Finite Group Databasebeta

← C2×#37710 #7635121

Subgroups — C2×#37710 #7635121

Order 512.

8284 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic, complemented
2C22×#533C211yesmaximal, characteristic, complemented
3C2×#2681C214yesmaximal, complemented
4C2×#2676C214yesmaximal, complemented
5#37710C2132yesmaximal, direct product factor
6C2×#2856C211yesmaximal, characteristic, complemented
7C2×#2491C212yesmaximal, complemented
8C2×#2414C212yesmaximal, complemented
9C2×#2426C214yesmaximal, complemented
10C2×#2458C214yesmaximal, complemented
11C2×#2452C212yesmaximal, complemented
12C2×#2641C211yesmaximal, characteristic, complemented
13C2×#2629C211yesmaximal, characteristic, complemented
14C2×#2363C212yesmaximal, complemented
15C2×#2330C211yesmaximal, characteristic, complemented
16C2×#2290C211yesmaximal, characteristic, complemented
17C2×#540—216no—
18C2×#540C22116yescomplemented
19C2×#533—216no—
20C2×#533C22116yescomplemented
21C22×C4⋊Q8C2212yes—
22C22×#127C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#124C2214yes—
26C22×#123C2214yes—
27C22×C22⋊Q8C2214yes—
28C22×C22⋊Q8C2214yes—
29C22×#121C2212yes—
30C22×#121C2214yes—
31C22×#121C2212yes—
32C22×#121C2214yes—
33C22×C4×Q8C2214yes—
34C22×C4×D4C2212yes—
35C22×C4×D4C2211yescharacteristic
36C22×#117C2211yescharacteristic
37C22×#117C2212yes—
38C23×C42C2211yescharacteristic
39#2856C22116yescomplemented
40#2681C22164yescomplemented
41#2676C22164yescomplemented
42#2641C22116yescomplemented
43#2629C22116yescomplemented
44#2491C22132yescomplemented
45#2458C22164yescomplemented
46#2452C22132yescomplemented
47#2426C22164yescomplemented
48#2414C22132yescomplemented
49C2×#398C2214yes—
50C2×#396C2214yes—
51C2×#396C2214yes—
52C2×#394C2214yes—
53C2×#394C2214yes—
54C2×#394C2214yes—
55C2×#394C2214yes—
56C2×#393C2214yes—
57C2×#393C2214yes—
58C2×#392C2214yes—
59#2363C22132yescomplemented
60#2330C22116yescomplemented
61C2×#390C2212yes—
62C2×#390C2214yes—
63C2×#390C2212yes—
64C2×#390C2212yes—
65C2×#390C2214yes—
66C2×#390C2214yes—
67C2×#389C2212yes—
68C2×#389C2214yes—
69C2×#389C2212yes—
70#2290C22116yescomplemented
71C2×#388C2214yes—
72C2×#388C2214yes—
73C2×#388C2214yes—
74C2×#388C2212yes—
75C2×#387C2214yes—
76C2×#387C2212yes—
77C2×#386C2214yes—
78C2×#386C2212yes—
79C2×#385C2214yes—
80C2×#385C2212yes—
81C2×#384C2211yescharacteristic
82C2×#380C2212yes—
83C2×C4×C4⋊C4C2212yes—
84C2×C4×C4⋊C4C2211yescharacteristic
85C2×C4×C22⋊C4C2211yescharacteristic
86C2×C4×C22⋊C4C2214yes—
87C2×C4×C22⋊C4C2211yescharacteristic
88C2×#376C2211yescharacteristic
89C23×Q8C2314yes—
90C23×D4C2312yes—
91C23×D4C2311yescharacteristic
92C23×D4C2314yes—
93C23×D4C2314yes—
94C23×D4C2314yes—
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2314yes—
99#540—2128no—
100#533—2128no—
101C2×C4⋊Q8—216no—
102C2×C4⋊Q8C23116yes—
103C2×#127—28no—
104C2×#127C2318yes—
105C2×#125—28no—
106C2×#125D4116yescomplemented
107C2×#125C2318yes—
108C2×#125C2318yes—
109C2×#124—416no—
110C2×#124C23132yes—
111C2×#123—232no—
112C2×#123C23132yes—
113C2×C22⋊Q8—232no—
114C2×C22⋊Q8—416no—
115C2×C22⋊Q8C23132yes—
116C2×C22⋊Q8C23132yes—
117C2×#121—216no—
118C2×#121—232no—
119C2×#121—216no—
120C2×#121—416no—
121C2×#121C23116yes—
122C2×#121C23116yes—
123C2×#121C23132yes—
124C2×#121C23132yes—
125C2×C4×Q8—232no—
126C2×C4×Q8C23132yes—
127C2×C4×D4—28no—
128C2×C4×D4—216no—
129C2×C4×D4C23116yes—
130C2×C4×D4C2318yes—
131C2×#117—216no—
132C2×#117—28no—
133C2×#117C23116yes—
134C2×#117C2318yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2314yes—
143C22×C4⋊C4C2314yes—
144C22×C4⋊C4C2314yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2314yes—
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2311yescharacteristic
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2311yescharacteristic
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2314yes—
158C22×C22⋊C4C2314yes—
159C22×C22⋊C4C2314yes—
160C22×C22⋊C4C2314yes—
161C22×C22⋊C4C2314yes—
162C22×C22⋊C4C2312yes—
163C22×C42—28no—
164C22×C42C2318yes—
165C22×C42C2312yes—
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168C22×C42C2314yes—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171C22×C42C2314yes—
172C22×C42C2311yescharacteristic
173#398C23132yes—
174#396C23132yes—
175#396C23132yes—
176#394C23132yes—
177#394C23132yes—
178#394C23132yes—
179#394C23132yes—
180#393C23132yes—
181#393C23132yes—
182#392C23132yes—
183#390C23132yes—
184#390C23116yes—
185#390C23116yes—
186#390C23132yes—
187#390C23116yes—
188#390C23132yes—
189#389C23116yes—
190#389C23132yes—
191#389C23116yes—
192#388C23132yes—
193#388C23132yes—
194#388C23132yes—
195#388C23116yes—
196#387C23132yes—
197#387C23116yes—
198#386C23132yes—
199#386C23116yes—
200#385C23132yes—
201#385C23116yes—
202#384C2318yes—
203#380C23116yes—
204C4×C4⋊C4C23116yes—
205C4×C4⋊C4C2318yes—
206C4×C22⋊C4C2318yes—
207C4×C22⋊C4C23132yes—
208C4×C22⋊C4C2318yes—
209#376C2318yes—
210C2×#95C2314yes—
211C2×#95C2312yes—
212C2×#95C2312yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2314yes—
216C2×#95C2312yes—
217C2×#95C2312yes—
218C2×#95C2312yes—
219C25C2411yescharacteristic
220C25C2414yes—
221C25C2411yescharacteristic
222C22×Q8—216no—
223C22×Q8—216no—
224C22×Q8—48no—
225C22×Q8C24116yes—
226C22×D4—24no—
227C22×D4—216no—
228C22×D4—28no—
229C22×D4—216no—
230C22×D4—216no—
231C22×D4—28no—
232C22×D4—28no—
233C22×D4—44no—
234C22×D4—48no—
235C22×D4—48no—
236C22×D4—48no—
237C22×D4—48no—
238C22×D4—48no—
239C22×D4—48no—
240C22×D4C2418yes—
241C22×D4C2414yes—
242C22×D4C24116yes—
243C22×D4C24116yes—
244C22×D4C24116yes—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—28no—
251C23×C4—24no—
252C23×C4—216no—
253C23×C4—44no—
254C23×C4—48no—
255C23×C4—48no—
256C23×C4#4118yes—
257C23×C4C2414yes—
258C23×C4C2411yescharacteristic
259C23×C4C2414yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2414yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2418yes—
266C23×C4C2414yes—
267C23×C4C2412yes—
268C23×C4C2414yes—
269C23×C4C2411yescharacteristic
270C23×C4C2412yes—
271C23×C4C24116yes—
272C23×C4C2411yescharacteristic
273C4⋊Q8—264no—
274#127—232no—
275#125—232no—
276#125C2×D4164yescomplemented
277#124—464no—
278#123—2128no—
279C22⋊Q8—2128no—
280C22⋊Q8—464no—
281#121—264no—
282#121—2128no—
283#121—264no—
284#121—464no—
285C4×Q8—2128no—
286C4×D4—232no—
287C4×D4—264no—
288#117—264no—
289#117—232no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—216no—
293C2×C4⋊C4—216no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4C2×D4116yes—
301C2×C4⋊C4C2×D4132yes—
302C2×C4⋊C4C2418yes—
303C2×C4⋊C4C2418yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C24116yes—
306C2×C4⋊C4C24116yes—
307C2×C4⋊C4C2418yes—
308C2×C4⋊C4C2418yes—
309C2×C4⋊C4C24116yes—
310C2×C4⋊C4C24116yes—
311C2×C4⋊C4C24116yes—
312C2×C4⋊C4C2418yes—
313C2×C4⋊C4C24116yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—216no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—216no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—216no—
322C2×C22⋊C4—216no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4C2418yes—
331C2×C22⋊C4C24116yes—
332C2×C22⋊C4C24116yes—
333C2×C22⋊C4C2414yes—
334C2×C22⋊C4C2414yes—
335C2×C22⋊C4C24116yes—
336C2×C22⋊C4C24116yes—
337C2×C22⋊C4C24116yes—
338C2×C22⋊C4C24116yes—
339C2×C22⋊C4C2418yes—
340C2×C22⋊C4C24116yes—
341C2×C22⋊C4C24116yes—
342C2×C22⋊C4C24116yes—
343C2×C22⋊C4C2418yes—
344C2×C22⋊C4C24116yes—
345C2×C22⋊C4C24116yes—
346C2×C42—24no—
347C2×C42—216no—
348C2×C42—28no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—232no—
353C2×C42—48no—
354C2×C42C2×D418yes—
355C2×C42C2414yes—
356C2×C42C24116yes—
357C2×C42C2414yes—
358C2×C42C2418yes—
359C2×C42C2414yes—
360C2×C42C2414yes—
361C2×C42C24116yes—
362C2×C42C2414yes—
363#95C2418yes—
364#95C2418yes—
365#95C2418yes—
366#95C2418yes—
367#95C24116yes—
368#95C2418yes—
369#95C24116yes—
370#95C24116yes—
371#95C24116yes—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24—22no—
377C24—28no—
378C24—22no—
379C24—28no—
380C24—24no—
381C24—28no—
382C24—28no—
383C24—22no—
384C24—28no—
385C24—28no—
386C24—28no—
387C24—24no—
388C24C2×#4114yes—
389C24C2512yes—
390C24C2518yes—
391C24C2512yes—
392C24C2511yescharacteristic
393C2×Q8—232no—
394C2×Q8—232no—
395C2×Q8—416no—
396C2×Q8—464no—
397C2×D4—28no—
398C2×D4—232no—
399C2×D4—216no—
400C2×D4—232no—
401C2×D4—232no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—48no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—416no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—432no—
413C2×D4—832no—
414C2×D4—832no—
415C2×D4—832no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—232no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—216no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—216no—
450C22×C4—28no—
451C22×C4—232no—
452C22×C4—48no—
453C22×C4—432no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—832no—
458C22×C4C22×D418yes—
459C22×C4C22×D4116yes—
460C22×C4C22×D414yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#41116yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)18yes—
465C22×C42−1+414yes—
466C22×C4C2512yes—
467C22×C4C2514yes—
468C22×C4C2512yes—
469C22×C4C2518yes—
470C22×C4C2514yes—
471C22×C4C2512yes—
472C22×C4C2518yes—
473C22×C4C2518yes—
474C22×C4C2514yes—
475C22×C4C2512yes—
476C22×C4C2514yes—
477C22×C4C2512yes—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—232no—
481C4⋊C4—232no—
482C4⋊C4—28no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4C22×D4132yes—
489C4⋊C4C22×D4164yes—
490C22⋊C4—28no—
491C22⋊C4—216no—
492C22⋊C4—232no—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—28no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—232no—
499C22⋊C4—216no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C42—216no—
507C42—28no—
508C42—28no—
509C42—232no—
510C42—28no—
511C42—28no—
512C42—416no—
513C42C22×D4116yes—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—24no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—416no—
537C23—44no—
538C23—416no—
539C23—416no—
540C23—48no—
541C23—416no—
542C23—416no—
543C23—44no—
544C23—48no—
545C23—416no—
546C23—416no—
547C23C23×D412yes—
548C23C22×#4114yes—
549C23C22×#4112yes—
550C23C2×H(5,𝔽2)14yes—
551C23C2×2−1+412yes—
552C23#58514yes—
553C23C2611yescharacteristic
554Q8—464no—
555D4—416no—
556D4—432no—
557D4—832no—
558D4—832no—
559D4—832no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—216no—
570C2×C4—216no—
571C2×C4—232no—
572C2×C4—232no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—432no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—416no—
597C2×C4—44no—
598C2×C4—416no—
599C2×C4—48no—
600C2×C4—416no—
601C2×C4—832no—
602C2×C4#548116yes—
603C2×C4C23×D418yes—
604C2×C4C23×D4116yes—
605C2×C4C23×D414yes—
606C2×C4C22×#4114yes—
607C2×C4C22×#4114yes—
608C2×C4C2×H(5,𝔽2)18yes—
609C2×C4C2×2−1+414yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—44no—
614C22—44no—
615C22—416no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—42no—
624C22—42no—
625C22—48no—
626C22—42no—
627C22—48no—
628C22—88no—
629C22—832no—
630C22#345514yes—
631C22C2×#54814yes—
632C22#348818yes—
633C22#350814yes—
634C22#354718yes—
635C22C24×D411yescharacteristic
636C22C23×#4111yescharacteristic
637C22C22×H(5,𝔽2)12yes—
638C22C22×2−1+411yescharacteristic
639C22C2×#58512yes—
640C4—28no—
641C4—28no—
642C4—216no—
643C4—24no—
644C4—44no—
645C4—44no—
646C4—48no—
647C4—42no—
648C4—48no—
649C4—42no—
650C4—42no—
651C4C2×#54818yes—
652C2—44no—
653C2—88no—
654C2—82no—
655C2#3771018yesdirect product factor
656C2C2×#345511yescharacteristic
657C2C22×#54811yescharacteristic
658C2C2×#354712yes—
659C2C2×#348812yes—
660C2C2×#350811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
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×#540C2semidirect
2. C22×#533C2semidirect
3. C2×#2681C2semidirect
4. C2×#2676C2semidirect
5. #37710C2direct (N × H)
6. C2×#2856C2semidirect
7. C2×#2491C2semidirect
8. C2×#2414C2semidirect
9. C2×#2426C2semidirect
10. C2×#2458C2semidirect
11. C2×#2452C2semidirect
12. C2×#2641C2semidirect
13. C2×#2629C2semidirect
14. C2×#2363C2semidirect
15. C2×#2330C2semidirect
16. C2×#2290C2semidirect
18. C2×#540C22semidirect
20. C2×#533C22semidirect
39. #2856C22semidirect
40. #2681C22semidirect
41. #2676C22semidirect
42. #2641C22semidirect
43. #2629C22semidirect
44. #2491C22semidirect
45. #2458C22semidirect
46. #2452C22semidirect
47. #2426C22semidirect
48. #2414C22semidirect
59. #2363C22semidirect
60. #2330C22semidirect
70. #2290C22semidirect
106. C2×#125D4semidirect
276. #125C2×D4semidirect
655. C2#37710direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5402
C22×#5332
C2×#26812
C2×#26762
#377102
C2×#28562
C2×#24912
C2×#24142
C2×#24262
C2×#24582
C2×#24522
C2×#26412
C2×#26292
C2×#23632
C2×#23302
C2×#22902