Finite Group Databasebeta

← C2×#37701 #7635129

Subgroups — C2×#37701 #7635129

Order 512.

8588 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×#541C211yesmaximal, characteristic, complemented
2C22×#532C211yesmaximal, characteristic, complemented
3C2×#2714C214yesmaximal, complemented
4C2×#2711C214yesmaximal, complemented
5#37701C2132yesmaximal, direct product factor
6C2×#2851C211yesmaximal, characteristic, complemented
7C2×#2407C212yesmaximal, complemented
8C2×#2448C212yesmaximal, complemented
9C2×#2426C212yesmaximal, complemented
10C2×#2467C214yesmaximal, complemented
11C2×#2460C214yesmaximal, complemented
12C2×#2641C211yesmaximal, characteristic, complemented
13C2×#2619C211yesmaximal, characteristic, complemented
14C2×#2341C212yesmaximal, complemented
15C2×#2327C211yesmaximal, characteristic, complemented
16C2×#2290C211yesmaximal, characteristic, complemented
17C23×#41C2212yes—
18C2×#541—216no—
19C2×#541C22116yescomplemented
20C2×#532—216no—
21C2×#532C22116yescomplemented
22C22×C4⋊Q8C2211yescharacteristic
23C22×#127C2211yescharacteristic
24C22×#126C2214yes—
25C22×#125C2211yescharacteristic
26C22×#125C2214yes—
27C22×#124C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×#121C2214yes—
30C22×#121C2212yes—
31C22×C4×Q8C2212yes—
32C22×C4×Q8C2212yes—
33C22×C4×D4C2214yes—
34C22×C4×D4C2211yescharacteristic
35C22×#117C2211yescharacteristic
36C23×C42C2211yescharacteristic
37#2851C22116yescomplemented
38#2714C22164yescomplemented
39#2711C22164yescomplemented
40#2641C22116yescomplemented
41#2619C22116yescomplemented
42#2467C22164yescomplemented
43#2460C22164yescomplemented
44#2448C22132yescomplemented
45#2426C22132yescomplemented
46#2407C22132yescomplemented
47C2×#399C2214yes—
48C2×#397C2214yes—
49C2×#396C2214yes—
50C2×#394C2214yes—
51C2×#394C2214yes—
52C2×#394C2214yes—
53C2×#394C2214yes—
54C2×#394C2214yes—
55C2×#392C2214yes—
56C2×#392C2214yes—
57#2341C22132yescomplemented
58#2327C22116yescomplemented
59C2×#391C2212yes—
60C2×#390C2212yes—
61C2×#390C2212yes—
62C2×#390C2214yes—
63C2×#390C2214yes—
64C2×#390C2212yes—
65C2×#390C2214yes—
66C2×#389C2212yes—
67#2290C22116yescomplemented
68C2×#388C2214yes—
69C2×#388C2214yes—
70C2×#388C2212yes—
71C2×#388C2214yes—
72C2×#387C2214yes—
73C2×#387C2214yes—
74C2×#387C2214yes—
75C2×#387C2212yes—
76C2×#386C2214yes—
77C2×#386C2212yes—
78C2×#385C2214yes—
79C2×#385C2214yes—
80C2×#385C2212yes—
81C2×#382C2211yescharacteristic
82C2×#380C2212yes—
83C2×#380C2214yes—
84C2×C4×C4⋊C4C2211yescharacteristic
85C2×C4×C22⋊C4C2211yescharacteristic
86C2×C4×C22⋊C4C2214yes—
87C2×C4×C22⋊C4C2211yescharacteristic
88C2×#376C2211yescharacteristic
89C22×#41—216no—
90C22×#41—216no—
91C22×#41—48no—
92C22×#41C23116yes—
93C23×Q8C2312yes—
94C23×D4C2312yes—
95C23×D4C2314yes—
96C23×D4C2311yescharacteristic
97C23×D4C2314yes—
98C23×D4C2314yes—
99C24×C4C2311yescharacteristic
100C24×C4C2314yes—
101C24×C4C2311yescharacteristic
102C24×C4C2312yes—
103#541—2128no—
104#532—2128no—
105C2×C4⋊Q8D4116yescomplemented
106C2×C4⋊Q8C2318yes—
107C2×#127—28no—
108C2×#127C2318yes—
109C2×#126—232no—
110C2×#126C23132yes—
111C2×#125—232no—
112C2×#125—28no—
113C2×#125C2318yes—
114C2×#125C23132yes—
115C2×#124—216no—
116C2×#124C23116yes—
117C2×C22⋊Q8—216no—
118C2×C22⋊Q8C23116yes—
119C2×#121—232no—
120C2×#121—48no—
121C2×#121C23132yes—
122C2×#121C23116yes—
123C2×C4×Q8—216no—
124C2×C4×Q8—48no—
125C2×C4×Q8C23116yes—
126C2×C4×Q8C23116yes—
127C2×C4×D4—28no—
128C2×C4×D4—232no—
129C2×C4×D4C2318yes—
130C2×C4×D4C23132yes—
131C2×#117—28no—
132C2×#117C2318yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2314yes—
141C22×C4⋊C4C2314yes—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2314yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2311yescharacteristic
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2311yescharacteristic
150C22×C22⋊C4C2314yes—
151C22×C22⋊C4C2314yes—
152C22×C22⋊C4C2314yes—
153C22×C22⋊C4C2314yes—
154C22×C22⋊C4C2314yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2312yes—
158C22×C22⋊C4C2314yes—
159C22×C22⋊C4C2312yes—
160C22×C42—28no—
161C22×C42C2318yes—
162C22×C42C2312yes—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167C22×C42C2314yes—
168C22×C42C2314yes—
169C22×C42C2311yescharacteristic
170#399C23132yes—
171#397C23132yes—
172#396C23132yes—
173#394C23132yes—
174#394C23132yes—
175#394C23132yes—
176#394C23132yes—
177#394C23132yes—
178#392C23132yes—
179#392C23132yes—
180#391C23116yes—
181#390C23116yes—
182#390C23116yes—
183#390C23132yes—
184#390C23132yes—
185#390C23116yes—
186#390C23132yes—
187#389C23116yes—
188#388C23132yes—
189#388C23132yes—
190#388C23132yes—
191#388C23116yes—
192#387C23132yes—
193#387C23132yes—
194#387C23132yes—
195#387C23116yes—
196#386C23132yes—
197#386C23116yes—
198#385C23132yes—
199#385C23132yes—
200#385C23116yes—
201#382C2318yes—
202#380C23116yes—
203#380C23132yes—
204C4×C4⋊C4C2318yes—
205C4×C22⋊C4C2318yes—
206C4×C22⋊C4C23132yes—
207C4×C22⋊C4C2318yes—
208#376C2318yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2312yes—
212C2×#95C2312yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2312yes—
216C2×#95C2312yes—
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×Q8C2×D4116yes—
229C22×Q8C2418yes—
230C22×D4—24no—
231C22×D4—216no—
232C22×D4—216no—
233C22×D4—216no—
234C22×D4—28no—
235C22×D4—28no—
236C22×D4—28no—
237C22×D4—48no—
238C22×D4—48no—
239C22×D4—44no—
240C22×D4—48no—
241C22×D4—48no—
242C22×D4—48no—
243C22×D4—48no—
244C22×D4C2418yes—
245C22×D4C24116yes—
246C22×D4C24116yes—
247C22×D4C2414yes—
248C22×D4C24116yes—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—216no—
253C23×C4—28no—
254C23×C4—24no—
255C23×C4—216no—
256C23×C4—24no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—48no—
260C23×C4#4118yes—
261C23×C4C2414yes—
262C23×C4C2414yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C23×C4C2412yes—
268C23×C4C2414yes—
269C23×C4C2411yescharacteristic
270C23×C4C24116yes—
271C23×C4C2414yes—
272C23×C4C2412yes—
273C23×C4C2414yes—
274C23×C4C2411yescharacteristic
275C23×C4C2418yes—
276C23×C4C2411yescharacteristic
277C4⋊Q8C2×D4164yescomplemented
278#127—232no—
279#126—2128no—
280#125—2128no—
281#125—232no—
282#124—264no—
283C22⋊Q8—264no—
284#121—2128no—
285#121—432no—
286C4×Q8—264no—
287C4×Q8—432no—
288C4×D4—232no—
289C4×D4—2128no—
290#117—232no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—216no—
294C2×C4⋊C4—216no—
295C2×C4⋊C4—216no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4C2×D4132yes—
302C2×C4⋊C4C2418yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2418yes—
305C2×C4⋊C4C2418yes—
306C2×C4⋊C4C24116yes—
307C2×C4⋊C4C24116yes—
308C2×C4⋊C4C24116yes—
309C2×C4⋊C4C24116yes—
310C2×C4⋊C4C2418yes—
311C2×C4⋊C4C24116yes—
312C2×C4⋊C4C24116yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—216no—
316C2×C22⋊C4—216no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4C2418yes—
330C2×C22⋊C4C24116yes—
331C2×C22⋊C4C24116yes—
332C2×C22⋊C4C2414yes—
333C2×C22⋊C4C2414yes—
334C2×C22⋊C4C24116yes—
335C2×C22⋊C4C24116yes—
336C2×C22⋊C4C2418yes—
337C2×C22⋊C4C2418yes—
338C2×C22⋊C4C24116yes—
339C2×C22⋊C4C24116yes—
340C2×C22⋊C4C24116yes—
341C2×C22⋊C4C24116yes—
342C2×C22⋊C4C24116yes—
343C2×C22⋊C4C24116yes—
344C2×C22⋊C4C24116yes—
345C2×C42—24no—
346C2×C42—28no—
347C2×C42—216no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—232no—
352C2×C42—48no—
353C2×C42C2×D418yes—
354C2×C42C2414yes—
355C2×C42C2418yes—
356C2×C42C2414yes—
357C2×C42C2414yes—
358C2×C42C24116yes—
359C2×C42C2414yes—
360C2×C42C24116yes—
361C2×C42C2414yes—
362#95C2418yes—
363#95C2418yes—
364#95C2418yes—
365#95C24116yes—
366#95C2418yes—
367#95C2418yes—
368#95C24116yes—
369#95C24116yes—
370#95C24116yes—
371#95C24116yes—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—28no—
378C24—22no—
379C24—28no—
380C24—28no—
381C24—22no—
382C24—28no—
383C24—28no—
384C24—28no—
385C24—28no—
386C24—24no—
387C24—24no—
388C24C2×#4114yes—
389C24C2512yes—
390C24C2518yes—
391C24C2512yes—
392C24C2511yescharacteristic
393#41—4256no—
394C2×Q8—264no—
395C2×Q8—216no—
396C2×Q8—216no—
397C2×Q8C22×D4132yes—
398C2×D4—28no—
399C2×D4—232no—
400C2×D4—232no—
401C2×D4—232no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—464no—
410C2×D4—464no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—416no—
415C2×D4—432no—
416C2×D4—832no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—232no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—216no—
449C22×C4—28no—
450C22×C4—232no—
451C22×C4—28no—
452C22×C4—232no—
453C22×C4—28no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—464no—
458C22×C4—432no—
459C22×C4—416no—
460C22×C4C22×D418yes—
461C22×C4C22×D4116yes—
462C22×C4C22×D414yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#41116yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4C2512yes—
468C22×C4C2512yes—
469C22×C4C2518yes—
470C22×C4C2514yes—
471C22×C4C2514yes—
472C22×C4C2514yes—
473C22×C4C2518yes—
474C22×C4C2512yes—
475C22×C4C2514yes—
476C22×C4C2518yes—
477C22×C4C2512yes—
478C22×C4C2512yes—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—232no—
482C4⋊C4—232no—
483C4⋊C4—232no—
484C4⋊C4—28no—
485C4⋊C4—216no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4C22×D4164yes—
490C22⋊C4—28no—
491C22⋊C4—216no—
492C22⋊C4—232no—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—28no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
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—22no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—28no—
523C23—22no—
524C23—28no—
525C23—28no—
526C23—22no—
527C23—28no—
528C23—28no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—24no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—416no—
537C23—416no—
538C23—44no—
539C23—416no—
540C23—416no—
541C23—48no—
542C23—44no—
543C23—416no—
544C23—48no—
545C23—416no—
546C23—416no—
547C23C23×D412yes—
548C23C22×#4114yes—
549C23C22×#4112yes—
550C23C2×H(5,𝔽2)12yes—
551C23#58514yes—
552C23#58514yes—
553C23C2611yescharacteristic
554Q8—264no—
555D4—464no—
556D4—464no—
557D4—416no—
558D4—432no—
559D4—832no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—232no—
568C2×C4—216no—
569C2×C4—232no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—464no—
593C2×C4—432no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—416no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4#546116yes—
604C2×C4C23×D418yes—
605C2×C4C23×D4116yes—
606C2×C4C23×D414yes—
607C2×C4C22×#4114yes—
608C2×C4C22×#4114yes—
609C2×C4C2×H(5,𝔽2)14yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—44no—
614C22—416no—
615C22—44no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—42no—
623C22—48no—
624C22—42no—
625C22—48no—
626C22—48no—
627C22—42no—
628C22—88no—
629C22—832no—
630C22#345514yes—
631C22C2×#54614yes—
632C22#348718yes—
633C22#351214yes—
634C22#356218yes—
635C22C24×D411yescharacteristic
636C22C23×#4111yescharacteristic
637C22C22×H(5,𝔽2)11yescharacteristic
638C22C2×#58512yes—
639C22C2×#58512yes—
640C4—28no—
641C4—24no—
642C4—216no—
643C4—44no—
644C4—48no—
645C4—44no—
646C4—48no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4C2×#54618yes—
652C2—44no—
653C2—82no—
654C2—88no—
655C2#3770118yesdirect product factor
656C2C2×#345511yescharacteristic
657C2C22×#54611yescharacteristic
658C2C2×#356212yes—
659C2C2×#348712yes—
660C2C2×#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×#541C2semidirect
2. C22×#532C2semidirect
3. C2×#2714C2semidirect
4. C2×#2711C2semidirect
5. #37701C2direct (N × H)
6. C2×#2851C2semidirect
7. C2×#2407C2semidirect
8. C2×#2448C2semidirect
9. C2×#2426C2semidirect
10. C2×#2467C2semidirect
11. C2×#2460C2semidirect
12. C2×#2641C2semidirect
13. C2×#2619C2semidirect
14. C2×#2341C2semidirect
15. C2×#2327C2semidirect
16. C2×#2290C2semidirect
19. C2×#541C22semidirect
21. C2×#532C22semidirect
37. #2851C22semidirect
38. #2714C22semidirect
39. #2711C22semidirect
40. #2641C22semidirect
41. #2619C22semidirect
42. #2467C22semidirect
43. #2460C22semidirect
44. #2448C22semidirect
45. #2426C22semidirect
46. #2407C22semidirect
57. #2341C22semidirect
58. #2327C22semidirect
67. #2290C22semidirect
105. C2×C4⋊Q8D4semidirect
277. C4⋊Q8C2×D4semidirect
655. C2#37701direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5412
C22×#5322
C2×#27142
C2×#27112
#377012
C2×#28512
C2×#24072
C2×#24482
C2×#24262
C2×#24672
C2×#24602
C2×#26412
C2×#26192
C2×#23412
C2×#23272
C2×#22902