Finite Group Databasebeta

← #370618

Subgroups — #370618

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5#13759C211yesmaximal, characteristic
6#11386C211yesmaximal, characteristic
7#11386C211yesmaximal, characteristic
8#11209C211yesmaximal, characteristic
9#11209C211yesmaximal, characteristic
10#10879C211yesmaximal, characteristic
11#10878C211yesmaximal, characteristic
12#10852C211yesmaximal, characteristic
13#10851C211yesmaximal, characteristic
14#10840C211yesmaximal, characteristic
15#10701C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2552C412yescomplemented
23#2551C412yescomplemented
24#2550C412yescomplemented
25#2550C412yescomplemented
26#2549—22no—
27#2440—22no—
28#2399—22no—
29C4×C4⋊Q8—22no—
30#2296C2211yescharacteristic
31#2296C2211yescharacteristic
32#2285C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37C2×#376C2211yescharacteristic
38C2×C43C2211yescharacteristic
39#1995—24no—
40#1994—24no—
41#1940—22no—
42#1939—22no—
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1728C2211yescharacteristic
60#1728C2211yescharacteristic
61#1436—22no—
62#1436—22no—
63C4×#95C2211yescharacteristic
64C4×#95C2211yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69#533—22no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533—41no—
74#533—41no—
75#533—41no—
76#533—41no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8C2×C412yes—
79C2×C4⋊Q8C2×C412yes—
80C2×#125C2×C412yes—
81C2×#125C2×C412yes—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—21no—
91C22×C4⋊C4D412yes—
92C22×C42—22no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#423—22no—
101#423—22no—
102#423—22no—
103#423—22no—
104#423—41no—
105#423—41no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C4×M16—22no—
111C4×M16—22no—
112C2×#97—21no—
113C2×C4×C8—21no—
114#398—22no—
115#398—22no—
116#398—22no—
117#398—22no—
118#395—22no—
119#395—22no—
120#395—22no—
121#395—22no—
122#391—22no—
123#391—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389C2×C412yes—
129#389C2×C412yes—
130#389C2×C412yes—
131#389C2×C412yes—
132#385—22no—
133#385—22no—
134#384—22no—
135#384C2×C412yes—
136#382C2×C412yes—
137#380—21no—
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4C2×C412yes—
145C4×C4⋊C4C2×C412yes—
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C4⋊C4D412yes—
151C4×C4⋊C4D412yes—
152C4×C4⋊C4Q814yescomplemented
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155#376—21no—
156#376—22no—
157#376—22no—
158#376—41no—
159#376Q814yescomplemented
160C2×#95—21no—
161C2×#95—21no—
162C2×#95—21no—
163C2×#95—22no—
164C2×#95—22no—
165C2×#95—22no—
166C2×#95D412yes—
167C43—21no—
168C43—22no—
169C43—41no—
170#339—22no—
171#339—22no—
172#339—22no—
173#339—22no—
174#339—22no—
175#339—22no—
176#339—22no—
177#339—22no—
178#339—22no—
179#339—22no—
180#339—41no—
181#339—41no—
182#339—41no—
183#339—41no—
184#339D414yes—
185#339Q814yescomplemented
186#336—22no—
187#336—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
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×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16C2×D412yes—
222C2×M16C2×Q812yes—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—48no—
232#125—24no—
233#125—24no—
234#125—42no—
235#125—42no—
236C22⋊Q8—44no—
237C22⋊Q8—44no—
238C22⋊Q8—44no—
239C22⋊Q8—44no—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×Q8—44no—
243C4×Q8—44no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117C2×D412yes—
257#117C2×Q812yes—
258#117#4112yes—
259#117#4112yes—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C22⋊C412yes—
276C2×C4⋊C4C22⋊C412yes—
277C2×C4⋊C4C22⋊C412yes—
278C2×C4⋊C4C22⋊C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C42—21no—
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—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×Q812yes—
334C2×C42C2×Q812yes—
335C2×C42C2×Q812yes—
336C2×C42C2×Q812yes—
337C2×C42C2×Q812yes—
338C2×C42C2×Q812yes—
339C2×C42C2×Q812yes—
340C2×C42#4112yes—
341C2×C42#4112yes—
342C4⋊C8—44no—
343C4⋊C8—44no—
344C4⋊C8—42no—
345C4⋊C8—42no—
346#97—42no—
347C4×C8—42no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C22⋊C412yes—
363#95C22⋊C412yes—
364#95C22⋊C412yes—
365#95C22⋊C412yes—
366C24C22×D411yescharacteristic
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—82no—
372C2×Q8—82no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C412yes—
415C22×C4C22⋊Q812yes—
416C22×C4C22⋊Q812yes—
417C22×C4C22⋊Q812yes—
418C22×C4C22⋊Q812yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q811yescharacteristic
423C22×C4C22×Q811yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C42−1+411yescharacteristic
432C22×C42−1+411yescharacteristic
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C4×D414yes—
485C4⋊C4C4×Q814yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—44no—
520C42—416no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—82no—
532C42C4×Q812yes—
533C42C4×Q812yes—
534C42C4×Q812yes—
535C42C4×Q812yes—
536C42#12412yes—
537C42#12412yes—
538C42C4⋊Q812yes—
539C42C4⋊Q812yes—
540C42C4⋊Q814yescomplemented
541C23—22no—
542C23—22no—
543C23C22×C22⋊C411yescharacteristic
544C23C2×C22⋊Q811yescharacteristic
545C23C2×C22⋊Q811yescharacteristic
546C23D4211yescharacteristic
547C23#54611yescharacteristic
548C23#54811yescharacteristic
549C23D4×Q811yescharacteristic
550Q8—88no—
551Q8—88no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4#39112yes—
608C2×C4#39112yes—
609C2×C4#39112yes—
610C2×C4#39112yes—
611C2×C4#42114yes—
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C2×C4×Q811yescharacteristic
614C2×C4C2×C4×Q811yescharacteristic
615C2×C4C2×C4×Q811yescharacteristic
616C2×C4#51811yescharacteristic
617C2×C4#51911yescharacteristic
618C2×C4C2×#12411yescharacteristic
619C2×C4C2×C4⋊Q811yescharacteristic
620C2×C4C2×C4⋊Q811yescharacteristic
621C2×C4#53211yescharacteristic
622C2×C4#53311yescharacteristic
623C2×C4#53311yescharacteristic
624C2×C4#54211yescharacteristic
625C2×C4#54211yescharacteristic
626C2×C4#54411yescharacteristic
627C2×C4#54411yescharacteristic
628C2×C4D4211yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54811yescharacteristic
632C2×C4D4×Q811yescharacteristic
633C2×C4D4×Q811yescharacteristic
634C2×C4#55411yescharacteristic
635C2×C4#55411yescharacteristic
636C2×C4#55511yescharacteristic
637C2×C4#55611yescharacteristic
638C2×C4#55711yescharacteristic
639C2×C4Q8211yescharacteristic
640C8—42no—
641C8—42no—
642C8—81no—
643C8—81no—
644C22—22no—
645C22—22no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—22no—
651C22—22no—
652C22C2×#39111yescharacteristic
653C22D4×C22⋊C411yescharacteristic
654C22Q8×C22⋊C411yescharacteristic
655C22#243911yescharacteristic
656C22#244011yescharacteristic
657C22#245111yescharacteristic
658C22#245211yescharacteristic
659C22C2×#42112yes—
660C22#290112yes—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—24no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4C4×C4⋊Q811yescharacteristic
678C4#233111yescharacteristic
679C4#236511yescharacteristic
680C4#236911yescharacteristic
681C4#249111yescharacteristic
682C4#249211yescharacteristic
683C4#255011yescharacteristic
684C4#255111yescharacteristic
685C2—21no—
686C2—21no—
687C2—21no—
688C2—21no—
689C2#1405311yescharacteristic
690C2#2132911yescharacteristic
691C2#2133011yescharacteristic
692C2#2243412yes—
693C2#2252912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #2552C4semidirect
23. #2551C4semidirect
24. #2550C4semidirect
25. #2550C4semidirect
152. C4×C4⋊C4Q8semidirect
159. #376Q8semidirect
185. #339Q8semidirect
540. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
C2×#19952
C2×#19942
#211462
#137592
#113862
#112092
#108792
#108782
#108522
#108512
#108402
#107012