Finite Group Databasebeta

← #358202

Subgroups — #358202

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic, complemented
3C2×C4×#103C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5#13785C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#11066C212yesmaximal
9#11023C212yesmaximal
10#10703C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#440—24no—
16C2×#440—24no—
17C2×#440C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2320—24no—
26#2320—24no—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#383—21no—
32C2×#382—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×#376C2211yescharacteristic
36C4×#103—24no—
37C4×#102—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1437—21no—
45#1437—21no—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—24no—
54C22×SD16—24no—
55C22×SD16C2×C414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×C4×Q8D414yes—
64C2×C4×Q8D414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#440—232no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107C2×C414yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103C2×C414yes—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—24no—
100C2×#102C2×C414yes—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97C2×C414yes—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107#391—24no—
108#391—24no—
109#390—24no—
110#390—24no—
111#389—24no—
112#389—24no—
113#387—216no—
114#383—28no—
115#382—28no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D414yes—
120#376—22no—
121#376—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D412yes—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137#336—24no—
138C25—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C22×C412yes—
147C22×Q8C2×D412yes—
148C22×Q8C2×D412yes—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4C22×C412yes—
158C22×D4#4114yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—82no—
168C23×C4—82no—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C2×SD16—216no—
175C2×SD16—216no—
176C2×SD16—28no—
177C2×SD16—28no—
178C2×SD16—28no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C22×C412yes—
193C22×C8C22×C412yes—
194C22×C8#4114yes—
195C22×C8#4114yes—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×Q8—28no—
199C4×Q8C22⋊C4116yescomplemented
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—44no—
203C4×D4—44no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
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—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—82no—
255C2×C42—82no—
256C2×C42C22×C412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—28no—
263#103—216no—
264#103—28no—
265#102—216no—
266#102—28no—
267#97—28no—
268C4×C8—44no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—42no—
277#95—82no—
278#95—82no—
279C24—22no—
280C24—24no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8C2×C22⋊C414yes—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4#11714yes—
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—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
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×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4C4×D412yes—
378C22×C4C4×D412yes—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4111yescharacteristic
393SD16—232no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8#11714yes—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—216no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—88no—
428C4⋊C4C2×C22⋊C414yes—
429C22⋊C4—48no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C42—216no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—84no—
438C42—84no—
439C42C2×C22⋊C414yes—
440C42C4×D414yes—
441C42#12114yes—
442C42#12314yes—
443C23—24no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—81no—
459C23—81no—
460C23—82no—
461C23—81no—
462C23—81no—
463C23—82no—
464C23C2×C4×D411yescharacteristic
465C23C2×#12011yescharacteristic
466C23C2×#12111yescharacteristic
467C23#52512yes—
468C23#53212yes—
469C23#57612yes—
470C23#57712yes—
471C23#57712yes—
472C23#57812yes—
473Q8—24no—
474Q8—24no—
475Q8—44no—
476Q8—44no—
477Q8—42no—
478Q8—42no—
479D4—24no—
480D4—24no—
481D4—44no—
482D4—44no—
483D4—42no—
484D4—42no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—816no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#38614yes—
538C2×C4#38614yes—
539C2×C4C22×C22⋊C411yescharacteristic
540C2×C4C2×#11711yescharacteristic
541C2×C4C2×#11711yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C4×#4111yescharacteristic
544C2×C4C4×#4111yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4#52512yes—
548C2×C4#52512yes—
549C2×C4#52512yes—
550C2×C4#52512yes—
551C2×C4#52912yes—
552C2×C4#52912yes—
553C2×C4#52912yes—
554C2×C4#52912yes—
555C2×C4#53212yes—
556C2×C4#53212yes—
557C8—24no—
558C8—44no—
559C8—42no—
560C22—44no—
561C22—48no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22C2×#38611yescharacteristic
570C22#229011yescharacteristic
571C22#229311yescharacteristic
572C22#238812yes—
573C22#239212yes—
574C22#294612yes—
575C22#294712yes—
576C22#300811yescharacteristic
577C22#301011yescharacteristic
578C22#301312yes—
579C22#304311yescharacteristic
580C22#304411yescharacteristic
581C22#304512yes—
582C22#311812yes—
583C22#312012yes—
584C22#312312yes—
585C22#312412yes—
586C22#317412yes—
587C22#317512yes—
588C22#317812yes—
589C22#317912yes—
590C4—24no—
591C4—24no—
592C4—44no—
593C4—44no—
594C4—82no—
595C4—84no—
596C4—82no—
597C4—84no—
598C4—82no—
599C4#228011yescharacteristic
600C4#229011yescharacteristic
601C4#229311yescharacteristic
602C4#229711yescharacteristic
603C4#239212yes—
604C4#239512yes—
605C2—84no—
606C2#1383911yescharacteristic
607C2#2093011yescharacteristic
608C2#2093211yescharacteristic
609C2#2112512yes—
610C2#2129112yes—
611C2#2357912yes—
612C2#2358412yes—
613C2#2369012yes—
614C2#2369512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2320C2semidirect
3. C2×C4×#103C2semidirect
6. #11219C2semidirect
7. #11190C2semidirect
10. #10703C2semidirect
11. #10694C2semidirect
17. C2×#440C4semidirect
199. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#23202
C2×C4×#1032
C2×C4×#1022
#137852
#112192
#111902
#110662
#110232
#107032
#106942