Finite Group Databasebeta

← C2×#34291 #7625604

Subgroups — C2×#34291 #7625604

Order 512.

11114 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C22×#519C211yesmaximal, characteristic, complemented
3C23×C4×Q8C211yesmaximal, characteristic, complemented
4#34291C2132yesmaximal, direct product factor
5C2×#2365C213yesmaximal, complemented
6C2×#2362C213yesmaximal, complemented
7C2×#2360C213yesmaximal, complemented
8C2×#2352C213yesmaximal
9C2×#2351C211yesmaximal, characteristic, complemented
10C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
11C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
12C2×#2327C213yesmaximal, complemented
13C2×#2323C213yesmaximal, complemented
14C2×C4×C22⋊Q8C213yesmaximal, complemented
15C2×#2290C213yesmaximal
16C2×#2276C211yesmaximal, characteristic, complemented
17C24×Q8C2211yescharacteristic
18C24×Q8C2211yescharacteristic
19C2×D4×Q8C4132yescomplemented
20C2×D4×Q8C22116yes—
21C22×C4⋊Q8C2213yes—
22C22×C22⋊Q8C2213yes—
23C22×C22⋊Q8C2213yes—
24C2×#519—216no—
25C2×#519C22116yescomplemented
26C22×C4×Q8—216no—
27C22×C4×Q8C2213yes—
28C22×C4×Q8C2213yes—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2213yes—
33C22×C4×Q8C2213yes—
34C22×C4×Q8C2211yescharacteristic
35C22×C4×Q8C2211yescharacteristic
36C22×C4×Q8C22116yescomplemented
37C22×C4×D4C2213yes—
38C22×#117C2213yes—
39C22×#117C2213yes—
40C23×C4⋊C4C2213yes—
41C23×C42C2213yes—
42#2365C22148yescomplemented
43#2362C22148yescomplemented
44#2360C22148yescomplemented
45#2352C22148yes—
46#2351C22116yescomplemented
47Q8×C22⋊C4C22116yescomplemented
48Q8×C22⋊C4C22116yescomplemented
49#2327C22148yescomplemented
50#2323C22148yescomplemented
51C4×C22⋊Q8C22148yescomplemented
52C2×#391C2213yes—
53C2×#391C2213yes—
54C2×#391C2213yes—
55C2×#391C2213yes—
56C2×#389C2213yes—
57C2×#389C2216yes—
58C2×#389C2213yes—
59#2290C22148yes—
60C2×#388C2213yes—
61C2×#388C2216yes—
62C2×#388C2213yes—
63C2×#387C2216yes—
64C2×#387C2213yes—
65C2×#387C2216yes—
66C2×#387C2216yes—
67C2×#387C2213yes—
68C2×#386C2213yes—
69C2×#386C2211yescharacteristic
70C2×#385C2213yes—
71C2×#385C2213yes—
72C2×#382C2213yes—
73C2×#380C2213yes—
74C2×#380C2213yes—
75#2276C22116yescomplemented
76C2×C4×C4⋊C4C2213yes—
77C2×C4×C4⋊C4C2213yes—
78C2×C4×C4⋊C4C2213yes—
79C2×C4×C4⋊C4C2213yes—
80C2×C4×C4⋊C4C2216yes—
81C2×C4×C22⋊C4C2213yes—
82C2×C4×C22⋊C4C2213yes—
83C2×C4×C22⋊C4C2216yes—
84C2×#376C2213yes—
85C23×Q8—28no—
86C23×Q8—28no—
87C23×Q8—28no—
88C23×Q8—28no—
89C23×Q8C2×C4116yes—
90C23×Q8C2×C4116yescomplemented
91C23×Q8C2318yes—
92C23×Q8C2313yes—
93C23×Q8C2313yes—
94C23×Q8C2318yes—
95C23×Q8C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2313yes—
99C24×C4C2313yes—
100C24×C4C2313yes—
101D4×Q8C2×C41256yescomplemented
102C2×C4⋊Q8C2×C4148yescomplemented
103C2×C4⋊Q8C23124yes—
104C2×C22⋊Q8C2×C4148yes—
105C2×C22⋊Q8C2×C4148yescomplemented
106C2×C22⋊Q8C23124yes—
107C2×C22⋊Q8C23124yes—
108#519—2128no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—2128no—
113C2×C4×Q8—224no—
114C2×C4×Q8—224no—
115C2×C4×Q8—224no—
116C2×C4×Q8—224no—
117C2×C4×Q8C2×C4116yescomplemented
118C2×C4×Q8D4116yescomplemented
119C2×C4×Q8C23124yes—
120C2×C4×Q8C23124yes—
121C2×C4×Q8C2318yes—
122C2×C4×Q8C2318yes—
123C2×C4×Q8C2318yes—
124C2×C4×Q8C23124yes—
125C2×C4×Q8C23124yes—
126C2×C4×Q8C2318yes—
127C2×C4×Q8C2318yes—
128C2×C4×D4C2×C4148yes—
129C2×C4×D4C23124yes—
130C2×#117—224no—
131C2×#117—224no—
132C2×#117C23124yes—
133C2×#117C23124yes—
134C22×C4⋊C4—224no—
135C22×C4⋊C4C23124yes—
136C22×C4⋊C4C2313yes—
137C22×C4⋊C4C2316yes—
138C22×C4⋊C4C2313yes—
139C22×C4⋊C4C2313yes—
140C22×C4⋊C4C2313yes—
141C22×C4⋊C4C2313yes—
142C22×C4⋊C4C2313yes—
143C22×C4⋊C4C2313yes—
144C22×C4⋊C4C2313yes—
145C22×C4⋊C4C2313yes—
146C22×C4⋊C4C2316yes—
147C22×C4⋊C4C2313yes—
148C22×C22⋊C4C2311yescharacteristic
149C22×C22⋊C4C2313yes—
150C22×C22⋊C4C2313yes—
151C22×C22⋊C4C2311yescharacteristic
152C22×C22⋊C4C2311yescharacteristic
153C22×C22⋊C4C2313yes—
154C22×C22⋊C4C2313yes—
155C22×C22⋊C4C2313yes—
156C22×C42—224no—
157C22×C42C2313yes—
158C22×C42C2313yes—
159C22×C42C2313yes—
160C22×C42C23124yes—
161C22×C42C2316yes—
162C22×C42C2313yes—
163C22×C42C2313yes—
164C22×C42C2316yes—
165C22×C42C2313yes—
166C22×C42C2313yes—
167#391C23124yes—
168#391C23124yes—
169#391C23124yes—
170#391C23124yes—
171#389C23124yes—
172#389C23148yes—
173#389C23124yes—
174#388C23124yes—
175#388C23148yes—
176#388C23124yes—
177#387C23148yes—
178#387C23124yes—
179#387C23148yes—
180#387C23148yes—
181#387C23124yes—
182#386C23124yes—
183#386C2318yes—
184#385C23124yes—
185#385C23124yes—
186#382C23124yes—
187#380C23124yes—
188#380C23124yes—
189C4×C4⋊C4C23124yes—
190C4×C4⋊C4C23124yes—
191C4×C4⋊C4C23124yes—
192C4×C4⋊C4C23124yes—
193C4×C4⋊C4C23148yes—
194C4×C22⋊C4C23124yes—
195C4×C22⋊C4C23124yes—
196C4×C22⋊C4C23148yes—
197#376C23124yes—
198C2×#95C2311yescharacteristic
199C2×#95C2313yes—
200C2×#95C2313yes—
201C2×#95C2313yes—
202C2×#95C2313yes—
203C2×#95C2316yes—
204C2×#95C2316yes—
205C2×#95C2313yes—
206C2×#95C2313yes—
207C2×#95C2313yes—
208C2×#95C2311yescharacteristic
209C2×#95C2316yes—
210C2×#95C2313yes—
211C25C2411yescharacteristic
212C25C2411yescharacteristic
213C22×Q8—2128no—
214C22×Q8—2128no—
215C22×Q8—232no—
216C22×Q8—232no—
217C22×Q8—232no—
218C22×Q8—232no—
219C22×Q8—212no—
220C22×Q8—212no—
221C22×Q8—212no—
222C22×Q8—212no—
223C22×Q8C22×C4124yes—
224C22×Q8C22×C4124yes—
225C22×Q8C22×C4164yes—
226C22×Q8C22×C418yes—
227C22×Q8C22×C4164yescomplemented
228C22×Q8C2×D418yes—
229C22×Q8#4118yes—
230C22×Q8C24112yes—
231C22×Q8C24112yes—
232C22×Q8C2414yes—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4C22×C418yes—
236C22×D4C2414yes—
237C23×C4—24no—
238C23×C4—212no—
239C23×C4—212no—
240C23×C4—24no—
241C23×C4—212no—
242C23×C4—212no—
243C23×C4—24no—
244C23×C4—212no—
245C23×C4—212no—
246C23×C4—46no—
247C23×C4C22×C4124yes—
248C23×C4C22×C4124yes—
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2413yes—
252C23×C4C2413yes—
253C23×C4C2411yescharacteristic
254C23×C4C2413yes—
255C23×C4C24112yes—
256C23×C4C24112yes—
257C23×C4C24112yes—
258C23×C4C2413yes—
259C23×C4C2413yes—
260C23×C4C2413yes—
261C23×C4C2411yescharacteristic
262C23×C4C2413yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2413yes—
266C4⋊Q8C22×C41192yescomplemented
267C22⋊Q8C22×C41192yes—
268C22⋊Q8C22×C41192yescomplemented
269C4×Q8—232no—
270C4×Q8—232no—
271C4×Q8—232no—
272C4×Q8—296no—
273C4×Q8—296no—
274C4×Q8—296no—
275C4×Q8—296no—
276C4×Q8C22×C4164yescomplemented
277C4×Q8C2×D4164yescomplemented
278C4×D4C22×C41192yes—
279#117—296no—
280#117—296no—
281C2×C4⋊C4—224no—
282C2×C4⋊C4—212no—
283C2×C4⋊C4—212no—
284C2×C4⋊C4—212no—
285C2×C4⋊C4—212no—
286C2×C4⋊C4—212no—
287C2×C4⋊C4—212no—
288C2×C4⋊C4—296no—
289C2×C4⋊C4—46no—
290C2×C4⋊C4C22×C4124yes—
291C2×C4⋊C4C22×C4124yes—
292C2×C4⋊C4C22×C4148yes—
293C2×C4⋊C4C2×D4124yes—
294C2×C4⋊C4C24112yes—
295C2×C4⋊C4C24124yes—
296C2×C4⋊C4C24112yes—
297C2×C4⋊C4C24112yes—
298C2×C4⋊C4C24112yes—
299C2×C4⋊C4C24112yes—
300C2×C4⋊C4C24112yes—
301C2×C4⋊C4C24112yes—
302C2×C4⋊C4C24112yes—
303C2×C4⋊C4C24112yes—
304C2×C4⋊C4C24124yes—
305C2×C4⋊C4C24112yes—
306C2×C22⋊C4—212no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—212no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—46no—
312C2×C22⋊C4C22×C4124yes—
313C2×C22⋊C4C22×C4124yes—
314C2×C22⋊C4C24112yes—
315C2×C22⋊C4C2414yes—
316C2×C22⋊C4C24112yes—
317C2×C22⋊C4C2414yes—
318C2×C22⋊C4C2414yes—
319C2×C22⋊C4C24112yes—
320C2×C22⋊C4C24112yes—
321C2×C22⋊C4C24112yes—
322C2×C42—224no—
323C2×C42—212no—
324C2×C42—212no—
325C2×C42—212no—
326C2×C42—212no—
327C2×C42—212no—
328C2×C42—296no—
329C2×C42—412no—
330C2×C42C22×C4124yes—
331C2×C42C2×D4124yes—
332C2×C42C24124yes—
333C2×C42C24112yes—
334C2×C42C24112yes—
335C2×C42C24112yes—
336C2×C42C24112yes—
337C2×C42C24112yes—
338C2×C42C24124yes—
339C2×C42C24112yes—
340C2×C42C24112yes—
341#95C2414yes—
342#95C24112yes—
343#95C24112yes—
344#95C24112yes—
345#95C24112yes—
346#95C24124yes—
347#95C24124yes—
348#95C24112yes—
349#95C24112yes—
350#95C24112yes—
351#95C2414yes—
352#95C24124yes—
353#95C24112yes—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24C23×C414yes—
365C24C23×C414yes—
366C242−1+414yes—
367C242−1+414yes—
368C24C2512yes—
369C24C2512yes—
370C24C2511yescharacteristic
371C2×Q8—2256no—
372C2×Q8—2256no—
373C2×Q8—296no—
374C2×Q8—296no—
375C2×Q8—224no—
376C2×Q8—224no—
377C2×Q8—224no—
378C2×Q8—224no—
379C2×Q8C4×D4164yes—
380C2×Q8C23×C4148yes—
381C2×Q8C23×C4148yes—
382C2×Q8C23×C4116yes—
383C2×Q8C22×D4116yes—
384C2×Q8C2×#41116yes—
385C2×D4—232no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4C23×C4116yes—
389C22×C4—232no—
390C22×C4—22no—
391C22×C4—26no—
392C22×C4—26no—
393C22×C4—26no—
394C22×C4—26no—
395C22×C4—26no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—26no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—26no—
404C22×C4—26no—
405C22×C4—296no—
406C22×C4—296no—
407C22×C4—28no—
408C22×C4—224no—
409C22×C4—224no—
410C22×C4—224no—
411C22×C4—224no—
412C22×C4—28no—
413C22×C4—26no—
414C22×C4—22no—
415C22×C4—26no—
416C22×C4—22no—
417C22×C4—26no—
418C22×C4—26no—
419C22×C4—22no—
420C22×C4—26no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—26no—
424C22×C4—224no—
425C22×C4—26no—
426C22×C4—26no—
427C22×C4—26no—
428C22×C4—22no—
429C22×C4—224no—
430C22×C4—412no—
431C22×C4—448no—
432C22×C4C23×C414yes—
433C22×C4C23×C4112yes—
434C22×C4C23×C4112yes—
435C22×C4C23×C4112yes—
436C22×C4C23×C4148yes—
437C22×C4C23×C4148yes—
438C22×C4C23×C4112yes—
439C22×C4C22×D414yes—
440C22×C4C22×D4112yes—
441C22×C4C22×D4112yes—
442C22×C4C2×#41112yes—
443C22×C42−1+414yes—
444C22×C4C2512yes—
445C22×C4C2516yes—
446C22×C4C2516yes—
447C22×C4C2516yes—
448C22×C4C2512yes—
449C22×C4C2512yes—
450C22×C4C2516yes—
451C22×C4C2516yes—
452C22×C4C2516yes—
453C22×C4C2512yes—
454C22×C4C2516yes—
455C22×C4C2512yes—
456C22×C4C2516yes—
457C4⋊C4—248no—
458C4⋊C4—224no—
459C4⋊C4—224no—
460C4⋊C4—224no—
461C4⋊C4—224no—
462C4⋊C4—224no—
463C4⋊C4—224no—
464C4⋊C4—412no—
465C4⋊C4C23×C4148yes—
466C4⋊C4C23×C4148yes—
467C4⋊C4C23×C4196yes—
468C4⋊C4C22×D4148yes—
469C22⋊C4—224no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—224no—
473C22⋊C4—28no—
474C22⋊C4—412no—
475C22⋊C4C23×C4148yes—
476C22⋊C4C23×C4148yes—
477C42—224no—
478C42—224no—
479C42—224no—
480C42—248no—
481C42—224no—
482C42—224no—
483C42—424no—
484C42C23×C4148yes—
485C42C22×D4148yes—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23#519116yescomplemented
510C23C24×C414yes—
511C23C24×C414yes—
512C23C24×C412yes—
513C23C23×D412yes—
514C23C22×#4112yes—
515C23C2×2−1+414yes—
516C23C2×2−1+414yes—
517C23C2×2−1+412yes—
518C23C2×2−1+412yes—
519C23C2×2−1+412yes—
520C23#58512yes—
521C23C2611yescharacteristic
522Q8—296no—
523Q8—296no—
524Q8C2×C4×D4164yes—
525D4—232no—
526C2×C4—232no—
527C2×C4—22no—
528C2×C4—26no—
529C2×C4—26no—
530C2×C4—26no—
531C2×C4—26no—
532C2×C4—26no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—26no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—224no—
542C2×C4—224no—
543C2×C4—28no—
544C2×C4—224no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—224no—
548C2×C4—296no—
549C2×C4—296no—
550C2×C4—26no—
551C2×C4—22no—
552C2×C4—26no—
553C2×C4—22no—
554C2×C4—26no—
555C2×C4—26no—
556C2×C4—26no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—26no—
561C2×C4—26no—
562C2×C4—26no—
563C2×C4—26no—
564C2×C4—22no—
565C2×C4—448no—
566C2×C4—412no—
567C2×C4—44no—
568C2×C4—412no—
569C2×C4—412no—
570C2×C4—44no—
571C2×C4C2×C4×D4148yes—
572C2×C4C24×C414yes—
573C2×C4C24×C4112yes—
574C2×C4C24×C4112yes—
575C2×C4C24×C4112yes—
576C2×C4C24×C4112yes—
577C2×C4C23×D414yes—
578C2×C4C23×D4112yes—
579C2×C4C23×D4112yes—
580C2×C4C22×#41112yes—
581C2×C4C2×2−1+414yes—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—232no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—416no—
597C22C22×C4×D414yes—
598C22C2×#51914yes—
599C22C2×#51918yescomplemented
600C22C4×2−1+414yes—
601C22#348614yes—
602C22#348914yes—
603C22#351614yes—
604C22#352314yes—
605C22C25×C411yescharacteristic
606C22C24×D411yescharacteristic
607C22C23×#4111yescharacteristic
608C22C22×2−1+411yescharacteristic
609C22C22×2−1+411yescharacteristic
610C22C22×2−1+411yescharacteristic
611C22C2×#58511yescharacteristic
612C4—24no—
613C4—24no—
614C4—212no—
615C4—212no—
616C4—212no—
617C4—24no—
618C4—212no—
619C4—46no—
620C4—46no—
621C4—42no—
622C4—42no—
623C4—46no—
624C4C22×C4×D4124yes—
625C2—28no—
626C2—44no—
627C2#3429118yesdirect product factor
628C2C23×C4×D411yescharacteristic
629C2C22×#51911yescharacteristic
630C2C2×C4×2−1+411yescharacteristic
631C2C2×#348611yescharacteristic
632C2C2×#351611yescharacteristic
633C2C2×#352311yescharacteristic
634C2C2×#348911yescharacteristic
subgroupG/H
center, socleC24C25
commutatorC22C25×C4
Frattini, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#519C2semidirect
3. C23×C4×Q8C2semidirect
4. #34291C2direct (N × H)
5. C2×#2365C2semidirect
6. C2×#2362C2semidirect
7. C2×#2360C2semidirect
9. C2×#2351C2semidirect
10. C2×Q8×C22⋊C4C2semidirect
11. C2×Q8×C22⋊C4C2semidirect
12. C2×#2327C2semidirect
13. C2×#2323C2semidirect
14. C2×C4×C22⋊Q8C2semidirect
16. C2×#2276C2semidirect
19. C2×D4×Q8C4semidirect
25. C2×#519C22semidirect
36. C22×C4×Q8C22semidirect
42. #2365C22semidirect
43. #2362C22semidirect
44. #2360C22semidirect
46. #2351C22semidirect
47. Q8×C22⋊C4C22semidirect
48. Q8×C22⋊C4C22semidirect
49. #2327C22semidirect
50. #2323C22semidirect
51. C4×C22⋊Q8C22semidirect
75. #2276C22semidirect
90. C23×Q8C2×C4semidirect
101. D4×Q8C2×C4semidirect
102. C2×C4⋊Q8C2×C4semidirect
105. C2×C22⋊Q8C2×C4semidirect
117. C2×C4×Q8C2×C4semidirect
118. C2×C4×Q8D4semidirect
227. C22×Q8C22×C4semidirect
266. C4⋊Q8C22×C4semidirect
268. C22⋊Q8C22×C4semidirect
276. C4×Q8C22×C4semidirect
277. C4×Q8C2×D4semidirect
509. C23#519semidirect
599. C22C2×#519semidirect
627. C2#34291direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C22×#5192
C23×C4×Q82
#342912
C2×#23652
C2×#23622
C2×#23602
C2×#23522
C2×#23512
C2×Q8×C22⋊C42
C2×#23272
C2×#23232
C2×C4×C22⋊Q82
C2×#22902
C2×#22762