Finite Group Databasebeta

← C2×#36433 #7630113

Subgroups — C2×#36433 #7630113

Order 512.

9680 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C22×C4×#41C211yesmaximal, characteristic, complemented
3C2×#2854C213yesmaximal, complemented
4#36433C2132yesmaximal, direct product factor
5C2×#2500C213yesmaximal, complemented
6C2×#2485C213yesmaximal, complemented
7C2×#2420C213yesmaximal, complemented
8C2×#2445C213yesmaximal, complemented
9C2×#2445C213yesmaximal, complemented
10C2×#2422C213yesmaximal, complemented
11C2×#2351C211yesmaximal, characteristic, complemented
12C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
13C2×#2341C211yesmaximal, characteristic, complemented
14C2×#2341C211yesmaximal, characteristic, complemented
15C2×#2341C211yesmaximal, characteristic, complemented
16C2×C4×#123C213yesmaximal, complemented
17C23×#41C2211yescharacteristic
18C23×#41C2211yescharacteristic
19C2×#550—216no—
20C2×#550C22116yes—
21C22×#127C2213yes—
22C22×#123C2213yes—
23C22×#123C2216yes—
24C22×#121C2213yes—
25C22×#121C2213yes—
26C2×C4×#41—216no—
27C2×C4×#41C22116yescomplemented
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33C22×C4×D4C2213yes—
34C22×C4×D4C2213yes—
35C22×C4×D4C2213yes—
36C22×#117C2213yes—
37C23×C42C2213yes—
38#2854C22148yescomplemented
39#2500C22148yescomplemented
40#2485C22148yescomplemented
41#2445C22148yescomplemented
42#2445C22148yescomplemented
43#2422C22148yescomplemented
44#2420C22148yescomplemented
45C2×#397C2216yes—
46C2×#394C2216yes—
47C2×#394C2216yes—
48C2×#394C2216yes—
49#2351C22116yescomplemented
50Q8×C4⋊C4C22116yescomplemented
51#2341C22116yescomplemented
52#2341C22116yescomplemented
53#2341C22116yescomplemented
54C4×#123C22148yescomplemented
55C2×#390C2213yes—
56C2×#390C2213yes—
57C2×#390C2213yes—
58C2×#390C2213yes—
59C2×#390C2213yes—
60C2×#390C2213yes—
61C2×#389C2213yes—
62C2×#389C2213yes—
63C2×#389C2213yes—
64C2×#389C2213yes—
65C2×#388C2213yes—
66C2×#388C2216yes—
67C2×#388C2213yes—
68C2×#387C2213yes—
69C2×#387C2211yescharacteristic
70C2×#386C2216yes—
71C2×#386C2213yes—
72C2×#385C2213yes—
73C2×#384C2213yes—
74C2×#382C2213yes—
75C2×#381C2213yes—
76C2×#380C2213yes—
77C2×#380C2213yes—
78C2×#380C2213yes—
79C2×C4×C4⋊C4C2213yes—
80C2×C4×C4⋊C4C2213yes—
81C2×C4×C22⋊C4C2213yes—
82C2×C4×C22⋊C4C2213yes—
83C2×C4×C22⋊C4C2213yes—
84C22×#41—28no—
85C22×#41—28no—
86C22×#41—28no—
87C22×#41—28no—
88C22×#41—28no—
89C22×#41D4116yes—
90C22×#41C2318yes—
91C22×#41C2318yes—
92C23×Q8C2311yescharacteristic
93C23×D4C2313yes—
94C23×D4C2313yes—
95C23×D4C2313yes—
96C23×D4C2316yes—
97C24×C4C2313yes—
98C24×C4C2313yes—
99C24×C4C2313yes—
100C24×C4C2313yes—
101#550—2128no—
102C2×#127—224no—
103C2×#127C23124yes—
104C2×#123—224no—
105C2×#123—424no—
106C2×#123C23124yes—
107C2×#123C23148yes—
108C2×#121—224no—
109C2×#121—224no—
110C2×#121C23124yes—
111C2×#121C23124yes—
112C4×#41—2128no—
113C2×C4×Q8—28no—
114C2×C4×Q8—28no—
115C2×C4×Q8—28no—
116C2×C4×Q8—28no—
117C2×C4×Q8—28no—
118C2×C4×Q8C2318yes—
119C2×C4×Q8C2318yes—
120C2×C4×Q8C2318yes—
121C2×C4×Q8C2318yes—
122C2×C4×Q8C2318yes—
123C2×C4×D4—224no—
124C2×C4×D4—224no—
125C2×C4×D4—224no—
126C2×C4×D4C23124yes—
127C2×C4×D4C23124yes—
128C2×C4×D4C23124yes—
129C2×#117—224no—
130C2×#117C23124yes—
131C22×C4⋊C4C2313yes—
132C22×C4⋊C4C2313yes—
133C22×C4⋊C4C2313yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2313yes—
136C22×C4⋊C4C2313yes—
137C22×C4⋊C4C2313yes—
138C22×C4⋊C4C2313yes—
139C22×C4⋊C4C2313yes—
140C22×C4⋊C4C2316yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4C2316yes—
143C22×C22⋊C4C2313yes—
144C22×C22⋊C4C2313yes—
145C22×C22⋊C4C2313yes—
146C22×C22⋊C4C2313yes—
147C22×C22⋊C4C2313yes—
148C22×C22⋊C4C2313yes—
149C22×C22⋊C4C2316yes—
150C22×C22⋊C4C2313yes—
151C22×C22⋊C4C2313yes—
152C22×C22⋊C4C2316yes—
153C22×C22⋊C4C2316yes—
154C22×C42—224no—
155C22×C42C23124yes—
156C22×C42C2311yescharacteristic
157C22×C42C2313yes—
158C22×C42C2313yes—
159C22×C42C2313yes—
160C22×C42C2313yes—
161C22×C42C2313yes—
162C22×C42C2313yes—
163#397C23148yes—
164#394C23148yes—
165#394C23148yes—
166#394C23148yes—
167#390C23124yes—
168#390C23124yes—
169#390C23124yes—
170#390C23124yes—
171#390C23124yes—
172#390C23124yes—
173#389C23124yes—
174#389C23124yes—
175#389C23124yes—
176#389C23124yes—
177#388C23124yes—
178#388C23148yes—
179#388C23124yes—
180#387C23124yes—
181#387C2318yes—
182#386C23148yes—
183#386C23124yes—
184#385C23124yes—
185#384C23124yes—
186#382C23124yes—
187#381C23124yes—
188#380C23124yes—
189#380C23124yes—
190#380C23124yes—
191C4×C4⋊C4C23124yes—
192C4×C4⋊C4C23124yes—
193C4×C22⋊C4C23124yes—
194C4×C22⋊C4C23124yes—
195C4×C22⋊C4C23124yes—
196C2×#95C2313yes—
197C2×#95C2313yes—
198C2×#95C2316yes—
199C2×#95C2313yes—
200C2×#95C2313yes—
201C2×#95C2311yescharacteristic
202C2×#95C2311yescharacteristic
203C2×#95C2311yescharacteristic
204C2×#95C2313yes—
205C2×#95C2311yescharacteristic
206C2×#95C2313yes—
207C25C2413yes—
208C25C2413yes—
209C2×#41—2128no—
210C2×#41—232no—
211C2×#41—232no—
212C2×#41—232no—
213C2×#41—232no—
214C2×#41—232no—
215C2×#41—464no—
216C2×#41C2×D4164yes—
217C22×Q8C2×D418yes—
218C22×Q8#4118yes—
219C22×Q8#4118yes—
220C22×Q8C2414yes—
221C22×D4—212no—
222C22×D4—224no—
223C22×D4—212no—
224C22×D4—212no—
225C22×D4—212no—
226C22×D4—212no—
227C22×D4—212no—
228C22×D4—212no—
229C22×D4—46no—
230C22×D4—412no—
231C22×D4—412no—
232C22×D4C2×D4124yes—
233C22×D4C24112yes—
234C22×D4C24112yes—
235C22×D4C24112yes—
236C22×D4C24124yes—
237C23×C4—212no—
238C23×C4—212no—
239C23×C4—212no—
240C23×C4—212no—
241C23×C4—212no—
242C23×C4—212no—
243C23×C4—212no—
244C23×C4—212no—
245C23×C4—212no—
246C23×C4—46no—
247C23×C4—46no—
248C23×C4C2×D4124yes—
249C23×C4C24112yes—
250C23×C4C2413yes—
251C23×C4C2413yes—
252C23×C4C2411yescharacteristic
253C23×C4C2413yes—
254C23×C4C2413yes—
255C23×C4C2411yescharacteristic
256C23×C4C2413yes—
257C23×C4C24112yes—
258C23×C4C24112yes—
259C23×C4C24112yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2413yes—
266#127—296no—
267#123—296no—
268#123—496no—
269#121—296no—
270#121—296no—
271C4×Q8—232no—
272C4×Q8—232no—
273C4×Q8—232no—
274C4×Q8—232no—
275C4×Q8—232no—
276C4×D4—296no—
277C4×D4—296no—
278C4×D4—296no—
279#117—296no—
280C2×C4⋊C4—212no—
281C2×C4⋊C4—212no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—212no—
284C2×C4⋊C4—212no—
285C2×C4⋊C4—212no—
286C2×C4⋊C4—212no—
287C2×C4⋊C4—212no—
288C2×C4⋊C4—46no—
289C2×C4⋊C4—412no—
290C2×C4⋊C4#4118yescomplemented
291C2×C4⋊C4C24112yes—
292C2×C4⋊C4C24112yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C24112yes—
295C2×C4⋊C4C24112yes—
296C2×C4⋊C4C24112yes—
297C2×C4⋊C4C24112yes—
298C2×C4⋊C4C24112yes—
299C2×C4⋊C4C24124yes—
300C2×C4⋊C4C24112yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—224no—
303C2×C22⋊C4—212no—
304C2×C22⋊C4—212no—
305C2×C22⋊C4—212no—
306C2×C22⋊C4—212no—
307C2×C22⋊C4—212no—
308C2×C22⋊C4—212no—
309C2×C22⋊C4—46no—
310C2×C22⋊C4—412no—
311C2×C22⋊C4—46no—
312C2×C22⋊C4—412no—
313C2×C22⋊C4—412no—
314C2×C22⋊C4C24124yes—
315C2×C22⋊C4C24112yes—
316C2×C22⋊C4C24112yes—
317C2×C22⋊C4C24112yes—
318C2×C22⋊C4C24112yes—
319C2×C22⋊C4C24112yes—
320C2×C22⋊C4C24112yes—
321C2×C22⋊C4C24124yes—
322C2×C22⋊C4C24124yes—
323C2×C22⋊C4C24112yes—
324C2×C22⋊C4C24112yes—
325C2×C22⋊C4C24124yes—
326C2×C42—212no—
327C2×C42—212no—
328C2×C42—212no—
329C2×C42—212no—
330C2×C42—296no—
331C2×C42—24no—
332C2×C42—212no—
333C2×C42—212no—
334C2×C42C24112yes—
335C2×C42C24112yes—
336C2×C42C24112yes—
337C2×C42C24112yes—
338C2×C42C24112yes—
339C2×C42C24112yes—
340C2×C42C2414yes—
341#95C2414yes—
342#95C2414yes—
343#95C2414yes—
344#95C24112yes—
345#95C24112yes—
346#95C24112yes—
347#95C2414yes—
348#95C24112yes—
349#95C24112yes—
350#95C24112yes—
351#95C24124yes—
352C24—26no—
353C24—26no—
354C24—26no—
355C24—26no—
356C24—26no—
357C24—26no—
358C24—26no—
359C24—26no—
360C24—26no—
361C24—26no—
362C24—26no—
363C24—26no—
364C24—26no—
365C24C22×D4112yes—
366C24C2516yes—
367C24C2516yes—
368C24C2511yescharacteristic
369#41—2256no—
370#41—4128no—
371C2×Q8#123164yescomplemented
372C2×Q8C22×D4116yes—
373C2×Q8C2×#41116yes—
374C2×Q8C2×#41116yes—
375C2×D4—224no—
376C2×D4—248no—
377C2×D4—296no—
378C2×D4—224no—
379C2×D4—224no—
380C2×D4—224no—
381C2×D4—224no—
382C2×D4—224no—
383C2×D4—224no—
384C2×D4—412no—
385C2×D4—448no—
386C2×D4—424no—
387C2×D4—424no—
388C2×D4—448no—
389C2×D4—848no—
390C2×D4C22×D4148yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—296no—
402C22×C4—224no—
403C22×C4—224no—
404C22×C4—26no—
405C22×C4—26no—
406C22×C4—224no—
407C22×C4—224no—
408C22×C4—26no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—26no—
412C22×C4—26no—
413C22×C4—26no—
414C22×C4—26no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—26no—
418C22×C4—26no—
419C22×C4—26no—
420C22×C4—26no—
421C22×C4—26no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—26no—
425C22×C4—26no—
426C22×C4—26no—
427C22×C4—224no—
428C22×C4—224no—
429C22×C4—224no—
430C22×C4—224no—
431C22×C4—224no—
432C22×C4—412no—
433C22×C4—412no—
434C22×C4—448no—
435C22×C4—448no—
436C22×C4—448no—
437C22×C4C22×D414yes—
438C22×C4C22×D4148yes—
439C22×C4C22×D4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#41112yes—
444C22×C4C2×#41112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4C2512yes—
447C22×C4C2512yes—
448C22×C4C2516yes—
449C22×C4C2516yes—
450C22×C4C2516yes—
451C22×C4C2516yes—
452C22×C4C2516yes—
453C22×C4C2512yes—
454C22×C4C2512yes—
455C22×C4C2512yes—
456C22×C4C2512yes—
457C22×C4C2512yes—
458C22×C4C2516yes—
459C4⋊C4—224no—
460C4⋊C4—224no—
461C4⋊C4—28no—
462C4⋊C4—224no—
463C4⋊C4—224no—
464C4⋊C4—224no—
465C4⋊C4—224no—
466C4⋊C4—224no—
467C4⋊C4—412no—
468C4⋊C4—424no—
469C4⋊C4C2×#41116yescomplemented
470C22⋊C4—248no—
471C22⋊C4—224no—
472C22⋊C4—224no—
473C22⋊C4—224no—
474C22⋊C4—224no—
475C22⋊C4—224no—
476C22⋊C4—224no—
477C22⋊C4—412no—
478C22⋊C4—424no—
479C22⋊C4—412no—
480C22⋊C4—424no—
481C22⋊C4—424no—
482C42—28no—
483C42—224no—
484C42—224no—
485C42—224no—
486C42—224no—
487C42—224no—
488C42—224no—
489C23—26no—
490C23—26no—
491C23—26no—
492C23—26no—
493C23—26no—
494C23—26no—
495C23—224no—
496C23—224no—
497C23—224no—
498C23—26no—
499C23—26no—
500C23—26no—
501C23—26no—
502C23—26no—
503C23—26no—
504C23—26no—
505C23—412no—
506C23—412no—
507C23—412no—
508C23—412no—
509C23—412no—
510C23—412no—
511C23—412no—
512C23—412no—
513C23—412no—
514C23—412no—
515C23—412no—
516C23C23×D4112yes—
517C23C23×D412yes—
518C23C22×#4112yes—
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)12yes—
522C23#58512yes—
523C23#58512yes—
524C23C2611yescharacteristic
525Q8C2×#123164yescomplemented
526D4—296no—
527D4—448no—
528D4—448no—
529D4—848no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—296no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—26no—
552C2×C4—26no—
553C2×C4—26no—
554C2×C4—26no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—26no—
558C2×C4—26no—
559C2×C4—26no—
560C2×C4—26no—
561C2×C4—26no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—26no—
567C2×C4—448no—
568C2×C4—448no—
569C2×C4—44no—
570C2×C4—412no—
571C2×C4—412no—
572C2×C4—412no—
573C2×C4—412no—
574C2×C4—412no—
575C2×C4—44no—
576C2×C4—448no—
577C2×C4C2×#123148yes—
578C2×C4C23×D414yes—
579C2×C4C23×D4112yes—
580C2×C4C22×#4114yes—
581C2×C4C22×#4114yes—
582C2×C4C22×#4114yes—
583C2×C4C22×#41112yes—
584C2×C4C22×#41112yes—
585C2×C4C2×H(5,𝔽2)14yes—
586C22—212no—
587C22—212no—
588C22—212no—
589C22—46no—
590C22—46no—
591C22—448no—
592C22—46no—
593C22—46no—
594C22—46no—
595C22—46no—
596C22—46no—
597C22—46no—
598C22—46no—
599C22—46no—
600C22—46no—
601C22—824no—
602C22C22×#12314yes—
603C22D4×#4114yes—
604C22#348014yes—
605C22#348314yes—
606C22#348814yes—
607C22#351714yes—
608C22#352414yes—
609C22C24×D411yescharacteristic
610C22C23×#4111yescharacteristic
611C22C23×#4111yescharacteristic
612C22C23×#4111yescharacteristic
613C22C22×H(5,𝔽2)11yescharacteristic
614C22C2×#58511yescharacteristic
615C22C2×#58511yescharacteristic
616C4—24no—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—24no—
621C4—46no—
622C4—42no—
623C4—46no—
624C4—46no—
625C4—42no—
626C4—46no—
627C4—46no—
628C4C22×#123124yes—
629C2—412no—
630C2—86no—
631C2#3643318yesdirect product factor
632C2C23×#12311yescharacteristic
633C2C2×D4×#4111yescharacteristic
634C2C2×#348011yescharacteristic
635C2C2×#348311yescharacteristic
636C2C2×#351711yescharacteristic
637C2C2×#352411yescharacteristic
638C2C2×#348811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC22×#550C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×#41C2semidirect
3. C2×#2854C2semidirect
4. #36433C2direct (N × H)
5. C2×#2500C2semidirect
6. C2×#2485C2semidirect
7. C2×#2420C2semidirect
8. C2×#2445C2semidirect
9. C2×#2445C2semidirect
10. C2×#2422C2semidirect
11. C2×#2351C2semidirect
12. C2×Q8×C4⋊C4C2semidirect
13. C2×#2341C2semidirect
14. C2×#2341C2semidirect
15. C2×#2341C2semidirect
16. C2×C4×#123C2semidirect
27. C2×C4×#41C22semidirect
38. #2854C22semidirect
39. #2500C22semidirect
40. #2485C22semidirect
41. #2445C22semidirect
42. #2445C22semidirect
43. #2422C22semidirect
44. #2420C22semidirect
49. #2351C22semidirect
50. Q8×C4⋊C4C22semidirect
51. #2341C22semidirect
52. #2341C22semidirect
53. #2341C22semidirect
54. C4×#123C22semidirect
290. C2×C4⋊C4#41semidirect
371. C2×Q8#123semidirect
469. C4⋊C4C2×#41semidirect
525. Q8C2×#123semidirect
631. C2#36433direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5502
C22×C4×#412
C2×#28542
#364332
C2×#25002
C2×#24852
C2×#24202
C2×#24452
C2×#24222
C2×#23512
C2×Q8×C4⋊C42
C2×#23412
C2×C4×#1232