Finite Group Databasebeta

← #391795

Subgroups — #391795

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic, complemented
3#14343C211yesmaximal, characteristic, complemented
4#11373C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11077C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#11009C211yesmaximal, characteristic, complemented
9#11009C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal, complemented
11#10969C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×C22⋊C8C2212yes—
22#2620—24no—
23#2620C418yescomplemented
24C2×#396—22no—
25C2×#394—22no—
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#380C2212yes—
30#1798—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1440—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
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⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D412yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102—42no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97#396—216no—
98#394—216no—
99#391—22no—
100#391C2×C414yes—
101#390—22no—
102#390C2×C414yes—
103#384—22no—
104#384C2×C414yes—
105#380—42no—
106#380—42no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
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—22no—
127C22×Q8—22no—
128C22×Q8C22×C412yes—
129C22×D4—24no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—82no—
140C22×D4—82no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2411yescharacteristic
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—
172C22×C8—22no—
173#124—44no—
174#124—44no—
175#124—48no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178#121—48no—
179#121—48no—
180#117—44no—
181#117—44no—
182#117—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
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—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22⋊C418yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C22⋊C418yes—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C4⋊C8—48no—
239#103—48no—
240#103—48no—
241#102—48no—
242#102—48no—
243C22⋊C8—48no—
244#95—28no—
245#95—44no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250C24—22no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24C2×C22⋊C412yes—
268C24C22×D411yescharacteristic
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—48no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—88no—
303C2×D4—84no—
304C2×D4—84no—
305C2×D4—88no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
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—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
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—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
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×C4C2×C22⋊C414yes—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C42−1+412yes—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—48no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23C22×C22⋊C411yescharacteristic
462C23#53412yes—
463C23#53511yescharacteristic
464C23#53511yescharacteristic
465C23#53712yes—
466C23#57612yes—
467C23#57712yes—
468C23#57712yes—
469C23#57812yes—
470Q8—44no—
471Q8—44no—
472Q8—82no—
473Q8—82no—
474D4—44no—
475D4—44no—
476D4—816no—
477D4—82no—
478D4—82no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
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—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
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—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—82no—
531C2×C4#51811yescharacteristic
532C2×C4#51811yescharacteristic
533C2×C4#51812yes—
534C2×C4#52012yes—
535C2×C4#53412yes—
536C2×C4#53512yes—
537C2×C4#53511yescharacteristic
538C2×C4#53611yescharacteristic
539C2×C4#56012yes—
540C2×C4#56012yes—
541C2×C4#56012yes—
542C2×C4#56012yes—
543C2×C4#56212yes—
544C2×C4#56212yes—
545C2×C4#56212yes—
546C2×C4#56212yes—
547C8—82no—
548C8—82no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—84no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—162no—
564C22—162no—
565C22—162no—
566C22—162no—
567C22#230811yescharacteristic
568C22#230911yescharacteristic
569C22#231111yescharacteristic
570C22#281312yes—
571C22#281312yes—
572C22#289712yes—
573C22#289812yes—
574C22#319812yes—
575C22#320112yes—
576C22#320312yes—
577C22#320412yes—
578C22#322011yescharacteristic
579C22#322111yescharacteristic
580C22#322211yescharacteristic
581C22#322311yescharacteristic
582C22#322412yes—
583C22#322512yes—
584C22#325712yes—
585C22#325812yes—
586C22#326212yes—
587C22#326312yes—
588C4—44no—
589C4—44no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—162no—
595C4—162no—
596C4#237712yes—
597C4#237712yes—
598C4#281512yes—
599C4#281612yes—
600C2—82no—
601C2—162no—
602C2#1465911yescharacteristic
603C2#2177212yes—
604C2#2181311yescharacteristic
605C2#2181411yescharacteristic
606C2#2189612yes—
607C2#3273412yes—
608C2#3273512yes—
609C2#3273612yes—
610C2#3273712yes—
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×#1798C2semidirect
3. #14343C2semidirect
4. #11373C2semidirect
5. #11371C2semidirect
6. #11077C2semidirect
7. #11076C2semidirect
8. #11009C2semidirect
9. #11009C2semidirect
10. #10970C2semidirect
23. #2620C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
C2×#17982
#143432
#113732
#113712
#110772
#110762
#110092
#109702
#109692