Finite Group Databasebeta

← #367023

Subgroups — #367023

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2#21259C211yesmaximal, characteristic
3C2×#1979C212yesmaximal
4C2×#1966C211yesmaximal, characteristic
5C2×#1950C211yesmaximal, characteristic
6C2×#1801C211yesmaximal, characteristic
7#20431C211yesmaximal, characteristic
8C4×#419C211yesmaximal, characteristic
9#11128C212yesmaximal
10#11029C212yesmaximal
11#10942C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#462—24no—
16C2×#462—21no—
17C2×#462—21no—
18C2×#462—22no—
19C2×#462C418yescomplemented
20C22×#108C2212yes—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#419C2212yes—
25C2×#419C2211yescharacteristic
26C2×#419C2211yescharacteristic
27C2×#419C2212yes—
28C22×#103C2212yes—
29C22×#101C2212yes—
30C22×#101C2212yes—
31C22×#101C2212yes—
32C2×#405C2212yes—
33C2×C4×M16C2211yescharacteristic
34C4×#117C2211yescharacteristic
35#1981—22no—
36#1981—22no—
37#1979—28no—
38#1966—24no—
39#1950—24no—
40#1801—24no—
41#1768—22no—
42#1767—21no—
43C4×C4⋊C8—21no—
44#1763—22no—
45C4×#103—22no—
46C2×#343C2212yes—
47C2×#341C2212yes—
48#1736C2211yescharacteristic
49#1736C2211yescharacteristic
50#1729C2212yes—
51C23×Q8C2312yes—
52C23×Q8C2311yescharacteristic
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—22no—
56C22×Q16—22no—
57C22×Q16—24no—
58C22×Q16—42no—
59C22×Q16—42no—
60C22×Q16C2×C418yes—
61C22×M16C2312yes—
62C22×M16C2311yescharacteristic
63C22×M16C2312yes—
64C22×M16C2311yescharacteristic
65C23×C8C2312yes—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C414yes—
74C2×C4×Q8—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—22no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C414yes—
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#462—28no—
91#462—28no—
92#462—216no—
93C2×#108—24no—
94C2×#108—24no—
95C2×#108—42no—
96#423—22no—
97#423—22no—
98#423—22no—
99#423D414yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C414yes—
105C2×C4⋊C8D414yes—
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8D412yes—
109#419—22no—
110#419—22no—
111#419—22no—
112#419—24no—
113#419—24no—
114#419—42no—
115#419—42no—
116#419D414yes—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—24no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—24no—
123C2×#103—24no—
124C2×#103C2×C418yes—
125C2×#101—24no—
126C2×#101—24no—
127C2×#101—24no—
128C2×#101—24no—
129C2×#101—24no—
130C2×#101—24no—
131C2×#101—42no—
132C2×#101—42no—
133C2×#101—42no—
134#405—24no—
135#405—42no—
136C4×M16—22no—
137C4×M16—22no—
138C2×#97—22no—
139C2×#97—21no—
140C2×C4×C8—22no—
141C2×C4×C8—21no—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4D412yes—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146#376—21no—
147C43—21no—
148#343—24no—
149#341—24no—
150#340—22no—
151#340—22no—
152#336—22no—
153#336—22no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8C22×C414yes—
179C22×Q8C22×C412yes—
180C22×Q8C2×D412yes—
181C22×Q8#4112yes—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4#4111yescharacteristic
186C23×C4#4111yescharacteristic
187C23×C4C2411yescharacteristic
188C2×Q16—216no—
189C2×Q16—28no—
190C2×Q16—28no—
191C2×Q16—48no—
192C2×Q16—44no—
193C2×Q16—44no—
194C2×Q16—48no—
195C2×Q16—44no—
196C2×Q16—44no—
197C2×Q16—48no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—22no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—22no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—22no—
219C2×M16C2×D414yes—
220C2×M16C2×D412yes—
221C2×M16#4112yes—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—22no—
230C22×C8C22×C414yes—
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233C22×C8C2×D414yes—
234C4⋊Q8—24no—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—42no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—42no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117C2×D412yes—
257#117#4112yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4C22×C412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—21no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—41no—
286C2×C42—41no—
287C2×C42—42no—
288C2×C42C22×C412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42#4112yes—
296#108—48no—
297C4⋊C8—24no—
298C4⋊C8—24no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8C22⋊C414yes—
302C4⋊C8C22⋊C414yes—
303C4⋊C8C22⋊C418yes—
304#103—28no—
305#103—28no—
306#103—28no—
307#103—28no—
308#103—48no—
309#103—48no—
310#101—48no—
311#101—48no—
312#101—48no—
313#97—44no—
314#97—42no—
315C4×C8—44no—
316C4×C8—42no—
317#95—22no—
318#95—22no—
319C24C2×#4111yescharacteristic
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—42no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8C4×D414yes—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12412yes—
405C22×C4#12412yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C42−1+411yescharacteristic
420Q16—416no—
421Q16—416no—
422M16—24no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—24no—
427M16—22no—
428M16—42no—
429M16—42no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—42no—
435M16—42no—
436M16—42no—
437M16—42no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8C2×C22⋊C414yes—
459C2×C8C2×C22⋊C414yes—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4#12114yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C42—22no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—44no—
483C42—44no—
484C42—44no—
485C42—82no—
486C42C2×C22⋊C412yes—
487C42C2×C22⋊C412yes—
488C42C4×D412yes—
489C42C4×D412yes—
490C42#12112yes—
491C42#12112yes—
492C42#12712yes—
493C42#12712yes—
494C23—22no—
495C23—22no—
496C23C2×C4×D411yescharacteristic
497C23C2×#12111yescharacteristic
498C23C2×#12411yescharacteristic
499C23#54711yescharacteristic
500C23#55311yescharacteristic
501C23#55311yescharacteristic
502C23#55511yescharacteristic
503Q8—24no—
504Q8—24no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—44no—
512Q8—44no—
513Q8—44no—
514Q8—44no—
515Q8—82no—
516Q8—82no—
517Q8—82no—
518Q8—82no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#39012yes—
551C2×C4#39012yes—
552C2×C4#39012yes—
553C2×C4#39012yes—
554C2×C4#47314yes—
555C2×C4C22×C22⋊C411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C4×#4111yescharacteristic
559C2×C4#51811yescharacteristic
560C2×C4#52011yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52911yescharacteristic
565C2×C4C2×#12711yescharacteristic
566C2×C4#53211yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53711yescharacteristic
569C2×C4#54011yescharacteristic
570C2×C4#54211yescharacteristic
571C2×C4D4211yescharacteristic
572C2×C4D4211yescharacteristic
573C2×C4D4211yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54611yescharacteristic
577C2×C4#54811yescharacteristic
578C2×C4#54811yescharacteristic
579C2×C4#54811yescharacteristic
580C2×C4#55011yescharacteristic
581C2×C4#55011yescharacteristic
582C2×C4#55011yescharacteristic
583C8—24no—
584C8—24no—
585C8—44no—
586C8—42no—
587C8—42no—
588C8—82no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22C2×#39011yescharacteristic
598C22#236111yescharacteristic
599C22#236311yescharacteristic
600C22#241411yescharacteristic
601C22#241611yescharacteristic
602C22#249011yescharacteristic
603C22#249111yescharacteristic
604C22C2×#47312yes—
605C22#306912yes—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—44no—
610C4—44no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4#197914yescomplemented
615C4C4×#12111yescharacteristic
616C4#232411yescharacteristic
617C4D4×C22⋊C411yescharacteristic
618C4#234111yescharacteristic
619C4#242611yescharacteristic
620C4#242811yescharacteristic
621C4#254011yescharacteristic
622C4#254111yescharacteristic
623C2—21no—
624C2—21no—
625C2—21no—
626C2—21no—
627C2#1399811yescharacteristic
628C2C2×#197911yescharacteristic
629C2#2125011yescharacteristic
630C2#2467212yes—
631C2#2477212yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
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
19. C2×#462C4semidirect
614. C4#1979semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
#212592
C2×#19792
C2×#19662
C2×#19502
C2×#18012
#204312
C4×#4192
#111282
#110292
#109422