Finite Group Databasebeta

← #391776

Subgroups — #391776

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic, complemented
3#14340C211yesmaximal, characteristic, complemented
4#11357C212yesmaximal, complemented
5#11052C211yesmaximal, characteristic, complemented
6#11051C211yesmaximal, characteristic, complemented
7#11050C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic, complemented
9#11009C212yesmaximal, complemented
10#10970C212yesmaximal, complemented
11#10969C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2212yes—
21#2620—24no—
22#2620C418yescomplemented
23C2×#396—22no—
24C2×#394—22no—
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31#1818—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1441—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#124—22no—
49C2×#124—42no—
50C2×#124—42no—
51C2×#124C2×C414yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8C2×C414yes—
56C2×#121—22no—
57C2×#121—42no—
58C2×#121—42no—
59C2×#121C2×C414yes—
60C2×#117—42no—
61C2×#117—42no—
62C2×#117C2×C414yes—
63C2×#117D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95#396—216no—
96#394—216no—
97#391—22no—
98#391C2×C414yes—
99#390—22no—
100#390C2×C414yes—
101#384—22no—
102#384—42no—
103#384C2×C414yes—
104#382—42no—
105#380—42no—
106#380—42no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336—24no—
117#336—22no—
118#336—22no—
119C25—21no—
120C25C2×D411yescharacteristic
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8C22×C412yes—
131C22×D4—24no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—82no—
144C22×D4—82no—
145C22×D4C22×C412yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179#124—48no—
180#124—48no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183#121—48no—
184#121—48no—
185#117—48no—
186#117—48no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22⋊C418yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4C22⋊C418yes—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C22×C412yes—
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#103—48no—
246#103—48no—
247#102—48no—
248#102—48no—
249C22⋊C8—48no—
250#95—28no—
251#95—44no—
252#95—44no—
253#95—82no—
254#95—82no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24C2×C22⋊C412yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—48no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—88no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—88no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
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—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4C2×C22⋊C414yes—
381C22×C4C2×C22⋊C414yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C42−1+411yescharacteristic
392C22×C42−1+411yescharacteristic
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—416no—
426C22⋊C4—48no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—81no—
462C23—81no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—81no—
478C23—81no—
479C23C22×C22⋊C411yescharacteristic
480C23#53411yescharacteristic
481C23#53411yescharacteristic
482C23#53511yescharacteristic
483C23#53511yescharacteristic
484C23#53511yescharacteristic
485C23#53611yescharacteristic
486C23#57612yes—
487C23#57712yes—
488C23#57712yes—
489C23#57812yes—
490Q8—44no—
491Q8—44no—
492Q8—82no—
493Q8—82no—
494D4—44no—
495D4—44no—
496D4—816no—
497D4—82no—
498D4—82no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#51812yes—
556C2×C4#51812yes—
557C2×C4#52011yescharacteristic
558C2×C4#52011yescharacteristic
559C2×C4#53412yes—
560C2×C4#53512yes—
561C2×C4#53711yescharacteristic
562C2×C4#53711yescharacteristic
563C2×C4#56012yes—
564C2×C4#56012yes—
565C2×C4#56012yes—
566C2×C4#56012yes—
567C2×C4#56212yes—
568C2×C4#56212yes—
569C2×C4#56212yes—
570C2×C4#56212yes—
571C8—82no—
572C8—82no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—84no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—162no—
590C22—162no—
591C22—161no—
592C22—161no—
593C22—161no—
594C22—161no—
595C22#230811yescharacteristic
596C22#230911yescharacteristic
597C22#231111yescharacteristic
598C22#281511yescharacteristic
599C22#281511yescharacteristic
600C22#281611yescharacteristic
601C22#281611yescharacteristic
602C22#289712yes—
603C22#289812yes—
604C22#319812yes—
605C22#320112yes—
606C22#320312yes—
607C22#320412yes—
608C22#322011yescharacteristic
609C22#322111yescharacteristic
610C22#322211yescharacteristic
611C22#322311yescharacteristic
612C22#322412yes—
613C22#322412yes—
614C22#322512yes—
615C22#322512yes—
616C22#324312yes—
617C22#324412yes—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—162no—
625C4—162no—
626C4#237712yes—
627C4#237712yes—
628C4#281312yes—
629C4#281312yes—
630C2—82no—
631C2—162no—
632C2#1465911yescharacteristic
633C2#2177212yes—
634C2#2181311yescharacteristic
635C2#2181411yescharacteristic
636C2#2187112yes—
637C2#3276511yescharacteristic
638C2#3276711yescharacteristic
639C2#3276912yes—
640C2#3278112yes—
641C2#3278311yescharacteristic
642C2#3278511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#1818C2semidirect
3. #14340C2semidirect
4. #11357C2semidirect
5. #11052C2semidirect
6. #11051C2semidirect
7. #11050C2semidirect
8. #11049C2semidirect
9. #11009C2semidirect
10. #10970C2semidirect
22. #2620C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
C2×#18182
#143402
#113572
#110522
#110512
#110502
#110492
#110092
#109702
#109692