Finite Group Databasebeta

← C2×#36300 #7629951

Subgroups — C2×#36300 #7629951

Order 512.

6474 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C212yesmaximal, complemented
2C2×#2858C211yesmaximal, characteristic
3#36300C2132yesmaximal, direct product factor
4C2×#2567C211yesmaximal, characteristic, complemented
5C2×#2548C211yesmaximal, characteristic
6C2×#2513C212yesmaximal, complemented
7C2×#2512C212yesmaximal, complemented
8C2×#2521C214yesmaximal, complemented
9C2×#2474C214yesmaximal, complemented
10C2×#2448C212yesmaximal, complemented
11C2×#2454C212yesmaximal, complemented
12C2×#2369C211yesmaximal, characteristic, complemented
13C2×#2367C212yesmaximal, complemented
14C2×#2366C211yesmaximal, characteristic
15C2×#2354C212yesmaximal, complemented
16C2×C4×#125C211yesmaximal, characteristic, complemented
17C2×#2293C212yesmaximal, complemented
18C2×C42×Q8C211yesmaximal, characteristic, complemented
19C22×C4⋊Q8C2211yescharacteristic
20C22×#127C2211yescharacteristic
21C22×#126C2214yes—
22C22×#125C2211yescharacteristic
23C22×#124C2212yes—
24C22×#124C2212yes—
25C22×#124C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2212yes—
30C22×C4×D4C2212yes—
31C22×#117C2214yes—
32#2860C22132yescomplemented
33#2858C22116yes—
34#2567C22116yescomplemented
35#2548C22116yes—
36#2521C22164yescomplemented
37#2513C22132yescomplemented
38#2512C22132yescomplemented
39#2474C22164yescomplemented
40#2454C22132yescomplemented
41#2448C22132yescomplemented
42C2×#399C2214yes—
43C2×#397C2214yes—
44C2×#397C2214yes—
45C2×#396C2214yes—
46C2×#395C2214yes—
47C2×#394C2214yes—
48#2369C22116yescomplemented
49#2367C22132yescomplemented
50#2366C22116yes—
51#2354C22132yescomplemented
52C4×#125C22116yescomplemented
53C2×#391C2212yes—
54C2×#391C2214yes—
55#2293C22132yescomplemented
56C2×#390C2212yes—
57C2×#390C2212yes—
58C2×#390C2212yes—
59C2×#390C2212yes—
60C2×#390C2212yes—
61C2×#390C2212yes—
62C2×#390C2212yes—
63C2×#390C2212yes—
64C2×#390C2212yes—
65C2×#389C2211yescharacteristic
66C2×#389C2212yes—
67C2×#389C2212yes—
68C2×#389C2212yes—
69C2×#389C2212yes—
70C2×#389C2211yescharacteristic
71C2×#388C2214yes—
72C2×#388C2214yes—
73C2×#388C2214yes—
74C2×#388C2214yes—
75C2×#388C2214yes—
76C2×#388C2214yes—
77C2×#388C2214yes—
78C2×#387C2212yes—
79C2×#387C2212yes—
80C2×#386C2214yes—
81C2×#385C2214yes—
82C2×#384C2212yes—
83C2×#382C2211yescharacteristic
84C2×#380C2214yes—
85C42×Q8C22116yescomplemented
86C2×C4×C4⋊C4C2211yescharacteristic
87C2×C4×C4⋊C4C2212yes—
88C2×C4×C4⋊C4C2212yes—
89C2×C4×C4⋊C4C2211yescharacteristic
90C2×C4×C4⋊C4C2211yescharacteristic
91C2×C4×C4⋊C4C2212yes—
92C2×C4×C4⋊C4C2211yescharacteristic
93C2×C4×C4⋊C4C2212yes—
94C2×C4×C4⋊C4C2212yes—
95C2×C4×C4⋊C4C2212yes—
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C4⋊C4C2211yescharacteristic
98C2×C4×C22⋊C4C2214yes—
99C2×C43C2211yescharacteristic
100C2×C43C2212yes—
101C23×Q8C2311yescharacteristic
102C23×Q8C2312yes—
103C23×D4C2312yes—
104C23×D4C2312yes—
105C23×D4C2312yes—
106C24×C4C2314yes—
107C2×C4⋊Q8—28no—
108C2×C4⋊Q8C2318yes—
109C2×#127—28no—
110C2×#127C2318yes—
111C2×#126—416no—
112C2×#126C23132yes—
113C2×#125—28no—
114C2×#125C2318yes—
115C2×#124—216no—
116C2×#124—48no—
117C2×#124—48no—
118C2×#124C23116yes—
119C2×#124C23116yes—
120C2×#124C23116yes—
121C2×C4×Q8—28no—
122C2×C4×Q8—28no—
123C2×C4×Q8—28no—
124C2×C4×Q8C2318yes—
125C2×C4×Q8C2318yes—
126C2×C4×Q8C2318yes—
127C2×C4×D4—216no—
128C2×C4×D4—216no—
129C2×C4×D4C23116yes—
130C2×C4×D4C23116yes—
131C2×#117—416no—
132C2×#117C23132yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2314yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2311yescharacteristic
151C22×C4⋊C4C2311yescharacteristic
152C22×C22⋊C4C2314yes—
153C22×C22⋊C4C2314yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2314yes—
158C22×C22⋊C4C2314yes—
159C22×C22⋊C4C2314yes—
160C22×C22⋊C4C2314yes—
161C22×C22⋊C4C2314yes—
162C22×C22⋊C4C2314yes—
163C22×C42C2312yes—
164C22×C42C2312yes—
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2311yescharacteristic
168C22×C42C2311yescharacteristic
169C22×C42C2312yes—
170C22×C42C2312yes—
171C22×C42C2311yescharacteristic
172C22×C42C2311yescharacteristic
173C22×C42C2312yes—
174C22×C42C2311yescharacteristic
175C22×C42C2312yes—
176C22×C42C2311yescharacteristic
177C22×C42C2314yes—
178C22×C42C2311yescharacteristic
179#399C23132yes—
180#397C23132yes—
181#397C23132yes—
182#396C23132yes—
183#395C23132yes—
184#394C23132yes—
185#391C23116yes—
186#391C23132yes—
187#390C23116yes—
188#390C23116yes—
189#390C23116yes—
190#390C23116yes—
191#390C23116yes—
192#390C23116yes—
193#390C23116yes—
194#390C23116yes—
195#390C23116yes—
196#389C2318yes—
197#389C2318yes—
198#389C23116yes—
199#389C23116yes—
200#389C23116yes—
201#389C23116yes—
202#388C23132yes—
203#388C23132yes—
204#388C23132yes—
205#388C23132yes—
206#388C23132yes—
207#388C23132yes—
208#388C23132yes—
209#387C23116yes—
210#387C23116yes—
211#386C23132yes—
212#385C23132yes—
213#384C23116yes—
214#382C2318yes—
215#380C23132yes—
216C4×C4⋊C4C2318yes—
217C4×C4⋊C4C23116yes—
218C4×C4⋊C4C23116yes—
219C4×C4⋊C4C2318yes—
220C4×C4⋊C4C2318yes—
221C4×C4⋊C4C23116yes—
222C4×C4⋊C4C2318yes—
223C4×C4⋊C4C23116yes—
224C4×C4⋊C4C23116yes—
225C4×C4⋊C4C23116yes—
226C4×C4⋊C4C2318yes—
227C4×C4⋊C4C23116yes—
228C4×C22⋊C4C23132yes—
229C2×#95C2312yes—
230C2×#95C2312yes—
231C2×#95C2314yes—
232C2×#95C2314yes—
233C2×#95C2312yes—
234C2×#95C2312yes—
235C2×#95C2312yes—
236C2×#95C2314yes—
237C2×#95C2312yes—
238C2×#95C2312yes—
239C2×#95C2312yes—
240C2×#95C2314yes—
241C2×#95C2312yes—
242C43C2318yes—
243C43C23116yes—
244C25C2414yes—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—28no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8#4118yes—
251C22×Q8C2414yes—
252C22×Q8C2418yes—
253C22×D4—28no—
254C22×D4—28no—
255C22×D4—28no—
256C22×D4—28no—
257C22×D4—28no—
258C22×D4—28no—
259C22×D4—28no—
260C22×D4—44no—
261C22×D4—44no—
262C22×D4C2418yes—
263C22×D4C2418yes—
264C22×D4C2418yes—
265C23×C4—216no—
266C23×C4—216no—
267C23×C4—48no—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2414yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2412yes—
276C23×C4C24116yes—
277C23×C4C2412yes—
278C23×C4C2412yes—
279C23×C4C2412yes—
280C23×C4C2411yescharacteristic
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C23×C4C2411yescharacteristic
285C4⋊Q8—232no—
286#127—232no—
287#126—464no—
288#125—232no—
289#124—264no—
290#124—432no—
291#124—432no—
292C4×Q8—232no—
293C4×Q8—232no—
294C4×Q8—232no—
295C4×D4—264no—
296C4×D4—264no—
297#117—464no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4#4118yes—
316C2×C4⋊C4#4118yes—
317C2×C4⋊C4C2418yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2418yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2418yes—
322C2×C4⋊C4C2418yes—
323C2×C4⋊C4C2418yes—
324C2×C4⋊C4C2418yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2418yes—
327C2×C4⋊C4C2418yes—
328C2×C4⋊C4C2418yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C24116yes—
332C2×C4⋊C4C24116yes—
333C2×C4⋊C4C24116yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2414yes—
336C2×C22⋊C4—216no—
337C2×C22⋊C4—216no—
338C2×C22⋊C4—216no—
339C2×C22⋊C4—216no—
340C2×C22⋊C4—216no—
341C2×C22⋊C4—216no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—48no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4C24116yes—
348C2×C22⋊C4C24116yes—
349C2×C22⋊C4C24116yes—
350C2×C22⋊C4C24116yes—
351C2×C22⋊C4C24116yes—
352C2×C22⋊C4C24116yes—
353C2×C22⋊C4C24116yes—
354C2×C22⋊C4C24116yes—
355C2×C22⋊C4C24116yes—
356C2×C22⋊C4C24116yes—
357C2×C22⋊C4C24116yes—
358C2×C42—28no—
359C2×C42—28no—
360C2×C42—28no—
361C2×C42—28no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—28no—
365C2×C42—28no—
366C2×C42—24no—
367C2×C42—28no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—28no—
371C2×C42—24no—
372C2×C42—48no—
373C2×C42#4118yes—
374C2×C42C2418yes—
375C2×C42C2418yes—
376C2×C42C2418yes—
377C2×C42C2418yes—
378C2×C42C2414yes—
379C2×C42C2414yes—
380C2×C42C2418yes—
381C2×C42C2418yes—
382C2×C42C2414yes—
383C2×C42C2414yes—
384C2×C42C2418yes—
385C2×C42C2414yes—
386C2×C42C2418yes—
387C2×C42C2414yes—
388C2×C42C24116yes—
389C2×C42C2414yes—
390#95C2418yes—
391#95C24116yes—
392#95C2418yes—
393#95C2418yes—
394#95C2418yes—
395#95C24116yes—
396#95C2418yes—
397#95C2418yes—
398#95C2418yes—
399#95C2418yes—
400#95C2418yes—
401#95C24116yes—
402#95C24116yes—
403C24—28no—
404C24—28no—
405C24—28no—
406C24—28no—
407C24—28no—
408C24—28no—
409C24—28no—
410C24C2518yes—
411C24C2511yescharacteristic
412C2×Q8—232no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—216no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—816no—
419C2×Q8C2×#41116yes—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—216no—
423C2×D4—216no—
424C2×D4—216no—
425C2×D4—216no—
426C2×D4—216no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—432no—
430C2×D4—432no—
431C2×D4—816no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—232no—
469C22×C4—232no—
470C22×C4—416no—
471C22×C4—464no—
472C22×C4C2×#4118yes—
473C22×C4C2×#4118yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4118yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C42−1+414yes—
484C22×C4C2514yes—
485C22×C4C2514yes—
486C22×C4C2514yes—
487C22×C4C2512yes—
488C22×C4C2512yes—
489C22×C4C2514yes—
490C22×C4C2518yes—
491C22×C4C2514yes—
492C22×C4C2512yes—
493C22×C4C2514yes—
494C22×C4C2514yes—
495C22×C4C2514yes—
496C22×C4C2512yes—
497C22×C4C2512yes—
498C22×C4C2512yes—
499C22×C4C2512yes—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4C2×#41116yes—
518C4⋊C4C2×#41116yes—
519C22⋊C4—232no—
520C22⋊C4—232no—
521C22⋊C4—232no—
522C22⋊C4—232no—
523C22⋊C4—232no—
524C22⋊C4—232no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C42—216no—
531C42—216no—
532C42—216no—
533C42—216no—
534C42—28no—
535C42—28no—
536C42—216no—
537C42—216no—
538C42—28no—
539C42—216no—
540C42—28no—
541C42—28no—
542C42—216no—
543C42—28no—
544C42—416no—
545C42C2×#41116yes—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—416no—
554C23—416no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—416no—
559C23—416no—
560C23C22×#4112yes—
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C22×#4112yes—
564C23C2×H(5,𝔽2)12yes—
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23C2611yescharacteristic
568Q8—232no—
569Q8—816no—
570D4—432no—
571D4—432no—
572D4—816no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—216no—
586C2×C4—216no—
587C2×C4—216no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—464no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—416no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4#529116yes—
625C2×C4#556116yes—
626C2×C4C22×#4118yes—
627C2×C4C22×#4118yes—
628C2×C4C22×#4114yes—
629C2×C4C22×#4118yes—
630C2×C4C22×#4114yes—
631C2×C4C22×#4114yes—
632C2×C4C22×#4114yes—
633C2×C4C22×#4114yes—
634C2×C4C22×#4114yes—
635C2×C4C2×H(5,𝔽2)14yes—
636C2×C4C2×H(5,𝔽2)14yes—
637C2×C4C2×2−1+414yes—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—832no—
646C22C2×#52914yes—
647C22C2×#55614yes—
648C22#348114yes—
649C22#348314yes—
650C22#350814yes—
651C22#352214yes—
652C22#352214yes—
653C22C23×#4111yescharacteristic
654C22C23×#4111yescharacteristic
655C22C23×#4111yescharacteristic
656C22C23×#4111yescharacteristic
657C22C22×H(5,𝔽2)11yescharacteristic
658C22C22×H(5,𝔽2)11yescharacteristic
659C22C22×2−1+411yescharacteristic
660C4—24no—
661C4—24no—
662C4—24no—
663C4—28no—
664C4—28no—
665C4—28no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—48no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—42no—
674C4C2×#52918yes—
675C4C2×#55618yes—
676C2—88no—
677C2#3630018yesdirect product factor
678C2C22×#52911yescharacteristic
679C2C22×#55611yescharacteristic
680C2C2×#348111yescharacteristic
681C2C2×#348311yescharacteristic
682C2C2×#352211yescharacteristic
683C2C2×#352211yescharacteristic
684C2C2×#350811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2860C2semidirect
3. #36300C2direct (N × H)
4. C2×#2567C2semidirect
6. C2×#2513C2semidirect
7. C2×#2512C2semidirect
8. C2×#2521C2semidirect
9. C2×#2474C2semidirect
10. C2×#2448C2semidirect
11. C2×#2454C2semidirect
12. C2×#2369C2semidirect
13. C2×#2367C2semidirect
15. C2×#2354C2semidirect
16. C2×C4×#125C2semidirect
17. C2×#2293C2semidirect
18. C2×C42×Q8C2semidirect
32. #2860C22semidirect
34. #2567C22semidirect
36. #2521C22semidirect
37. #2513C22semidirect
38. #2512C22semidirect
39. #2474C22semidirect
40. #2454C22semidirect
41. #2448C22semidirect
48. #2369C22semidirect
49. #2367C22semidirect
51. #2354C22semidirect
52. C4×#125C22semidirect
55. #2293C22semidirect
85. C42×Q8C22semidirect
677. C2#36300direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28602
C2×#28582
#363002
C2×#25672
C2×#25482
C2×#25132
C2×#25122
C2×#25212
C2×#24742
C2×#24482
C2×#24542
C2×#23692
C2×#23672
C2×#23662
C2×#23542
C2×C4×#1252
C2×#22932
C2×C42×Q82