Finite Group Databasebeta

← #355739

Subgroups — #355739

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C211yesmaximal, characteristic
2C2×#1929C211yesmaximal, characteristic
3#20872C211yesmaximal, characteristic
4#20436C211yesmaximal, characteristic, complemented
5#20375C211yesmaximal, characteristic
6#10934C211yesmaximal, characteristic
7#10929C211yesmaximal, characteristic
8#10911C211yesmaximal, characteristic, complemented
9#10909C211yesmaximal, characteristic, complemented
10#10886C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10860C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10790C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2296C2211yescharacteristic
33#1938C418yescomplemented
34#1929C418yescomplemented
35#1865C412yescomplemented
36#1864C412yescomplemented
37#1863C412yescomplemented
38#1862C412yescomplemented
39#1771—21no—
40#1771—21no—
41#1770—21no—
42#1770—21no—
43#1750—21no—
44#1750—21no—
45#1749—21no—
46#1749—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C4×#95C2211yescharacteristic
65C22×#41C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C4×#41—41no—
75C4×#41—41no—
76C4×#41—41no—
77C4×#41—41no—
78C2×C4×Q8C2×C412yes—
79C2×C4×D4—22no—
80C2×C4×D4C2×C412yes—
81C2×#117—21no—
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#108—41no—
90C2×#108—41no—
91C2×#108C2×C414yescomplemented
92#427—41no—
93#427—41no—
94C2×#107C2×C412yes—
95C2×#106C2×C412yes—
96#424—22no—
97#424C2×C414yescomplemented
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×#104—41no—
101C2×#104—41no—
102C2×#104—41no—
103C2×#104—41no—
104C2×#104C2×C414yes—
105C2×#104C2×C414yes—
106#416—41no—
107#416—41no—
108#416C2×C414yes—
109#416C2×C414yes—
110C2×#103C2×C412yes—
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C412yes—
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#408—22no—
117#408—22no—
118#408C2×C414yes—
119#408C2×C414yes—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124#405—22no—
125#405—41no—
126#405—41no—
127#405C2×C414yescomplemented
128C4×M16—41no—
129C4×M16—41no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97—41no—
134C2×#97—41no—
135C2×#97—41no—
136C2×#97—41no—
137C2×#97C2×C414yescomplemented
138C2×#97D411yescharacteristic
139C2×#97D411yescharacteristic
140C2×#97D411yescharacteristic
141C2×#97D411yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8—22no—
145C2×C4×C8—21no—
146C2×C4×C8—41no—
147C2×C4×C8—41no—
148#389C2×C412yes—
149#389C2×C412yes—
150#385—22no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C2×#95—22no—
154#341—41no—
155#341—41no—
156#341C2×C414yescomplemented
157#340C2×C412yes—
158#340C2×C412yes—
159#339—41no—
160#339—41no—
161#339—41no—
162#339—41no—
163#339C2×C414yescomplemented
164#339C2×C414yescomplemented
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—22no—
172#336—21no—
173#336—21no—
174#336C2×C412yes—
175#336C2×C412yes—
176#336C2×C414yescomplemented
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41C22×C412yes—
188C2×#41C22×C412yes—
189C22×Q8C22×C411yescharacteristic
190C22×D4—21no—
191C22×D4C22×C411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16C22×C412yes—
219C2×M16C22×C412yes—
220C2×M16C22×C412yes—
221C2×M16C22×C412yes—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8C22×C412yes—
245C22×C8C22×C411yescharacteristic
246C22×C8C22×C411yescharacteristic
247C22×C8C22×C412yes—
248C22×C8C22×C412yes—
249C22×C8C22×C412yes—
250C22×C8C22×C412yes—
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8#4112yes—
256C22×C8#4112yes—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—44no—
262C4×D4—44no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—42no—
274#117—42no—
275#117C22×C412yes—
276#117C22×C412yes—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42C22×C412yes—
309C2×C42C22×C412yes—
310C2×C42C22×C412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313#108—44no—
314#107—42no—
315#106—42no—
316C4⋊C8—24no—
317C4⋊C8—24no—
318#104—44no—
319#104—44no—
320#103—42no—
321#103C4214yes—
322#103C4214yes—
323#102—42no—
324#102C4214yes—
325#102C4214yes—
326C22⋊C8—24no—
327C22⋊C8—24no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330#97—24no—
331#97—44no—
332#97—41no—
333#97—41no—
334#97—41no—
335#97—41no—
336#97—42no—
337#97—42no—
338C4×C8—24no—
339C4×C8—42no—
340C4×C8—44no—
341C4×C8—42no—
342#95—22no—
343#95—22no—
344#95—44no—
345C24—21no—
346C24C2×#4111yescharacteristic
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—22no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8C2×C4212yes—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4C2×C4212yes—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C23×C411yescharacteristic
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8C2×C4212yes—
485C2×C8C2×C4212yes—
486C2×C8C4×D412yes—
487C2×C8C4×D412yes—
488C2×C8C4×D412yes—
489C2×C8C4×D412yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—44no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×C4212yes—
500C4⋊C4C2×C4212yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—44no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—44no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23C2×C4×D411yescharacteristic
537C23C4×#4111yescharacteristic
538C23C4×#4111yescharacteristic
539C23#53811yescharacteristic
540C23#53811yescharacteristic
541C23#54111yescharacteristic
542C23#54211yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547D4—22no—
548D4—22no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—41no—
554D4—41no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—88no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4C22×C4211yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C4×#4111yescharacteristic
595C2×C4#51811yescharacteristic
596C2×C4#51811yescharacteristic
597C2×C4#51811yescharacteristic
598C2×C4#51811yescharacteristic
599C2×C4#51911yescharacteristic
600C2×C4#52011yescharacteristic
601C2×C4#52011yescharacteristic
602C2×C4#52011yescharacteristic
603C2×C4#52011yescharacteristic
604C2×C4#52011yescharacteristic
605C2×C4#52011yescharacteristic
606C2×C4#52011yescharacteristic
607C2×C4#53411yescharacteristic
608C2×C4#53511yescharacteristic
609C2×C4#53611yescharacteristic
610C2×C4#53711yescharacteristic
611C2×C4#53811yescharacteristic
612C2×C4#53911yescharacteristic
613C2×C4#53911yescharacteristic
614C2×C4#53911yescharacteristic
615C2×C4#54011yescharacteristic
616C2×C4#54211yescharacteristic
617C2×C4#54211yescharacteristic
618C2×C4#54211yescharacteristic
619C8—22no—
620C8—22no—
621C8—22no—
622C8—22no—
623C8—22no—
624C8—22no—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—42no—
629C8—42no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—82no—
642C22—82no—
643C22C42×D411yescharacteristic
644C22#232111yescharacteristic
645C22#232311yescharacteristic
646C22#233411yescharacteristic
647C22#233411yescharacteristic
648C22#233511yescharacteristic
649C22#233711yescharacteristic
650C22#295612yes—
651C22#295712yes—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#227411yescharacteristic
662C4#227511yescharacteristic
663C4#232211yescharacteristic
664C4#232411yescharacteristic
665C4#232511yescharacteristic
666C4#232611yescharacteristic
667C4#233211yescharacteristic
668C4#233511yescharacteristic
669C2—21no—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—82no—
674C2#1378911yescharacteristic
675C2#2051211yescharacteristic
676C2#2051311yescharacteristic
677C2#2206212yes—
678C2#2213212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20436C2semidirect
8. #10911C2semidirect
9. #10909C2semidirect
10. #10886C2semidirect
11. #10882C2semidirect
12. #10860C2semidirect
13. #10809C2semidirect
15. #10696C2semidirect
33. #1938C4semidirect
34. #1929C4semidirect
35. #1865C4semidirect
36. #1864C4semidirect
37. #1863C4semidirect
38. #1862C4semidirect
91. C2×#108C2×C4semidirect
97. #424C2×C4semidirect
127. #405C2×C4semidirect
137. C2×#97C2×C4semidirect
156. #341C2×C4semidirect
163. #339C2×C4semidirect
164. #339C2×C4semidirect
176. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
C2×#19292
#208722
#204362
#203752
#109342
#109292
#109112
#109092
#108862
#108822
#108602
#108092
#107902
#106962