Finite Group Databasebeta

← #414459

Subgroups — #414459

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C211yesmaximal, characteristic
2C2×#2085C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4C2×#2057C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#14276C211yesmaximal, characteristic
7#11325C211yesmaximal, characteristic
8#11325C211yesmaximal, characteristic
9#11324C211yesmaximal, characteristic
10#11323C211yesmaximal, characteristic
11#10863C211yesmaximal, characteristic
12#10863C211yesmaximal, characteristic
13#10862C211yesmaximal, characteristic
14#10861C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#2085—24no—
34#2085—24no—
35#2058—24no—
36#2057—24no—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49#1446—22no—
50#1445—22no—
51#1441—22no—
52#1441—22no—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C23×Q8—21no—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—41no—
69C2×C4⋊Q8—41no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—41no—
94C2×#107—41no—
95C2×#107—41no—
96C2×#107—41no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106—41no—
106C2×#106—41no—
107C2×#106—41no—
108C2×#106—41no—
109C2×#97—41no—
110C2×#97—41no—
111C2×#97Q814yescomplemented
112C2×C4×C8—41no—
113C2×C4×C8—41no—
114C2×C4×C8Q814yescomplemented
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#384—22no—
120#384—22no—
121#382—22no—
122#382—22no—
123#382—41no—
124#382—41no—
125#376—41no—
126#376—41no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D412yes—
139C2×#95D412yes—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#336D414yes—
149#336D414yes—
150#336Q814yescomplemented
151#336Q814yescomplemented
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—82no—
156C22×Q8—82no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×Q812yes—
191C22×C8C2×Q812yes—
192C22×C8C2×Q812yes—
193C22×C8C2×Q812yes—
194C22×C8C2×Q812yes—
195C22×C8C2×Q812yes—
196C22×C8C2×Q812yes—
197C22×C8C2×Q812yes—
198C4⋊Q8—44no—
199C4⋊Q8—44no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4#4112yes—
257C2×C4⋊C4#4112yes—
258C2×C4⋊C4#4112yes—
259C2×C4⋊C4#4112yes—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C2×C42C2×Q812yes—
276C2×C42C2×Q812yes—
277C2×C42C2×Q812yes—
278#107—44no—
279#107—44no—
280#107—44no—
281#107—44no—
282#106—44no—
283#106—44no—
284#106—44no—
285#106—44no—
286#97—44no—
287C4×C8—44no—
288#95—42no—
289#95—42no—
290#95—44no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—82no—
296#95—82no—
297#95—82no—
298#95—82no—
299#95—82no—
300#95—82no—
301C24C22×D411yescharacteristic
302C2×Q8—48no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×Q8—88no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4#12712yes—
368C22×C4#12712yes—
369C22×C4C4⋊Q814yes—
370C22×C4C4⋊Q814yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q811yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C42−1+411yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8C4⋊Q814yes—
410C2×C8C4⋊Q814yes—
411C2×C8C4⋊Q814yes—
412C2×C8C4⋊Q814yes—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—48no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C23#52511yescharacteristic
462C23#52511yescharacteristic
463C23C2×#12711yescharacteristic
464C23C2×C4⋊Q811yescharacteristic
465C23C2×C4⋊Q811yescharacteristic
466C23#53411yescharacteristic
467C23#53711yescharacteristic
468C23#57612yes—
469C23#57712yes—
470C23#57712yes—
471C23#57812yes—
472Q8—816no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4C2×C4⋊Q811yescharacteristic
538C2×C4C2×C4⋊Q811yescharacteristic
539C2×C4C2×C4⋊Q811yescharacteristic
540C2×C4C2×C4⋊Q811yescharacteristic
541C2×C4#53311yescharacteristic
542C2×C4#53311yescharacteristic
543C2×C4#53311yescharacteristic
544C2×C4#53311yescharacteristic
545C2×C4#54311yescharacteristic
546C2×C4#54311yescharacteristic
547C2×C4#54411yescharacteristic
548C2×C4#54411yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4#54611yescharacteristic
551C2×C4#54611yescharacteristic
552C2×C4#54611yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54611yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4#54811yescharacteristic
557C2×C4#55411yescharacteristic
558C2×C4#55411yescharacteristic
559C2×C4#55411yescharacteristic
560C2×C4#55411yescharacteristic
561C2×C4#55411yescharacteristic
562C2×C4#55411yescharacteristic
563C2×C4#55711yescharacteristic
564C2×C4#55711yescharacteristic
565C8—24no—
566C8—24no—
567C8—24no—
568C8—24no—
569C22#239911yescharacteristic
570C22#253911yescharacteristic
571C22#254011yescharacteristic
572C22#258311yescharacteristic
573C22#258411yescharacteristic
574C22#265911yescharacteristic
575C22#286811yescharacteristic
576C22#311812yes—
577C22#311812yes—
578C22#312112yes—
579C22#312112yes—
580C22#315411yescharacteristic
581C22#315411yescharacteristic
582C22#315511yescharacteristic
583C22#315611yescharacteristic
584C22#316312yes—
585C22#316312yes—
586C22#316412yes—
587C22#316412yes—
588C22#319812yes—
589C22#320112yes—
590C22#325712yes—
591C22#325812yes—
592C4—44no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—161no—
601C4—161no—
602C4—161no—
603C4—161no—
604C4#254911yescharacteristic
605C4#254911yescharacteristic
606C4#254911yescharacteristic
607C4#254911yescharacteristic
608C4#255011yescharacteristic
609C4#255011yescharacteristic
610C4#255011yescharacteristic
611C4#255011yescharacteristic
612C2#1517311yescharacteristic
613C2#2384212yes—
614C2#2832111yescharacteristic
615C2#2832411yescharacteristic
616C2#2839311yescharacteristic
617C2#2839611yescharacteristic
618C2#2930812yes—
619C2#2932512yes—
620C2#3029912yes—
621C2#3331811yescharacteristic
622C2#3331911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
111. C2×#97Q8semidirect
114. C2×C4×C8Q8semidirect
150. #336Q8semidirect
151. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20852
C2×#20582
C2×#20572
#142822
#142762
#113252
#113242
#113232
#108632
#108622
#108612
#106942