398C2214yes—
59C2×#397C2214yes—
60C2×#396C2214yes—
61#2365C22116yescomplemented
62#2365C22132yescomplemented
63#2351C22132yescomplemented
64C4×C22⋊Q8C22116yes—
65C4×C22⋊Q8C22116yes—
66C2×#391C2211yescharacteristic
67C2×#391C2212yes—
68C2×#391C2212yes—
69C2×#391C2212yes—
70C2×#391C2212yes—
71C2×#391C2212yes—
72C2×#391C2211yescharacteristic
73C2×#389C2211yescharacteristic
74C2×#389C2212yes—
75C2×#389C2214yes—
76C2×#389C2214yes—
77C2×#389C2212yes—
78C2×#389C2214yes—
79C2×#389C2212yes—
80C2×#389C2211yescharacteristic
81C2×#388C2214yes—
82C2×#388C2214yes—
83C2×#387C2214yes—
84C2×#387C2214yes—
85C2×#387C2214yes—
86C2×#387C2214yes—
87C2×#387C2212yes—
88C2×#387C2212yes—
89C2×#385C2212yes—
90C2×#385C2212yes—
91C2×#382C2212yes—
92C2×#382C2214yes—
93C2×#380C2211yescharacteristic
94C2×#380C2211yescharacteristic
95C2×C4×C4⋊C4C2214yes—
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C4⋊C4C2212yes—
98C2×C4×C4⋊C4C2211yescharacteristic
99C2×C4×C4⋊C4C2211yescharacteristic
100C2×C4×C4⋊C4C2212yes—
101C2×C4×C22⋊C4C2212yes—
102C2×C4×C22⋊C4C2212yes—
103C23×Q8C2312yes—
104C23×Q8C2311yescharacteristic
105C23×Q8C2312yes—
106C23×Q8C2311yescharacteristic
107C24×C4C2311yescharacteristic
108C24×C4C2311yescharacteristic
109C24×C4C2311yescharacteristic
110#557—2128no—
111#533—2128no—
112C2×C4⋊Q8—216no—
113C2×C4⋊Q8—216no—
114C2×C4⋊Q8—28no—
115C2×C4⋊Q8D4116yes—
116C2×C4⋊Q8C2318yes—
117C2×C4⋊Q8C23116yes—
118C2×C4⋊Q8C23116yes—
119C2×C4⋊Q8C2318yes—
120C2×#125—216no—
121C2×#125—232no—
122C2×#125—216no—
123C2×#125C23116yes—
124C2×#125C23116yes—
125C2×#125C23132yes—
126C2×C22⋊Q8—216no—
127C2×C22⋊Q8—28no—
128C2×C22⋊Q8—216no—
129C2×C22⋊Q8—28no—
130C2×C22⋊Q8C23116yes—
131C2×C22⋊Q8C23116yes—
132C2×C22⋊Q8C2318yes—
133C2×C22⋊Q8C2318yes—
134C2×C4×Q8—216no—
135C2×C4×Q8—216no—
136C2×C4×Q8—216no—
137C2×C4×Q8—216no—
138C2×C4×Q8—216no—
139C2×C4×Q8—28no—
140C2×C4×Q8—28no—
141C2×C4×Q8C2318yes—
142C2×C4×Q8C23116yes—
143C2×C4×Q8C23116yes—
144C2×C4×Q8C23116yes—
145C2×C4×Q8C23116yes—
146C2×C4×Q8C23116yes—
147C2×C4×Q8C2318yes—
148C2×#117—216no—
149C2×#117C23116yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2314yes—
152C22×C4⋊C4C2314yes—
153C22×C4⋊C4C2314yes—
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2312yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2312yes—
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2314yes—
160C22×C4⋊C4C2314yes—
161C22×C4⋊C4C2314yes—
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2311yescharacteristic
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2311yescharacteristic
166C22×C4⋊C4C2311yescharacteristic
167C22×C4⋊C4C2311yescharacteristic
168C22×C4⋊C4C2312yes—
169C22×C4⋊C4C2311yescharacteristic
170C22×C4⋊C4C2312yes—
171C22×C4⋊C4C2312yes—
172C22×C4⋊C4C2312yes—
173C22×C4⋊C4C2311yescharacteristic
174C22×C22⋊C4C2312yes—
175C22×C22⋊C4C2312yes—
176C22×C22⋊C4C2314yes—
177C22×C22⋊C4C2312yes—
178C22×C22⋊C4C2312yes—
179C22×C42—28no—
180C22×C42C2311yescharacteristic
181C22×C42C2311yescharacteristic
182C22×C42C2311yescharacteristic
183C22×C42C2318yes—
184C22×C42C2314yes—
185C22×C42C2312yes—
186C22×C42C2314yes—
187C22×C42C2312yes—
188C22×C42C2312yes—
189C22×C42C2314yes—
190C22×C42C2312yes—
191C22×C42C2311yescharacteristic
192#400C23132yes—
193#400C23132yes—
194#400C23132yes—
195#398C23132yes—
196#398C23132yes—
197#398C23132yes—
198#397C23132yes—
199#396C23132yes—
200#391C23116yes—
201#391C23116yes—
202#391C23116yes—
203#391C2318yes—
204#391C23116yes—
205#391C2318yes—
206#391C23116yes—
207#389C2318yes—
208#389C23132yes—
209#389C23116yes—
210#389C23132yes—
211#389C23116yes—
212#389C23132yes—
213#389C23116yes—
214#389C2318yes—
215#388C23132yes—
216#388C23132yes—
217#387C23132yes—
218#387C23132yes—
219#387C23132yes—
220#387C23132yes—
221#387C23116yes—
222#387C23116yes—
223#385C23116yes—
224#385C23116yes—
225#382C23116yes—
226#382C23132yes—
227#380C2318yes—
228#380C2318yes—
229C4×C4⋊C4C2318yes—
230C4×C4⋊C4C23132yes—
231C4×C4⋊C4C23116yes—
232C4×C4⋊C4C23116yes—
233C4×C4⋊C4C23116yes—
234C4×C4⋊C4C2318yes—
235C4×C22⋊C4C23116yes—
236C4×C22⋊C4C23116yes—
237C2×#95C2314yes—
238C2×#95C2314yes—
239C2×#95C2314yes—
240C2×#95C2314yes—
241C2×#95C2312yes—
242C2×#95C2312yes—
243C2×#95C2312yes—
244C2×#95C2314yes—
245C2×#95C2314yes—
246C2×#95C2312yes—
247C2×#95C2312yes—
248C2×#95C2312yes—
249C2×#95C2314yes—
250C2×#95C2314yes—
251C2×#95C2312yes—
252C2×#95C2312yes—
253C2×#95C2312yes—
254C2×#95C2312yes—
255C25C2411yescharacteristic
256C22×Q8—24no—
257C22×Q8—28no—
258C22×Q8—24no—
259C22×Q8—28no—
260C22×Q8—24no—
261C22×Q8—28no—
262C22×Q8—28no—
263C22×Q8—24no—
264C22×Q8—24no—
265C22×Q8—28no—
266C22×Q8C2×D4116yes—
267C22×Q8#4118yes—
268C22×Q8C2418yes—
269C22×Q8C2414yes—
270C22×Q8C2418yes—
271C22×Q8C2414yes—
272C23×C4—24no—
273C23×C4—24no—
274C23×C4—24no—
275C23×C4—24no—
276C23×C4—24no—
277C23×C4—24no—
278C23×C4—24no—
279C23×C4—24no—
280C23×C4—24no—
281C23×C4C2414yes—
282C23×C4C2414yes—
283C23×C4C2414yes—
284C23×C4C2412yes—
285C23×C4C2414yes—
286C23×C4C2412yes—
287C23×C4C2412yes—
288C23×C4C2412yes—
289C23×C4C2412yes—
290C23×C4C2412yes—
291C23×C4C2411yescharacteristic
292C23×C4C2411yescharacteristic
293C23×C4C2414yes—
294C23×C4C2411yescharacteristic
295C23×C4C2412yes—
296C23×C4C2412yes—
297C23×C4C2411yescharacteristic
298C23×C4C2411yescharacteristic
299C23×C4C2411yescharacteristic
300C4⋊Q8—264no—
301C4⋊Q8—264no—
302C4⋊Q8—232no—
303C4⋊Q8C2×D4164yes—
304#125—264no—
305#125—2128no—
306#125—264no—
307C22⋊Q8—264no—
308C22⋊Q8—232no—
309C22⋊Q8—264no—
310C22⋊Q8—232no—
311C4×Q8—264no—
312C4×Q8—264no—
313C4×Q8—264no—
314C4×Q8—264no—
315C4×Q8—264no—
316C4×Q8—232no—
317C4×Q8—232no—
318#117—264no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—216no—
322C2×C4⋊C4—216no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—216no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—216no—
329C2×C4⋊C4—216no—
330C2×C4⋊C4—216no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—216no—
335C2×C4⋊C4—216no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D4116yes—
341C2×C4⋊C4C2×D4116yes—
342C2×C4⋊C4#4118yes—
343C2×C4⋊C4C2418yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C24116yes—
346C2×C4⋊C4C24116yes—
347C2×C4⋊C4C2418yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C24116yes—
350C2×C4⋊C4C2418yes—
351C2×C4⋊C4C2418yes—
352C2×C4⋊C4C24116yes—
353C2×C4⋊C4C24116yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2418yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C24116yes—
358C2×C4⋊C4C24116yes—
359C2×C4⋊C4C24116yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2418yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C2414yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—216no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4C24116yes—
373C2×C22⋊C4C2418yes—
374C2×C22⋊C4C2418yes—
375C2×C22⋊C4C2418yes—
376C2×C22⋊C4C2418yes—
377C2×C42—216no—
378C2×C42—216no—
379C2×C42—28no—
380C2×C42—28no—
381C2×C42—24no—
382C2×C42—24no—
383C2×C42—24no—
384C2×C42—216no—
385C2×C42—28no—
386C2×C42—28no—
387C2×C42—232no—
388C2×C42C2×D418yes—
389C2×C42C24116yes—
390C2×C42C2418yes—
391C2×C42C24116yes—
392C2×C42C2418yes—
393C2×C42C2414yes—
394C2×C42C2414yes—
395C2×C42C2418yes—
396C2×C42C24116yes—
397C2×C42C2414yes—
398C2×C42C2418yes—
399C2×C42C2414yes—
400#95C2418yes—
401#95C2418yes—
402#95C2418yes—
403#95C2418yes—
404#95C2418yes—
405#95C2418yes—
406#95C2418yes—
407#95C24116yes—
408#95C2418yes—
409#95C2418yes—
410#95C2418yes—
411#95C24116yes—
412#95C24116yes—
413#95C24116yes—
414#95C24116yes—
415#95C24116yes—
416#95C24116yes—
417#95C24116yes—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C242−1+414yes—
425C24C2512yes—
426C24C2511yescharacteristic
427C2×Q8—264no—
428C2×Q8—232no—
429C2×Q8—28no—
430C2×Q8—216no—
431C2×Q8—28no—
432C2×Q8—216no—
433C2×Q8—28no—
434C2×Q8—216no—
435C2×Q8—216no—
436C2×Q8—28no—
437C2×Q8—28no—
438C2×Q8—216no—
439C2×Q8—416no—
440C2×Q8—432no—
441C2×Q8C22×D4132yes—
442C2×Q8C2×#41116yes—
443C22×C4—232no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—28no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—28no—
488C22×C4—28no—
489C22×C4—28no—
490C22×C4—416no—
491C22×C4—416no—
492C22×C4C22×D4116yes—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C2×#4118yes—
497C22×C4C2×#4118yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4114yes—
500C22×C42−1+414yes—
501C22×C42−1+414yes—
502C22×C42−1+414yes—
503C22×C4C2514yes—
504C22×C4C2514yes—
505C22×C4C2514yes—
506C22×C4C2514yes—
507C22×C4C2512yes—
508C22×C4C2512yes—
509C22×C4C2514yes—
510C22×C4C2518yes—
511C22×C4C2514yes—
512C22×C4C2512yes—
513C22×C4C2518yes—
514C22×C4C2514yes—
515C22×C4C2514yes—
516C22×C4C2512yes—
517C22×C4C2512yes—
518C22×C4C2512yes—
519C4⋊C4—216no—
520C4⋊C4—28no—
521C4⋊C4—232no—
522C4⋊C4—232no—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—232no—
526C4⋊C4—216no—
527C4⋊C4—28no—
528C4⋊C4—232no—
529C4⋊C4—232no—
530C4⋊C4—232no—
531C4⋊C4—28no—
532C4⋊C4—216no—
533C4⋊C4—216no—
534C4⋊C4—232no—
535C4⋊C4—232no—
536C4⋊C4—216no—
537C4⋊C4—216no—
538C4⋊C4—28no—
539C4⋊C4—28no—
540C4⋊C4C22×D4132yes—
541C4⋊C4C22×D4132yes—
542C4⋊C4C2×#41116yes—
543C22⋊C4—216no—
544C22⋊C4—216no—
545C22⋊C4—232no—
546C22⋊C4—216no—
547C22⋊C4—216no—
548C42—28no—
549C42—28no—
550C42—28no—
551C42—232no—
552C42—232no—
553C42—216no—
554C42—216no—
555C42—232no—
556C42—216no—
557C42—216no—
558C42C22×D4116yes—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23C23×D412yes—
573C23C22×#4112yes—
574C23C22×#4112yes—
575C23C2×2−1+414yes—
576C23C2×2−1+412yes—
577C23C2×2−1+412yes—
578C23C2×2−1+412yes—
579C23C2×2−1+412yes—
580C23C2611yescharacteristic
581Q8—264no—
582Q8—232no—
583Q8—416no—
584Q8—432no—
585C2×C4—232no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—232no—
597C2×C4—216no—
598C2×C4—216no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—416no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—416no—
638C2×C4—416no—
639C2×C4#547116yes—
640C2×C4#548116yes—
641C2×C4C23×D4116yes—
642C2×C4C23×D414yes—
643C2×C4C23×D414yes—
644C2×C4C23×D414yes—
645C2×C4C22×#4118yes—
646C2×C4C22×#4118yes—
647C2×C4C22×#4114yes—
648C2×C4C22×#4114yes—
649C2×C4C2×2−1+414yes—
650C2×C4C2×2−1+414yes—
651C2×C4C2×2−1+414yes—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—416no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22C2×#54714yes—
661C22C2×#54814yes—
662C22#348214yes—
663C22#348614yes—
664C22#352314yes—
665C22#352314yes—
666C22#355614yes—
667C22C24×D411yescharacteristic
668C22C23×#4111yescharacteristic
669C22C23×#4111yescharacteristic
670C22C22×2−1+411yescharacteristic
671C22C22×2−1+411yescharacteristic
672C22C22×2−1+411yescharacteristic
673C22C22×2−1+411yescharacteristic
674C4—24no—
675C4—24no—
676C4—28no—
677C4—216no—
678C4—28no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—48no—
685C4—42no—
686C4—44no—
687C4—44no—
688C4C2×#54718yes—
689C4C2×#54818yes—
690C2—44no—
691C2#5907118yesdirect product factor
692C2C22×#54811yescharacteristic
693C2C22×#54711yescharacteristic
694C2C2×#348211yescharacteristic
695C2C2×#355611yescharacteristic
696C2C2×#348611yescharacteristic
697C2C2×#352311yescharacteristic
698C2C2×#352311yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#557C2semidirect
3. #59071C2direct (N × H)
4. C2×#2719C2semidirect
5. C2×#2568C2semidirect
6. C2×#2546C2semidirect
9. C2×#2453C2semidirect
11. C2×#2446C2semidirect
14. C2×#2365C2semidirect
15. C2×#2365C2semidirect
16. C2×#2351C2semidirect
20. C2×#557C22semidirect
43. #2719C22semidirect
44. #2568C22semidirect
45. #2546C22semidirect
48. #2453C22semidirect
49. #2446C22semidirect
61. #2365C22semidirect
62. #2365C22semidirect
63. #2351C22semidirect
691. C2#59071direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C22×#5332
#590712
C2×#27192
C2×#25682
C2×#25462
C2×#24892
C2×#24752
C2×#24532
C2×#24212
C2×#24462
C2×#24292
C2×#23652
C2×#23512
C2×C4×C22⋊Q82