Finite Group Databasebeta

← #366920

Subgroups — #366920

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#1998C211yesmaximal, characteristic, complemented
3#21325C211yesmaximal, characteristic, complemented
4C2×#1950C211yesmaximal, characteristic, complemented
5#20646C211yesmaximal, characteristic
6#20441C211yesmaximal, characteristic, complemented
7C2×C4×#100C211yesmaximal, characteristic
8#11366C211yesmaximal, characteristic, complemented
9#11365C211yesmaximal, characteristic, complemented
10#11127C211yesmaximal, characteristic
11#11127C211yesmaximal, characteristic
12#10943C211yesmaximal, characteristic
13#10943C211yesmaximal, characteristic
14#10870C211yesmaximal, characteristic, complemented
15#10869C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#478—24no—
19C2×#478—21no—
20C2×#478—21no—
21C2×#478—21no—
22C2×#478—21no—
23C2×#478C418yescomplemented
24C22×#108C2211yescharacteristic
25C22×#108C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#100C2211yescharacteristic
35C22×#100C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×C4×M16C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2285C2211yescharacteristic
41#1998—24no—
42#1992—22no—
43#1991—22no—
44#1950—24no—
45#1804—22no—
46#1802—22no—
47#1771—21no—
48#1771—21no—
49#1770—21no—
50C8×C4⋊C4—21no—
51C4×#100—24no—
52C2×#343C2211yescharacteristic
53C2×#343C2211yescharacteristic
54C2×#343C2211yescharacteristic
55C2×#343C2211yescharacteristic
56C2×#343C2211yescharacteristic
57C2×#343C2211yescharacteristic
58C2×#341C2211yescharacteristic
59C2×#341C2211yescharacteristic
60#1736C2211yescharacteristic
61#1736C2211yescharacteristic
62#1736C2211yescharacteristic
63#1736C2211yescharacteristic
64#1736C2211yescharacteristic
65#1736C2211yescharacteristic
66#1729C2211yescharacteristic
67#1729C2211yescharacteristic
68C23×D4C2311yescharacteristic
69C24×C4—21no—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#125—22no—
79C2×#125—21no—
80C2×#125—21no—
81C2×#125—22no—
82C2×#125—21no—
83C2×#125—21no—
84C2×#125—22no—
85C2×#125C2×C414yes—
86C2×C4×D4—22no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—22no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—22no—
93C2×C4×D4C2×C414yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C42C2311yescharacteristic
103#478—28no—
104#478—28no—
105#478—28no—
106#478—28no—
107C2×#108—22no—
108C2×#108—22no—
109C2×#108—22no—
110C2×#108—22no—
111C2×#108—41no—
112C2×#108—41no—
113C2×#107—21no—
114C2×#107—21no—
115C2×#107—22no—
116C2×#107C2×C414yes—
117C2×#106—21no—
118C2×#106—21no—
119C2×#106—22no—
120C2×#106C2×C414yes—
121#423—22no—
122#423—22no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8C2×C414yes—
126C2×C4⋊C8D414yes—
127C2×C4⋊C8D411yescharacteristic
128C2×C4⋊C8D411yescharacteristic
129C2×C4⋊C8D411yescharacteristic
130C2×C4⋊C8D411yescharacteristic
131#418—22no—
132#418—22no—
133#418—41no—
134#418—41no—
135C2×#102—21no—
136C2×#102—21no—
137C2×#102—22no—
138C2×#102—21no—
139C2×#102—21no—
140C2×#102—22no—
141C2×#102—22no—
142C2×#102—22no—
143C2×#102C2×C414yes—
144C2×#102C2×C414yes—
145C2×#100—22no—
146C2×#100—22no—
147C2×#100—22no—
148C2×#100—22no—
149C2×#100—22no—
150C2×#100Q814yes—
151#405—22no—
152#405—22no—
153#405—41no—
154#405—41no—
155C4×M16—22no—
156C4×M16—22no—
157C4×M16—22no—
158C4×M16D414yes—
159C2×#97—21no—
160C2×#97—21no—
161C2×#97—21no—
162C2×#97—21no—
163C2×C4×C8—21no—
164C2×C4×C8—21no—
165C2×C4×C8—21no—
166C2×C4×C8—21no—
167#384D412yes—
168#382D412yes—
169#380—21no—
170#380—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173#343—22no—
174#343—22no—
175#343—22no—
176#343—22no—
177#343—22no—
178#343—22no—
179#341—22no—
180#341—22no—
181#340—22no—
182#340—22no—
183#340—22no—
184#340—22no—
185#340—22no—
186#340—22no—
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191C25—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C22×C412yes—
204C22×D4C2×Q812yes—
205C22×D4#4112yes—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—24no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—42no—
213C23×C4—84no—
214C23×C4#4111yescharacteristic
215C23×C4#4111yescharacteristic
216C23×C4C2411yescharacteristic
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16C2×D412yes—
248C2×M16C2×D412yes—
249C2×M16C2×Q812yes—
250C2×M16C2×Q812yes—
251C2×M16#4112yes—
252C2×M16#4112yes—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8C22×C412yes—
268C22×C8C22×C412yes—
269C22×C8C2×D411yescharacteristic
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D412yes—
274C22×C8C2×D412yes—
275#125—24no—
276#125—24no—
277#125—24no—
278#125—24no—
279#125—42no—
280#125—42no—
281#125—42no—
282#125—42no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—41no—
319C2×C4⋊C4—41no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4C22×C412yes—
325C2×C4⋊C4C22×C412yes—
326C2×C4⋊C4C22×C412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—41no—
333C2×C22⋊C4—41no—
334C2×C22⋊C4—42no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42C22×C412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42#4112yes—
345#108—44no—
346#108—44no—
347#107—24no—
348#107—24no—
349#106—24no—
350#106—24no—
351C4⋊C8—42no—
352C4⋊C8—42no—
353C4⋊C8C22⋊C414yes—
354C4⋊C8C22⋊C414yes—
355C4⋊C8C22⋊C414yes—
356C4⋊C8C22⋊C414yes—
357#102—24no—
358#102—24no—
359#102—24no—
360#102—24no—
361#102—44no—
362#102—44no—
363#100—28no—
364#100—28no—
365#97—24no—
366#97—42no—
367#97—42no—
368#97—42no—
369C4×C8—24no—
370C4×C8—42no—
371C4×C8—42no—
372C4×C8—42no—
373#95—22no—
374#95—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—82no—
383C24—82no—
384C24C2×#4111yescharacteristic
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4C4×Q814yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4C4×D412yes—
436C22×C4C4×D412yes—
437C22×C4#12111yescharacteristic
438C22×C4#12111yescharacteristic
439C22×C4#12111yescharacteristic
440C22×C4#12111yescharacteristic
441C22×C4#12412yes—
442C22×C4#12412yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×Q811yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C42−1+411yescharacteristic
455C22×C42−1+411yescharacteristic
456C22×C42−1+411yescharacteristic
457M16—22no—
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478M16—41no—
479M16—41no—
480M16C22⋊Q814yes—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8C2×C22⋊C412yes—
516C2×C8C2×C22⋊C412yes—
517C2×C8C2×C22⋊C412yes—
518C2×C8C2×C22⋊C412yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—82no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42C2×C22⋊C412yes—
552C42C2×C22⋊C412yes—
553C42C4×D412yes—
554C42C4×D412yes—
555C42#12112yes—
556C42#12112yes—
557C42#12712yes—
558C42#12712yes—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—82no—
569C23—82no—
570C23—84no—
571C23—84no—
572C23—82no—
573C23—82no—
574C23—84no—
575C23—84no—
576C23—82no—
577C23—82no—
578C23C2×C4×D411yescharacteristic
579C23C2×#12111yescharacteristic
580C23C2×#12411yescharacteristic
581C23#52911yescharacteristic
582C23#53311yescharacteristic
583C23#54111yescharacteristic
584C23#54211yescharacteristic
585D4—24no—
586D4—24no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4#39012yes—
622C2×C4#39012yes—
623C2×C4#39012yes—
624C2×C4#39012yes—
625C2×C4C22×C22⋊C411yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×Q811yescharacteristic
628C2×C4C4×#4111yescharacteristic
629C2×C4#51911yescharacteristic
630C2×C4#52011yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×C22⋊Q811yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4C2×#12711yescharacteristic
635C2×C4#53611yescharacteristic
636C2×C4#53711yescharacteristic
637C2×C4#54611yescharacteristic
638C2×C4#54611yescharacteristic
639C2×C4#54611yescharacteristic
640C2×C4#54711yescharacteristic
641C2×C4#54711yescharacteristic
642C2×C4#54711yescharacteristic
643C2×C4#54811yescharacteristic
644C2×C4#54811yescharacteristic
645C2×C4#54811yescharacteristic
646C2×C4D4×Q811yescharacteristic
647C2×C4D4×Q811yescharacteristic
648C2×C4D4×Q811yescharacteristic
649C2×C4#55211yescharacteristic
650C2×C4#55211yescharacteristic
651C2×C4#55411yescharacteristic
652C2×C4#55611yescharacteristic
653C8—22no—
654C8—22no—
655C8—22no—
656C8—22no—
657C8—24no—
658C8—42no—
659C8—42no—
660C8—42no—
661C8—81no—
662C8—81no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—44no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—82no—
677C22—82no—
678C22—84no—
679C22—84no—
680C22C2×#39011yescharacteristic
681C22C4×C22⋊Q811yescharacteristic
682C22#232311yescharacteristic
683C22#242711yescharacteristic
684C22#242911yescharacteristic
685C22#248911yescharacteristic
686C22#249211yescharacteristic
687C22#306712yes—
688C22#306812yes—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#233911yescharacteristic
699C4Q8×C22⋊C411yescharacteristic
700C4#236011yescharacteristic
701C4#236211yescharacteristic
702C4#241511yescharacteristic
703C4#241711yescharacteristic
704C4#254011yescharacteristic
705C4#254211yescharacteristic
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2#1399711yescharacteristic
712C2#2124811yescharacteristic
713C2#2124911yescharacteristic
714C2#2536212yes—
715C2#2549312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1998C2semidirect
3. #21325C2semidirect
4. C2×#1950C2semidirect
6. #20441C2semidirect
8. #11366C2semidirect
9. #11365C2semidirect
14. #10870C2semidirect
15. #10869C2semidirect
23. C2×#478C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#19982
#213252
C2×#19502
#206462
#204412
C2×C4×#1002
#113662
#113652
#111272
#109432
#108702
#108692