Finite Group Databasebeta

← #374470

Subgroups — #374470

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1991C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#11200C212yesmaximal
7#11199C212yesmaximal
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2629—24no—
19#2629C418yescomplemented
20C2×#398—22no—
21C2×#396—22no—
22#2314—24no—
23#2314—24no—
24C2×#389C2212yes—
25C2×#388—22no—
26C2×#387—22no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2212yes—
33#1992—24no—
34#1991—24no—
35#1750—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125C2×C414yes—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117C2×C414yes—
62C2×#117D414yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—22no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90#398—216no—
91#396—216no—
92#389—24no—
93#389C2×C418yes—
94#388—216no—
95#387—216no—
96#384—22no—
97#384—22no—
98#384C2×C414yes—
99#384D414yes—
100#382—22no—
101#382D414yes—
102#380—24no—
103#380—24no—
104#380—24no—
105#380C2×C418yes—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#336—22no—
122#336—22no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8#4112yes—
171C22×C8#4112yes—
172C22×C8#4112yes—
173C22×C8#4112yes—
174#125—28no—
175#125—28no—
176#125—44no—
177#125—44no—
178#117—28no—
179#117—28no—
180#117—28no—
181#117—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
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—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22⋊C418yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C22×C414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×D414yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C22⋊C418yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—28no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C22⋊C8—28no—
246C22⋊C8—28no—
247#97—44no—
248C4×C8—44no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24C2×C22⋊C412yes—
265C24C22×D411yescharacteristic
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
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—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
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×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C414yes—
340C22×C4C2×C22⋊C414yes—
341C22×C4#12014yes—
342C22×C4#12314yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C42−1+411yescharacteristic
354C22×C42−1+411yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C4⋊C4—216no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
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—416no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C22⋊C4—216no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—88no—
414C42C4×D414yes—
415C42C4×D414yes—
416C42#12114yes—
417C42#12114yes—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×#12311yescharacteristic
444C23#53211yescharacteristic
445C23#53211yescharacteristic
446C23#53611yescharacteristic
447C23#53611yescharacteristic
448C23#57612yes—
449C23#57712yes—
450C23#57712yes—
451C23#57812yes—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#38618yes—
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C4×#4111yescharacteristic
512C2×C4C4×#4111yescharacteristic
513C2×C4#51911yescharacteristic
514C2×C4#51911yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4#52511yescharacteristic
518C2×C4#52511yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53611yescharacteristic
521C2×C4#54711yescharacteristic
522C2×C4#54711yescharacteristic
523C2×C4#54711yescharacteristic
524C2×C4#54711yescharacteristic
525C2×C4#54811yescharacteristic
526C2×C4#54811yescharacteristic
527C2×C4#54811yescharacteristic
528C2×C4#54811yescharacteristic
529C2×C4#55211yescharacteristic
530C2×C4#55211yescharacteristic
531C2×C4#55211yescharacteristic
532C2×C4#55211yescharacteristic
533C2×C4#55611yescharacteristic
534C2×C4#55611yescharacteristic
535C2×C4#55611yescharacteristic
536C2×C4#55611yescharacteristic
537C8—42no—
538C8—42no—
539C8—42no—
540C8—42no—
541C22—42no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—81no—
547C22—81no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—82no—
552C22C2×#38611yescharacteristic
553C22#229711yescharacteristic
554C22#231011yescharacteristic
555C22#243211yescharacteristic
556C22#243211yescharacteristic
557C22#249911yescharacteristic
558C22#249911yescharacteristic
559C22#289712yes—
560C22#289812yes—
561C22#301212yes—
562C22#301312yes—
563C22#309812yes—
564C22#309912yes—
565C22#317411yescharacteristic
566C22#317411yescharacteristic
567C22#317611yescharacteristic
568C22#317711yescharacteristic
569C22#317812yes—
570C22#317912yes—
571C22#324112yes—
572C22#324212yes—
573C22#324312yes—
574C22#324412yes—
575C4—28no—
576C4—42no—
577C4—44no—
578C4—84no—
579C4—84no—
580C4—84no—
581C4—84no—
582C4#236211yescharacteristic
583C4#236211yescharacteristic
584C4#236211yescharacteristic
585C4#236211yescharacteristic
586C4#241511yescharacteristic
587C4#241511yescharacteristic
588C4#241511yescharacteristic
589C4#241511yescharacteristic
590C2—82no—
591C2#1413811yescharacteristic
592C2#2091512yes—
593C2#2141311yescharacteristic
594C2#2141411yescharacteristic
595C2#2184312yes—
596C2#2568211yescharacteristic
597C2#2568311yescharacteristic
598C2#2568412yes—
599C2#2788411yescharacteristic
600C2#2788511yescharacteristic
601C2#2788812yes—
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
3. C2×#1992C2semidirect
4. C2×#1991C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect
19. #2629C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23142
C2×#19922
C2×#19912
C2×#17502
#112002
#111992
#108852
#108842
#108432
#108412