Finite Group Databasebeta

← #380986

Subgroups — #380986

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11372C212yesmaximal, complemented
6#11228C212yesmaximal, complemented
7#11227C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17#2628—24no—
18#2628C418yescomplemented
19C2×#400—21no—
20C2×#399—21no—
21C2×#395—21no—
22C2×#393—21no—
23C2×#389C2212yes—
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26#1798—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—21no—
41C22⋊#95—21no—
42C23×Q8—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—41no—
52C2×C4⋊Q8C2×C414yescomplemented
53C2×#117—22no—
54C2×#117—41no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117—41no—
58C2×#117—41no—
59C2×#117—41no—
60C2×#117—41no—
61C2×#117—41no—
62C2×#117C2×C414yes—
63C2×#117D414yes—
64C2×#117Q814yescomplemented
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—41no—
70C22×C4⋊C4—41no—
71C22×C4⋊C4—41no—
72C22×C4⋊C4—41no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4Q814yescomplemented
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8—41no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96#400—28no—
97#399—28no—
98#395—28no—
99#393—28no—
100#389—24no—
101#389C2×C418yescomplemented
102#382—22no—
103#382C2×C414yescomplemented
104#380—24no—
105#380C2×C418yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#340—42no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×Q8—82no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C22×C412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×Q812yes—
154C23×C4C2×Q812yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#117—44no—
177#117—44no—
178#117—44no—
179#117—44no—
180#117—44no—
181#117—44no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C22⋊C414yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×Q814yes—
226C2×C4⋊C4C2×Q814yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4C22⋊C414yes—
238C2×C22⋊C4C22⋊C414yes—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C22×C412yes—
251C2×C42C22×C412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C4⋊C8—44no—
255C4⋊C8—44no—
256C22⋊C8—48no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—82no—
262#95—82no—
263#95—82no—
264#95—82no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24C2×C22⋊C412yes—
274C24C22⋊Q812yes—
275C24C22⋊Q812yes—
276C24C22×D411yescharacteristic
277C2×Q8—48no—
278C2×Q8—88no—
279C2×Q8—88no—
280C2×Q8—84no—
281C2×Q8—84no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
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—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—48no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
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—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C4×Q814yes—
365C22×C4C4×Q814yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4#12414yes—
369C22×C4C4⋊Q814yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C22×Q811yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—88no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—88no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—48no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23#39114yes—
455C23C22×C22⋊C411yescharacteristic
456C23C2×C22⋊Q811yescharacteristic
457C23C2×C22⋊Q811yescharacteristic
458C23#53411yescharacteristic
459C23#53411yescharacteristic
460C23#53511yescharacteristic
461C23#53611yescharacteristic
462C23#57612yes—
463C23#57712yes—
464C23#57712yes—
465C23#57812yes—
466Q8—816no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—816no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#39114yes—
531C2×C4C2×C4×Q811yescharacteristic
532C2×C4C2×C4×Q811yescharacteristic
533C2×C4#51811yescharacteristic
534C2×C4#51811yescharacteristic
535C2×C4#52011yescharacteristic
536C2×C4#52011yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×C4⋊Q811yescharacteristic
539C2×C4#53411yescharacteristic
540C2×C4#53411yescharacteristic
541C2×C4#53711yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#53811yescharacteristic
544C2×C4#53811yescharacteristic
545C2×C4#53811yescharacteristic
546C2×C4#53811yescharacteristic
547C2×C4#53911yescharacteristic
548C2×C4#53911yescharacteristic
549C2×C4#53911yescharacteristic
550C2×C4#53911yescharacteristic
551C2×C4#54311yescharacteristic
552C2×C4#54311yescharacteristic
553C2×C4#54311yescharacteristic
554C2×C4#54311yescharacteristic
555C2×C4#54411yescharacteristic
556C2×C4#54411yescharacteristic
557C2×C4#54411yescharacteristic
558C2×C4#54411yescharacteristic
559C8—82no—
560C8—82no—
561C22—28no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22C2×#39111yescharacteristic
573C22#230811yescharacteristic
574C22#231111yescharacteristic
575C22#262611yescharacteristic
576C22#262611yescharacteristic
577C22#262711yescharacteristic
578C22#262711yescharacteristic
579C22#289712yes—
580C22#289812yes—
581C22#308112yes—
582C22#308212yes—
583C22#308312yes—
584C22#308412yes—
585C22#319812yes—
586C22#319911yescharacteristic
587C22#320011yescharacteristic
588C22#320011yescharacteristic
589C22#320112yes—
590C22#320211yescharacteristic
591C22#322412yes—
592C22#322512yes—
593C22#324312yes—
594C22#324412yes—
595C4—24no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—84no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4#232911yescharacteristic
605C4#232911yescharacteristic
606C4#232911yescharacteristic
607C4#232911yescharacteristic
608C4#263911yescharacteristic
609C4#263911yescharacteristic
610C4#265911yescharacteristic
611C4#265911yescharacteristic
612C2—44no—
613C2#1428111yescharacteristic
614C2#2131512yes—
615C2#2177011yescharacteristic
616C2#2177111yescharacteristic
617C2#2187112yes—
618C2#2995911yescharacteristic
619C2#2996011yescharacteristic
620C2#2996112yes—
621C2#2997512yes—
622C2#2997611yescharacteristic
623C2#2997711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11384C2semidirect
5. #11372C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
18. #2628C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
64. C2×#117Q8semidirect
75. C22×C4⋊C4Q8semidirect
101. #389C2×C4semidirect
103. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17982
#137952
#113842
#113722
#112282
#112272
#107312
#107302