Finite Group Databasebeta

← C2×#37700 #7635125

Subgroups — C2×#37700 #7635125

Order 512.

8804 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic, complemented
2C22×#532C211yesmaximal, characteristic, complemented
3C2×#2708C214yesmaximal, complemented
4C2×#2688C214yesmaximal, complemented
5#37700C2132yesmaximal, direct product factor
6C2×#2851C211yesmaximal, characteristic, complemented
7C2×#2407C212yesmaximal, complemented
8C2×#2447C212yesmaximal, complemented
9C2×#2427C212yesmaximal, complemented
10C2×#2444C214yesmaximal, complemented
11C2×#2419C214yesmaximal, complemented
12C2×#2642C211yesmaximal, characteristic, complemented
13C2×#2621C211yesmaximal, characteristic, complemented
14C2×#2339C212yesmaximal, complemented
15C2×#2328C211yesmaximal, characteristic, complemented
16C2×#2290C211yesmaximal, characteristic, complemented
17C23×#41C2212yes—
18C2×#540—216no—
19C2×#540C22116yescomplemented
20C2×#532—216no—
21C2×#532C22116yescomplemented
22C22×C4⋊Q8C2211yescharacteristic
23C22×#127C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#124C2212yes—
26C22×#123C2214yes—
27C22×C22⋊Q8C2212yes—
28C22×#121C2214yes—
29C22×#121C2212yes—
30C22×#121C2214yes—
31C22×C4×D4C2212yes—
32C22×C4×D4C2214yes—
33C22×C4×D4C2211yescharacteristic
34C22×C4×D4C2212yes—
35C22×#117C2211yescharacteristic
36C23×C42C2211yescharacteristic
37#2851C22116yescomplemented
38#2708C22164yescomplemented
39#2688C22164yescomplemented
40#2642C22116yescomplemented
41#2621C22116yescomplemented
42#2447C22132yescomplemented
43#2444C22164yescomplemented
44#2427C22132yescomplemented
45#2419C22164yescomplemented
46#2407C22132yescomplemented
47C2×#400C2214yes—
48C2×#400C2214yes—
49C2×#398C2214yes—
50C2×#397C2214yes—
51C2×#397C2214yes—
52C2×#397C2214yes—
53C2×#397C2214yes—
54C2×#394C2214yes—
55C2×#394C2214yes—
56C2×#392C2214yes—
57#2339C22132yescomplemented
58#2328C22116yescomplemented
59C2×#391C2212yes—
60C2×#391C2212yes—
61C2×#391C2212yes—
62C2×#390C2212yes—
63C2×#389C2212yes—
64#2290C22116yescomplemented
65C2×#388C2212yes—
66C2×#387C2214yes—
67C2×#387C2212yes—
68C2×#386C2214yes—
69C2×#386C2214yes—
70C2×#386C2214yes—
71C2×#386C2214yes—
72C2×#386C2214yes—
73C2×#386C2214yes—
74C2×#386C2212yes—
75C2×#385C2214yes—
76C2×#385C2214yes—
77C2×#385C2214yes—
78C2×#385C2212yes—
79C2×#382C2211yescharacteristic
80C2×#381C2214yes—
81C2×#381C2214yes—
82C2×#380C2212yes—
83C2×#380C2214yes—
84C2×C4×C4⋊C4C2211yescharacteristic
85C2×C4×C22⋊C4C2211yescharacteristic
86C2×C4×C22⋊C4C2211yescharacteristic
87C2×#376C2211yescharacteristic
88C22×#95C2214yes—
89C22×#41—216no—
90C22×#41—216no—
91C22×#41—48no—
92C22×#41C23116yes—
93C23×Q8C2312yes—
94C23×D4C2314yes—
95C23×D4C2312yes—
96C23×D4C2311yescharacteristic
97C23×D4C2314yes—
98C23×D4C2314yes—
99C24×C4C2311yescharacteristic
100C24×C4C2314yes—
101C24×C4C2311yescharacteristic
102C24×C4C2312yes—
103C24×C4C2314yes—
104C24×C4C2314yes—
105#540—2128no—
106#532—2128no—
107C2×C4⋊Q8—28no—
108C2×C4⋊Q8C2318yes—
109C2×#127D4116yescomplemented
110C2×#127C2318yes—
111C2×#125—28no—
112C2×#125C2318yes—
113C2×#124—216no—
114C2×#124C23116yes—
115C2×#123—232no—
116C2×#123C23132yes—
117C2×C22⋊Q8—48no—
118C2×C22⋊Q8C23116yes—
119C2×#121—232no—
120C2×#121—232no—
121C2×#121—216no—
122C2×#121C23116yes—
123C2×#121C23132yes—
124C2×#121C23132yes—
125C2×C4×D4—232no—
126C2×C4×D4—216no—
127C2×C4×D4—28no—
128C2×C4×D4—48no—
129C2×C4×D4C23116yes—
130C2×C4×D4C2318yes—
131C2×C4×D4C23132yes—
132C2×C4×D4C23116yes—
133C2×#117—28no—
134C2×#117C2318yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2314yes—
142C22×C4⋊C4C2314yes—
143C22×C4⋊C4C2312yes—
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2311yescharacteristic
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2314yes—
153C22×C22⋊C4C2314yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2312yes—
158C22×C42—28no—
159C22×C42C2318yes—
160C22×C42C2312yes—
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2311yescharacteristic
166#400C23132yes—
167#400C23132yes—
168#398C23132yes—
169#397C23132yes—
170#397C23132yes—
171#397C23132yes—
172#397C23132yes—
173#394C23132yes—
174#394C23132yes—
175#392C23132yes—
176#391C23116yes—
177#391C23116yes—
178#391C23116yes—
179#390C23116yes—
180#389C23116yes—
181#388C23116yes—
182#387C23132yes—
183#387C23116yes—
184#386C23132yes—
185#386C23132yes—
186#386C23132yes—
187#386C23132yes—
188#386C23132yes—
189#386C23132yes—
190#386C23116yes—
191#385C23132yes—
192#385C23132yes—
193#385C23132yes—
194#385C23116yes—
195#382C2318yes—
196#381C23132yes—
197#381C23132yes—
198#380C23116yes—
199#380C23132yes—
200C4×C4⋊C4C2318yes—
201C4×C22⋊C4C2318yes—
202C4×C22⋊C4C2318yes—
203#376C2318yes—
204C2×#95C23132yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2312yes—
210C2×#95C2312yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2312yes—
215C2×#95C2312yes—
216C2×#95C2314yes—
217C2×#95C2312yes—
218C2×#95C2314yes—
219C25C2411yescharacteristic
220C25C2414yes—
221C25C2411yescharacteristic
222C2×#41—264no—
223C2×#41—264no—
224C2×#41—432no—
225C2×#41—4128no—
226C22×Q8—28no—
227C22×Q8—28no—
228C22×Q8—28no—
229C22×Q8C2418yes—
230C22×D4—216no—
231C22×D4—24no—
232C22×D4—216no—
233C22×D4—216no—
234C22×D4—28no—
235C22×D4—28no—
236C22×D4—48no—
237C22×D4—48no—
238C22×D4—44no—
239C22×D4—48no—
240C22×D4—48no—
241C22×D4—48no—
242C22×D4C2×D4116yes—
243C22×D4C2×D4132yes—
244C22×D4C24116yes—
245C22×D4C2418yes—
246C22×D4C2414yes—
247C22×D4C24116yes—
248C22×D4C24116yes—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—216no—
252C23×C4—216no—
253C23×C4—24no—
254C23×C4—28no—
255C23×C4—24no—
256C23×C4—216no—
257C23×C4—24no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—48no—
261C23×C4—48no—
262C23×C4—48no—
263C23×C4—48no—
264C23×C4—48no—
265C23×C4—48no—
266C23×C4#4118yes—
267C23×C4C2414yes—
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2414yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C23×C4C2412yes—
275C23×C4C2414yes—
276C23×C4C2414yes—
277C23×C4C24116yes—
278C23×C4C2412yes—
279C23×C4C2414yes—
280C23×C4C2411yescharacteristic
281C23×C4C24116yes—
282C23×C4C2418yes—
283C23×C4C24116yes—
284C23×C4C2411yescharacteristic
285C4⋊Q8—232no—
286#127C2×D4164yescomplemented
287#125—232no—
288#124—264no—
289#123—2128no—
290C22⋊Q8—432no—
291#121—2128no—
292#121—2128no—
293#121—264no—
294C4×D4—2128no—
295C4×D4—264no—
296C4×D4—232no—
297C4×D4—432no—
298#117—232no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4C2418yes—
309C2×C4⋊C4C2418yes—
310C2×C4⋊C4C24116yes—
311C2×C4⋊C4C2418yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C24116yes—
314C2×C4⋊C4C24116yes—
315C2×C4⋊C4C2418yes—
316C2×C4⋊C4C24116yes—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—216no—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4—48no—
331C2×C22⋊C4C2418yes—
332C2×C22⋊C4C24116yes—
333C2×C22⋊C4C24116yes—
334C2×C22⋊C4C24116yes—
335C2×C22⋊C4C2414yes—
336C2×C22⋊C4C2414yes—
337C2×C22⋊C4C24116yes—
338C2×C22⋊C4C24116yes—
339C2×C22⋊C4C24116yes—
340C2×C22⋊C4C2418yes—
341C2×C22⋊C4C2418yes—
342C2×C22⋊C4C24116yes—
343C2×C22⋊C4C24116yes—
344C2×C22⋊C4C24116yes—
345C2×C42—24no—
346C2×C42—28no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—232no—
351C2×C42C2×D418yes—
352C2×C42C2414yes—
353C2×C42C2414yes—
354C2×C42C2418yes—
355C2×C42C2414yes—
356C2×C42C2414yes—
357C2×C42C2414yes—
358#95C2418yes—
359#95C24116yes—
360#95C24116yes—
361#95C24116yes—
362#95C2418yes—
363#95C2418yes—
364#95C2418yes—
365#95C2418yes—
366#95C24116yes—
367#95C24116yes—
368#95C24116yes—
369#95C24116yes—
370#95C24116yes—
371#95C24116yes—
372C24—22no—
373C24—28no—
374C24—28no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—28no—
379C24—28no—
380C24—28no—
381C24—28no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—24no—
387C24C22×D4116yes—
388C24C2×#4114yes—
389C24C2512yes—
390C24C2518yes—
391C24C2512yes—
392C24C2511yescharacteristic
393#41—4256no—
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8—216no—
397C2×Q8—432no—
398C2×D4—232no—
399C2×D4—28no—
400C2×D4—232no—
401C2×D4—232no—
402C2×D4—264no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—464no—
406C2×D4—416no—
407C2×D4—416no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—464no—
413C2×D4—416no—
414C2×D4—832no—
415C2×D4C22×D4132yes—
416C2×D4C22×D4164yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—232no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—232no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—232no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—232no—
455C22×C4—28no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—464no—
465C22×C4—432no—
466C22×C4—416no—
467C22×C4—832no—
468C22×C4—832no—
469C22×C4C22×D418yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#41116yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4C2512yes—
476C22×C4C2514yes—
477C22×C4C2512yes—
478C22×C4C2514yes—
479C22×C4C2518yes—
480C22×C4C2518yes—
481C22×C4C2518yes—
482C22×C4C2514yes—
483C22×C4C2512yes—
484C22×C4C2514yes—
485C22×C4C2512yes—
486C22×C4C2512yes—
487C4⋊C4—216no—
488C4⋊C4—216no—
489C4⋊C4—232no—
490C4⋊C4—232no—
491C4⋊C4—28no—
492C4⋊C4—216no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—232no—
499C22⋊C4—232no—
500C22⋊C4—28no—
501C22⋊C4—216no—
502C22⋊C4—28no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C42—216no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42C22×D4116yes—
516C23—22no—
517C23—28no—
518C23—28no—
519C23—22no—
520C23—24no—
521C23—28no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—232no—
529C23—232no—
530C23—232no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—416no—
538C23—48no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—48no—
543C23—44no—
544C23—44no—
545C23—48no—
546C23—44no—
547C23—48no—
548C23C23×D4116yes—
549C23C23×D412yes—
550C23C22×#4114yes—
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)12yes—
553C23#58514yes—
554C23#58514yes—
555C23C2611yescharacteristic
556Q8—432no—
557D4—264no—
558D4—464no—
559D4—464no—
560D4—416no—
561D4—832no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—216no—
568C2×C4—232no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—28no—
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—28no—
594C2×C4—464no—
595C2×C4—432no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—416no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—416no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—48no—
605C2×C4—416no—
606C2×C4—832no—
607C2×C4—832no—
608C2×C4#546116yes—
609C2×C4C23×D418yes—
610C2×C4C23×D414yes—
611C2×C4C22×#4114yes—
612C2×C4C22×#4114yes—
613C2×C4C2×H(5,𝔽2)14yes—
614C22—216no—
615C22—216no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—216no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—44no—
624C22—416no—
625C22—464no—
626C22—48no—
627C22—48no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—88no—
634C22#345514yes—
635C22C2×#54614yes—
636C22#348518yes—
637C22#351214yes—
638C22#356218yes—
639C22C24×D411yescharacteristic
640C22C23×#4111yescharacteristic
641C22C22×H(5,𝔽2)11yescharacteristic
642C22C2×#58512yes—
643C22C2×#58512yes—
644C4—28no—
645C4—24no—
646C4—48no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—48no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—48no—
655C4C2×#54618yes—
656C2—416no—
657C2—44no—
658C2—82no—
659C2#3770018yesdirect product factor
660C2C2×#345511yescharacteristic
661C2C22×#54611yescharacteristic
662C2C2×#356212yes—
663C2C2×#348512yes—
664C2C2×#351211yescharacteristic
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×#532C2semidirect
3. C2×#2708C2semidirect
4. C2×#2688C2semidirect
5. #37700C2direct (N × H)
6. C2×#2851C2semidirect
7. C2×#2407C2semidirect
8. C2×#2447C2semidirect
9. C2×#2427C2semidirect
10. C2×#2444C2semidirect
11. C2×#2419C2semidirect
12. C2×#2642C2semidirect
13. C2×#2621C2semidirect
14. C2×#2339C2semidirect
15. C2×#2328C2semidirect
16. C2×#2290C2semidirect
19. C2×#540C22semidirect
21. C2×#532C22semidirect
37. #2851C22semidirect
38. #2708C22semidirect
39. #2688C22semidirect
40. #2642C22semidirect
41. #2621C22semidirect
42. #2447C22semidirect
43. #2444C22semidirect
44. #2427C22semidirect
45. #2419C22semidirect
46. #2407C22semidirect
57. #2339C22semidirect
58. #2328C22semidirect
64. #2290C22semidirect
109. C2×#127D4semidirect
286. #127C2×D4semidirect
659. C2#37700direct (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×#5322
C2×#27082
C2×#26882
#377002
C2×#28512
C2×#24072
C2×#24472
C2×#24272
C2×#24442
C2×#24192
C2×#26422
C2×#26212
C2×#23392
C2×#23282
C2×#22902