Finite Group Databasebeta

← C2×#30633 #7501376

Subgroups — C2×#30633 #7501376

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2C22×#483C211yesmaximal, characteristic
3C2×#2429C211yesmaximal, characteristic
4C2×#2440C211yesmaximal, characteristic
5#30633C2116yesmaximal, direct product factor
6C2×#2096C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2058C211yesmaximal, characteristic, complemented
9C2×#2043C211yesmaximal, characteristic, complemented
10C2×#2035C211yesmaximal, characteristic
11C2×#2035C211yesmaximal, characteristic
12C2×#1941C211yesmaximal, characteristic, complemented
13C2×#1923C211yesmaximal, characteristic
14C2×#1872C211yesmaximal, characteristic
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1803C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C2×#487—28no—
25C2×#487C2218yescomplemented
26C2×#483—28no—
27C2×#483C2218yes—
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#103C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43C22×C4×C8C2211yescharacteristic
44#2440—24no—
45#2440C2218yes—
46#2429—24no—
47#2429C2218yes—
48C2×#398—21no—
49C2×#398—21no—
50C2×#396—21no—
51C2×#395—21no—
52C2×#391C2211yescharacteristic
53C2×#389—21no—
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#388—21no—
58C2×#385—21no—
59C2×#384C2211yescharacteristic
60C2×#380C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62#2096C2218yescomplemented
63#2066C2218yescomplemented
64#2058C2218yescomplemented
65#2043C2218yescomplemented
66#2035C2218yes—
67#2035C2218yes—
68#1941C2218yescomplemented
69#1923C2218yes—
70#1872C2218yes—
71#1865C2218yescomplemented
72#1803C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2314yes—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—42no—
102C2×C22⋊Q8—42no—
103C2×C22⋊Q8—48no—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—24no—
108C2×C4×Q8—42no—
109C2×C4×Q8C2314yes—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—42no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#487—232no—
141#483—232no—
142C2×#106—24no—
143C2×#106—24no—
144C2×#106—24no—
145C2×#106—42no—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8—42no—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103D418yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×C22⋊C8—24no—
169C2×C22⋊C8—42no—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172C2×C4×C8—24no—
173C2×C4×C8C2314yes—
174#398—28no—
175#398—28no—
176#396—28no—
177#395—28no—
178#391—22no—
179#391C2314yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—28no—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#388—28no—
188#385—28no—
189#384—22no—
190#384C2314yes—
191#380—22no—
192#380C2314yes—
193C4×C22⋊C4D414yes—
194C4×C22⋊C4C2314yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—21no—
212C22×Q8—24no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—82no—
227C22×Q8C2×D414yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C2×D414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—28no—
273C4⋊Q8—28no—
274C4⋊Q8—44no—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—432no—
285C4×Q8—28no—
286C4×Q8—44no—
287C4×Q8—48no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4#4114yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4SD1618yes—
347C2×C22⋊C4C2×D414yes—
348C2×C22⋊C4C2×D414yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359#106—28no—
360#106—28no—
361#106—28no—
362#106—44no—
363C4⋊C8—44no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103C2×D4116yes—
372C22⋊C8—28no—
373C22⋊C8—44no—
374C4×C8—28no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D414yes—
382#95C2×D414yes—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24C2×SD1614yes—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—84no—
420C2×Q8—816no—
421C2×Q8C22×D414yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—48no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—84no—
482C22×C4—816no—
483C22×C4C2×SD1614yes—
484C22×C4C2×SD1614yes—
485C22×C4#13714yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C42−1+412yes—
497C22×C42−1+412yes—
498C22×C4C2511yescharacteristic
499C2×C8—28no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8C22×D414yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—84no—
543C4⋊C4C22×D414yes—
544C4⋊C4C2×#4114yes—
545C22⋊C4—28no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4C2×SD16116yes—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—48no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—28no—
562C23—28no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23#53612yes—
571C23D4212yes—
572C23#54612yes—
573C23C22×SD1614yes—
574C23C22×SD1612yes—
575C23C2×#13712yes—
576C23#57712yes—
577C23#57812yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—816no—
589Q8—84no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
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—816no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4#53612yes—
646C2×C4#53714yes—
647C2×C4#54214yes—
648C2×C4D4212yes—
649C2×C4#54612yes—
650C2×C4#54714yes—
651C2×C4#56214yes—
652C2×C4C22×SD1614yes—
653C2×C4C22×SD1614yes—
654C2×C4C2×#13714yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—24no—
662C8—42no—
663C8—42no—
664C8—42no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—42no—
669C22—42no—
670C22—416no—
671C22—42no—
672C22—42no—
673C22#267214yes—
674C22#322914yes—
675C22#324214yes—
676C22#327014yes—
677C22D4×SD1614yes—
678C22#329814yes—
679C22#333114yes—
680C22C2×#53611yescharacteristic
681C22C2×D4211yescharacteristic
682C22C2×#54611yescharacteristic
683C22C23×SD1611yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57711yescharacteristic
686C22C2×#57811yescharacteristic
687C4—24no—
688C4—24no—
689C4—24no—
690C4—44no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4#267214yes—
701C4C2×#53711yescharacteristic
702C4C2×#54211yescharacteristic
703C4C2×#54711yescharacteristic
704C4C2×#56211yescharacteristic
705C2—44no—
706C2#3063318yesdirect product factor
707C2C2×#267211yescharacteristic
708C2C2×#322911yescharacteristic
709C2C2×#324211yescharacteristic
710C2C2×#327011yescharacteristic
711C2C2×D4×SD1611yescharacteristic
712C2C2×#329811yescharacteristic
713C2C2×#333111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#487C2semidirect
5. #30633C2direct (N × H)
6. C2×#2096C2semidirect
7. C2×#2066C2semidirect
8. C2×#2058C2semidirect
9. C2×#2043C2semidirect
12. C2×#1941C2semidirect
15. C2×#1865C2semidirect
16. C2×#1803C2semidirect
25. C2×#487C22semidirect
62. #2096C22semidirect
63. #2066C22semidirect
64. #2058C22semidirect
65. #2043C22semidirect
68. #1941C22semidirect
71. #1865C22semidirect
72. #1803C22semidirect
706. C2#30633direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4872
C22×#4832
C2×#24292
C2×#24402
#306332
C2×#20962
C2×#20662
C2×#20582
C2×#20432
C2×#20352
C2×#19412
C2×#19232
C2×#18722
C2×#18652
C2×#18032