Finite Group Databasebeta

← #400376

Subgroups — #400376

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#2086C211yesmaximal, characteristic, complemented
3C2×#2054C211yesmaximal, characteristic
4C2×#1916C211yesmaximal, characteristic, complemented
5#14018C212yesmaximal, complemented
6#11379C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#11022C211yesmaximal, characteristic
10#10820C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2212yes—
14C2×C4×D8—24no—
15C2×C4×D8—24no—
16C2×C4×D8—24no—
17C22×#107C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2086—24no—
29#2054—24no—
30#1916—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1439—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—22no—
45C23×D4C2312yes—
46C24×C4—22no—
47C22×D8—24no—
48C22×D8—24no—
49C22×D8—22no—
50C22×D8—22no—
51C22×D8D414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C4×D8—232no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107D414yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—42no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8D414yes—
95#390—24no—
96#390—24no—
97#390—24no—
98#390—42no—
99#389—24no—
100#389—42no—
101#385—48no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120C25—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C2×D414yes—
133C22×D4#4114yes—
134C22×D4#4114yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C2×D8—28no—
153C2×D8—28no—
154C2×D8—216no—
155C2×D8—216no—
156C2×D8—28no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C4×D4—216no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
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—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4D814yescomplemented
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238#107—28no—
239C4⋊C8—44no—
240#102—216no—
241#102—48no—
242#102—48no—
243C4×C8—28no—
244#95—22no—
245#95—22no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—82no—
250#95—82no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24C22×D411yescharacteristic
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4#12318yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—416no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
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—88no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4C2×D812yes—
354C22×C4C2×D812yes—
355C22×C4C2×D812yes—
356C22×C4#13512yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+412yes—
367D8—232no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8#12014yes—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—88no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—41no—
413C42—41no—
414C42—42no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C23—28no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23C2×#12111yescharacteristic
455C23#52512yes—
456C23D4211yescharacteristic
457C23D4211yescharacteristic
458C23#54812yes—
459C23C22×D811yescharacteristic
460C23C2×#13511yescharacteristic
461C23#57512yes—
462C23#57611yescharacteristic
463C23#57611yescharacteristic
464C23#57712yes—
465D4—28no—
466D4—28no—
467D4—44no—
468D4—44no—
469D4—44no—
470D4—44no—
471D4—44no—
472D4—44no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—832no—
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—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#46614yes—
529C2×C4C2×#12011yescharacteristic
530C2×C4C2×#12312yes—
531C2×C4#52511yescharacteristic
532C2×C4#52511yescharacteristic
533C2×C4#52512yes—
534C2×C4#52912yes—
535C2×C4#53211yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53912yes—
539C2×C4#53912yes—
540C2×C4#54611yescharacteristic
541C2×C4#54611yescharacteristic
542C2×C4#54712yes—
543C2×C4#54812yes—
544C2×C4#55112yes—
545C2×C4#55212yes—
546C2×C4#55312yes—
547C8—24no—
548C8—42no—
549C8—81no—
550C8—81no—
551C22—48no—
552C22—48no—
553C22—48no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22#239211yescharacteristic
567C22#240811yescharacteristic
568C22#244312yes—
569C22#246312yes—
570C22#253611yescharacteristic
571C22C2×#46611yescharacteristic
572C22#305411yescharacteristic
573C22#305811yescharacteristic
574C22#306011yescharacteristic
575C22#310012yes—
576C22#310212yes—
577C22#311812yes—
578C22#311912yes—
579C22#326712yes—
580C22#327112yes—
581C22D4×D811yescharacteristic
582C22D4×D811yescharacteristic
583C22#328011yescharacteristic
584C22#328011yescharacteristic
585C22#335812yes—
586C22#336012yes—
587C22#337012yes—
588C22#337012yes—
589C4—24no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—84no—
594C4—81no—
595C4—84no—
596C4—81no—
597C4—162no—
598C4#241912yes—
599C4#243211yescharacteristic
600C4#246012yes—
601C4#260311yescharacteristic
602C4#263311yescharacteristic
603C4#284911yescharacteristic
604C2—88no—
605C2#1480111yescharacteristic
606C2#2369611yescharacteristic
607C2#2370211yescharacteristic
608C2#2426711yescharacteristic
609C2#2427111yescharacteristic
610C2#2615812yes—
611C2#2616012yes—
612C2#2688912yes—
613C2#2689112yes—
614C2#2818511yescharacteristic
615C2#2819111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2086C2semidirect
4. C2×#1916C2semidirect
5. #14018C2semidirect
6. #11379C2semidirect
7. #11189C2semidirect
8. #11066C2semidirect
10. #10820C2semidirect
11. #10788C2semidirect
204. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#20862
C2×#20542
C2×#19162
#140182
#113792
#111892
#110662
#110222
#108202
#107882