Finite Group Databasebeta

← C2×#39353 #7636718

Subgroups — C2×#39353 #7636718

Order 512.

6946 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C212yesmaximal
2C2×#2722C214yesmaximal
3C2×#2719C214yesmaximal, complemented
4#39353C2132yesmaximal, direct product factor
5C2×#2857C211yesmaximal, characteristic
6C2×#2552C212yesmaximal, complemented
7C2×#2552C211yesmaximal, characteristic, complemented
8C2×#2546C211yesmaximal, characteristic, complemented
9C2×#2506C212yesmaximal, complemented
10C2×#2496C212yesmaximal
11C2×#2496C212yesmaximal
12C2×#2453C212yesmaximal, complemented
13C2×#2446C212yesmaximal, complemented
14C2×#2625C212yesmaximal
15C2×#2365C211yesmaximal, characteristic, complemented
16C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
17C2×C4×C22⋊Q8C211yesmaximal, characteristic
18C2×#2285C211yesmaximal, characteristic
19C2×#541—232no—
20C2×#541C22132yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×#126C2214yes—
24C22×#126C2214yes—
25C22×#125C2212yes—
26C22×#125C2214yes—
27C22×#125C2214yes—
28C22×C22⋊Q8C2212yes—
29C22×C22⋊Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2212yes—
32C22×C4×Q8C2212yes—
33C22×#117C2212yes—
34C23×C42C2211yescharacteristic
35#2857C22116yes—
36#2722C22164yes—
37#2719C22164yescomplemented
38#2625C22132yes—
39#2552C22116yescomplemented
40#2552C22132yescomplemented
41#2546C22116yescomplemented
42#2506C22132yescomplemented
43#2496C22132yes—
44#2496C22132yes—
45#2453C22132yescomplemented
46#2446C22132yescomplemented
47C2×#400C2214yes—
48C2×#400C2214yes—
49C2×#400C2214yes—
50C2×#398C2214yes—
51C2×#398C2214yes—
52C2×#398C2214yes—
53C2×#398C2214yes—
54C2×#398C2214yes—
55C2×#397C2214yes—
56C2×#397C2214yes—
57#2365C22116yescomplemented
58Q8×C4⋊C4C22116yescomplemented
59C4×C22⋊Q8C22116yes—
60C2×#391C2212yes—
61C2×#391C2212yes—
62C2×#391C2212yes—
63C2×#391C2211yescharacteristic
64C2×#389C2212yes—
65C2×#389C2212yes—
66C2×#389C2212yes—
67C2×#389C2212yes—
68C2×#389C2214yes—
69C2×#389C2212yes—
70C2×#389C2212yes—
71C2×#389C2212yes—
72C2×#389C2214yes—
73C2×#389C2212yes—
74C2×#389C2211yescharacteristic
75C2×#387C2214yes—
76C2×#387C2214yes—
77C2×#387C2214yes—
78C2×#387C2214yes—
79C2×#387C2212yes—
80C2×#385C2212yes—
81#2285C22116yes—
82C2×#384C2211yescharacteristic
83C2×#384C2212yes—
84C2×#384C2212yes—
85C2×#384C2211yescharacteristic
86C2×#384C2211yescharacteristic
87C2×#382C2212yes—
88C2×#380C2212yes—
89C2×#380C2212yes—
90C2×#380C2211yescharacteristic
91C2×#380C2212yes—
92C2×#380C2211yescharacteristic
93C2×#380C2211yescharacteristic
94C2×C4×C4⋊C4C2211yescharacteristic
95C2×C4×C4⋊C4C2212yes—
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C4⋊C4C2211yescharacteristic
98C2×C4×C4⋊C4C2211yescharacteristic
99C2×C4×C4⋊C4C2212yes—
100C2×C4×C4⋊C4C2212yes—
101C2×C4×C22⋊C4C2212yes—
102C23×Q8C2312yes—
103C23×Q8C2311yescharacteristic
104C24×C4C2311yescharacteristic
105C24×C4C2311yescharacteristic
106C24×C4C2311yescharacteristic
107#541—2256no—
108C2×C4⋊Q8—216no—
109C2×C4⋊Q8—28no—
110C2×C4⋊Q8C2318yes—
111C2×C4⋊Q8C23116yes—
112C2×#126—232no—
113C2×#126—232no—
114C2×#126C23132yes—
115C2×#126C23132yes—
116C2×#125—216no—
117C2×#125—232no—
118C2×#125—232no—
119C2×#125C23116yes—
120C2×#125C23132yes—
121C2×#125C23132yes—
122C2×C22⋊Q8—216no—
123C2×C22⋊Q8D4116yes—
124C2×C22⋊Q8C23116yes—
125C2×C22⋊Q8C2318yes—
126C2×C4×Q8—216no—
127C2×C4×Q8—216no—
128C2×C4×Q8—28no—
129C2×C4×Q8C23116yes—
130C2×C4×Q8C2318yes—
131C2×C4×Q8C23116yes—
132C2×#117—216no—
133C2×#117C23116yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2314yes—
142C22×C4⋊C4C2314yes—
143C22×C4⋊C4C2314yes—
144C22×C4⋊C4C2314yes—
145C22×C4⋊C4C2314yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2311yescharacteristic
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2314yes—
157C22×C4⋊C4C2314yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2312yes—
162C22×C22⋊C4C2312yes—
163C22×C22⋊C4C2312yes—
164C22×C22⋊C4C2314yes—
165C22×C22⋊C4C2312yes—
166C22×C22⋊C4C2312yes—
167C22×C42—28no—
168C22×C42C2318yes—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171C22×C42C2311yescharacteristic
172C22×C42C2312yes—
173C22×C42C2312yes—
174C22×C42C2314yes—
175C22×C42C2314yes—
176C22×C42C2312yes—
177C22×C42C2312yes—
178C22×C42C2311yescharacteristic
179#400C23132yes—
180#400C23132yes—
181#400C23132yes—
182#398C23132yes—
183#398C23132yes—
184#398C23132yes—
185#398C23132yes—
186#398C23132yes—
187#397C23132yes—
188#397C23132yes—
189#391C23116yes—
190#391C23116yes—
191#391C23116yes—
192#391C2318yes—
193#389C23116yes—
194#389C23116yes—
195#389C23116yes—
196#389C23132yes—
197#389C23116yes—
198#389C23116yes—
199#389C2318yes—
200#389C23132yes—
201#389C23116yes—
202#389C23116yes—
203#389C23116yes—
204#387C23132yes—
205#387C23132yes—
206#387C23132yes—
207#387C23132yes—
208#387C23116yes—
209#385C23116yes—
210#384C2318yes—
211#384C23116yes—
212#384C23116yes—
213#384C2318yes—
214#384C2318yes—
215#382C23116yes—
216#380C23116yes—
217#380C23116yes—
218#380C2318yes—
219#380C23116yes—
220#380C2318yes—
221#380C2318yes—
222C4×C4⋊C4C2318yes—
223C4×C4⋊C4C23116yes—
224C4×C4⋊C4C23116yes—
225C4×C4⋊C4C2318yes—
226C4×C4⋊C4C23116yes—
227C4×C4⋊C4C23116yes—
228C4×C4⋊C4C2318yes—
229C4×C22⋊C4C23116yes—
230C2×#95C2314yes—
231C2×#95C2314yes—
232C2×#95C2314yes—
233C2×#95C2314yes—
234C2×#95C2314yes—
235C2×#95C2314yes—
236C2×#95C2312yes—
237C2×#95C2312yes—
238C2×#95C2312yes—
239C2×#95C2312yes—
240C2×#95C2314yes—
241C2×#95C2312yes—
242C2×#95C2312yes—
243C2×#95C2312yes—
244C2×#95C2312yes—
245C2×#95C2314yes—
246C2×#95C2312yes—
247C25C2411yescharacteristic
248C22×Q8—28no—
249C22×Q8—24no—
250C22×Q8—28no—
251C22×Q8—24no—
252C22×Q8—28no—
253C22×Q8C2×D418yes—
254C22×Q8C2418yes—
255C22×Q8C2414yes—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—24no—
262C23×C4—24no—
263C23×C4—42no—
264C23×C4C2×D418yes—
265C23×C4C2414yes—
266C23×C4C2412yes—
267C23×C4C2414yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2414yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C23×C4C2414yes—
276C23×C4C2414yes—
277C23×C4C2412yes—
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C4⋊Q8—264no—
285C4⋊Q8—232no—
286#126—2128no—
287#126—2128no—
288#125—264no—
289#125—2128no—
290#125—2128no—
291C22⋊Q8—264no—
292C22⋊Q8C2×D4164yes—
293C4×Q8—264no—
294C4×Q8—264no—
295C4×Q8—232no—
296#117—264no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—216no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—216no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4C2×D418yes—
324C2×C4⋊C4C2×D4116yes—
325C2×C4⋊C4C2418yes—
326C2×C4⋊C4C2418yes—
327C2×C4⋊C4C2418yes—
328C2×C4⋊C4C24116yes—
329C2×C4⋊C4C24116yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2418yes—
332C2×C4⋊C4C24116yes—
333C2×C4⋊C4C24116yes—
334C2×C4⋊C4C24116yes—
335C2×C4⋊C4C24116yes—
336C2×C4⋊C4C24116yes—
337C2×C4⋊C4C2418yes—
338C2×C4⋊C4C2418yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2418yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2418yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2418yes—
347C2×C4⋊C4C2418yes—
348C2×C4⋊C4C2418yes—
349C2×C4⋊C4C24116yes—
350C2×C4⋊C4C24116yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2418yes—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—28no—
355C2×C22⋊C4—216no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4C2×D4116yes—
358C2×C22⋊C4C2418yes—
359C2×C22⋊C4C2418yes—
360C2×C22⋊C4C24116yes—
361C2×C22⋊C4C2418yes—
362C2×C22⋊C4C2418yes—
363C2×C42—28no—
364C2×C42—28no—
365C2×C42—216no—
366C2×C42—216no—
367C2×C42—28no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—28no—
372C2×C42—232no—
373C2×C42#4118yes—
374C2×C42C2418yes—
375C2×C42C2418yes—
376C2×C42C24116yes—
377C2×C42C24116yes—
378C2×C42C2418yes—
379C2×C42C2418yes—
380C2×C42C2414yes—
381C2×C42C2414yes—
382C2×C42C2414yes—
383C2×C42C2414yes—
384#95C2418yes—
385#95C2418yes—
386#95C2418yes—
387#95C2418yes—
388#95C2418yes—
389#95C2418yes—
390#95C2418yes—
391#95C2418yes—
392#95C2418yes—
393#95C24116yes—
394#95C24116yes—
395#95C24116yes—
396#95C24116yes—
397#95C24116yes—
398#95C24116yes—
399#95C24116yes—
400#95C24116yes—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24C22×D414yes—
408C24C2512yes—
409C24C2511yescharacteristic
410C2×Q8—232no—
411C2×Q8—216no—
412C2×Q8—28no—
413C2×Q8—216no—
414C2×Q8—28no—
415C2×Q8—216no—
416C2×Q8—432no—
417C2×Q8C22×D4116yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—232no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—44no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4C22×D418yes—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D4116yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C42−1+414yes—
475C22×C42−1+414yes—
476C22×C42−1+414yes—
477C22×C42−1+414yes—
478C22×C4C2512yes—
479C22×C4C2514yes—
480C22×C4C2514yes—
481C22×C4C2514yes—
482C22×C4C2514yes—
483C22×C4C2518yes—
484C22×C4C2514yes—
485C22×C4C2514yes—
486C22×C4C2518yes—
487C22×C4C2514yes—
488C22×C4C2514yes—
489C22×C4C2512yes—
490C22×C4C2512yes—
491C22×C4C2512yes—
492C22×C4C2512yes—
493C22×C4C2512yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—232no—
498C4⋊C4—232no—
499C4⋊C4—232no—
500C4⋊C4—232no—
501C4⋊C4—232no—
502C4⋊C4—232no—
503C4⋊C4—232no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—416no—
519C4⋊C4—416no—
520C4⋊C4C22×D4116yes—
521C4⋊C4C22×D4132yes—
522C22⋊C4—216no—
523C22⋊C4—216no—
524C22⋊C4—232no—
525C22⋊C4—48no—
526C22⋊C4C22×D4132yes—
527C42—216no—
528C42—216no—
529C42—232no—
530C42—232no—
531C42—28no—
532C42—28no—
533C42—28no—
534C42—216no—
535C42—216no—
536C42C2×#41116yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23C23×D414yes—
551C23C23×D412yes—
552C23C22×#4112yes—
553C23C2×2−1+412yes—
554C23C2×2−1+412yes—
555C23C2×2−1+412yes—
556C23C2×2−1+412yes—
557C23#58512yes—
558C23C2611yescharacteristic
559Q8—232no—
560Q8—432no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—216no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—232no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
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—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—416no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—416no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—416no—
615C2×C4#541116yescomplemented
616C2×C4#547116yes—
617C2×C4C23×D418yes—
618C2×C4C23×D418yes—
619C2×C4C23×D414yes—
620C2×C4C23×D414yes—
621C2×C4C22×#4114yes—
622C2×C4C22×#4114yes—
623C2×C4C22×#4114yes—
624C2×C4C2×2−1+414yes—
625C2×C4C2×2−1+414yes—
626C2×C4C2×2−1+414yes—
627C2×C4C2×2−1+414yes—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—416no—
635C22—42no—
636C22C2×#54114yes—
637C22C2×#54714yes—
638C22#349514yes—
639C22#349714yes—
640C22#349914yes—
641C22#355514yes—
642C22#355614yes—
643C22C24×D411yescharacteristic
644C22C23×#4111yescharacteristic
645C22C22×2−1+411yescharacteristic
646C22C22×2−1+411yescharacteristic
647C22C22×2−1+411yescharacteristic
648C22C22×2−1+411yescharacteristic
649C22C2×#58511yescharacteristic
650C4—28no—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—28no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—48no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—42no—
663C4—48no—
664C4C2×#54118yescomplemented
665C4C2×#54718yes—
666C2—44no—
667C2#3935318yesdirect product factor
668C2C22×#54111yescharacteristic
669C2C22×#54711yescharacteristic
670C2C2×#355511yescharacteristic
671C2C2×#355611yescharacteristic
672C2C2×#349711yescharacteristic
673C2C2×#349911yescharacteristic
674C2C2×#349511yescharacteristic
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
3. C2×#2719C2semidirect
4. #39353C2direct (N × H)
6. C2×#2552C2semidirect
7. C2×#2552C2semidirect
8. C2×#2546C2semidirect
9. C2×#2506C2semidirect
12. C2×#2453C2semidirect
13. C2×#2446C2semidirect
15. C2×#2365C2semidirect
16. C2×Q8×C4⋊C4C2semidirect
37. #2719C22semidirect
39. #2552C22semidirect
40. #2552C22semidirect
41. #2546C22semidirect
42. #2506C22semidirect
45. #2453C22semidirect
46. #2446C22semidirect
57. #2365C22semidirect
58. Q8×C4⋊C4C22semidirect
615. C2×C4#541semidirect
664. C4C2×#541semidirect
667. C2#39353direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#27222
C2×#27192
#393532
C2×#28572
C2×#25522
C2×#25462
C2×#25062
C2×#24962
C2×#24532
C2×#24462
C2×#26252
C2×#23652
C2×Q8×C4⋊C42
C2×C4×C22⋊Q82
C2×#22852