Finite Group Databasebeta

← #392922

Subgroups — #392922

Order 512.

1634 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×#405C211yesmaximal, characteristic
2C2×#1943C211yesmaximal, characteristic, complemented
3C2×#1941C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5C2×#1809C211yesmaximal, characteristic
6#14264C212yesmaximal
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×#405—24no—
20C2×#405—24no—
21C2×#405—24no—
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30#1943—24no—
31#1941—24no—
32#1809—24no—
33#1809C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1444—22no—
41#1442—22no—
42#1439—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16C2×C414yes—
52C22×M16D414yes—
53C22×M16Q814yes—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8Q814yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107C2×C414yes—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106C2×C414yes—
95#405—232no—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97D414yes—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8D414yes—
108#389—24no—
109#389—24no—
110#389—24no—
111#389—24no—
112#380—24no—
113#380—24no—
114#380—24no—
115#380C2×C418yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340C2×C418yes—
130C25C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×Q812yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—28no—
154C2×M16—28no—
155C2×M16—28no—
156C2×M16—28no—
157C2×M16—28no—
158C2×M16—28no—
159C2×M16C4⋊C4116yescomplemented
160C22×C8—28no—
161C22×C8—28no—
162C22×C8—28no—
163C22×C8—28no—
164C22×C8—28no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—28no—
178C22×C8—28no—
179C22×C8C22×C412yes—
180C22×C8C22×C412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×Q812yes—
188C22×C8C2×Q812yes—
189C22×C8C2×Q812yes—
190C22×C8C2×Q812yes—
191C22×C8#4112yes—
192C22×C8#4112yes—
193C22×C8#4112yes—
194C22×C8#4112yes—
195#117—28no—
196#117—28no—
197#117—28no—
198#117—28no—
199C2×C4⋊C4—24no—
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—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253#107—28no—
254#107—44no—
255#106—28no—
256#106—44no—
257#97—28no—
258#97—28no—
259C4×C8—28no—
260C4×C8—28no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—82no—
266#95—82no—
267#95—82no—
268#95—82no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24C4×D412yes—
278C24#12112yes—
279C24C4⋊Q812yes—
280C24C22×D411yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
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—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—22no—
311C22×C4—22no—
312C22×C4—22no—
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—24no—
322C22×C4—24no—
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×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×C4C2×C4⋊C414yes—
350C22×C4C4×D412yes—
351C22×C4C4×Q814yes—
352C22×C4#12114yes—
353C22×C4#12114yes—
354C22×C4#12112yes—
355C22×C4C22⋊Q814yes—
356C22×C4#12514yes—
357C22×C4C4⋊Q812yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372M16—232no—
373C2×C8—232no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8C2×C4⋊C414yes—
393C2×C8C2×C4⋊C414yes—
394C2×C8C4×D414yes—
395C2×C8C4×D414yes—
396C2×C8#12014yes—
397C2×C8C22⋊Q814yes—
398C2×C8C22⋊Q814yes—
399C2×C8C22⋊Q814yes—
400C2×C8C22⋊Q814yes—
401C2×C8#12314yes—
402C4⋊C4—216no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
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—42no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C22⋊C4—216no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23#38914yes—
455C23C2×C4×D411yescharacteristic
456C23C2×#12111yescharacteristic
457C23C2×#12111yescharacteristic
458C23C2×#12111yescharacteristic
459C23C2×C4⋊Q811yescharacteristic
460C23#53211yescharacteristic
461C23#53211yescharacteristic
462C23#57612yes—
463C23#57712yes—
464C23#57712yes—
465C23#57812yes—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
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—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
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—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
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#38914yes—
523C2×C4C22×C4⋊C411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×Q811yescharacteristic
527C2×C4C4×#4111yescharacteristic
528C2×C4C4×#4111yescharacteristic
529C2×C4C2×#12011yescharacteristic
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#52511yescharacteristic
538C2×C4#52511yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4C2×#12511yescharacteristic
543C2×C4#52911yescharacteristic
544C2×C4#52911yescharacteristic
545C2×C4#52911yescharacteristic
546C2×C4#52911yescharacteristic
547C2×C4#53311yescharacteristic
548C2×C4#53311yescharacteristic
549C2×C4#53311yescharacteristic
550C2×C4#53311yescharacteristic
551C8—24no—
552C8—24no—
553C8—24no—
554C8#38518yes—
555C22—24no—
556C22—28no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22C2×#38911yescharacteristic
567C22C4×#12111yescharacteristic
568C22C4×#12111yescharacteristic
569C22#284711yescharacteristic
570C22#285411yescharacteristic
571C22#285711yescharacteristic
572C22#285711yescharacteristic
573C22#294612yes—
574C22#294712yes—
575C22#305712yes—
576C22#306112yes—
577C22#306212yes—
578C22#306212yes—
579C22#306312yes—
580C22#306312yes—
581C22#316312yes—
582C22#316412yes—
583C22#317311yescharacteristic
584C22#317311yescharacteristic
585C22#317611yescharacteristic
586C22#317711yescharacteristic
587C22#317812yes—
588C22#317912yes—
589C4—24no—
590C4—44no—
591C4—44no—
592C4—84no—
593C4—84no—
594C4—162no—
595C4—162no—
596C4C2×#38511yescharacteristic
597C4#229611yescharacteristic
598C4C4×C22⋊Q811yescharacteristic
599C4C4×C22⋊Q811yescharacteristic
600C4#284811yescharacteristic
601C4#285111yescharacteristic
602C4#285111yescharacteristic
603C4#285511yescharacteristic
604C2—44no—
605C2#1469711yescharacteristic
606C2#2116012yes—
607C2#2153512yes—
608C2#2154111yescharacteristic
609C2#2154211yescharacteristic
610C2#3293812yes—
611C2#3314011yescharacteristic
612C2#3314111yescharacteristic
613C2#3321311yescharacteristic
614C2#3321611yescharacteristic
615C2#3321912yes—
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
2. C2×#1943C2semidirect
3. C2×#1941C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
9. #10863C2semidirect
10. #10862C2semidirect
33. #1809C4semidirect
159. C2×M16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#19432
C2×#19412
C2×#18092
#142642
#108812
#108802
#108632
#108622
#107212