Finite Group Databasebeta

← #361908

Subgroups — #361908

Order 512.

1602 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×C4×SD16C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic, complemented
3C2×#1805C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×#1770C211yesmaximal, characteristic, complemented
6#13942C211yesmaximal, characteristic, complemented
7#13941C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11023C212yesmaximal
10#10928C212yesmaximal, complemented
11#10927C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×SD16—24no—
16C2×C4×SD16—24no—
17C2×C4×SD16C418yescomplemented
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#97C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1835—24no—
32#1805—24no—
33#1804—24no—
34#1770—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C22⋊#95—22no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C22×SD16—22no—
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16C2×C414yes—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
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⋊C4D412yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—24no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C4×SD16—232no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106C2×C414yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8D414yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103C2×C414yes—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102C2×C414yes—
101C2×#97—24no—
102C2×#97—42no—
103C2×#97—42no—
104C2×C4×C8C2×C414yes—
105C2×C4×C8D414yes—
106C2×C4×C8D414yes—
107#391—24no—
108#391—42no—
109#390—24no—
110#390—42no—
111#382—22no—
112#382—22no—
113#382—22no—
114#382—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—24no—
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#336—24no—
130C25—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8C22×C412yes—
143C22×Q8#4112yes—
144C22×Q8#4112yes—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22×C412yes—
157C22×D4#4112yes—
158C22×D4#4112yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—84no—
168C23×C4C2×D414yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C2×SD16—28no—
173C2×SD16—28no—
174C2×SD16—28no—
175C2×SD16—28no—
176C2×SD16—28no—
177C2×SD16—28no—
178C2×SD16—28no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C22×C412yes—
194C22×C8C22×C412yes—
195C22×C8C2×D414yes—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—44no—
207C4×D4—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—84no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258#106—28no—
259C4⋊C8—28no—
260#103—28no—
261#103—48no—
262#102—28no—
263#102—48no—
264#97—48no—
265C4×C8C22⋊C4116yescomplemented
266#95—24no—
267#95—48no—
268#95—48no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—84no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—44no—
282C24—44no—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8#11714yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4#11714yes—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
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—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4#12112yes—
381C22×C4#12112yes—
382C22×C4#12712yes—
383C22×C4#12712yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398SD16—232no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8C2×C22⋊C414yes—
419C2×C8C2×C22⋊C414yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—48no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—48no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—816no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—48no—
445C42C2×C22⋊C414yes—
446C42C4×D414yes—
447C42#12114yes—
448C42#12414yes—
449C23—24no—
450C23—42no—
451C23—48no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—48no—
460C23—84no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—84no—
466C23C2×C4×D411yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×#12711yescharacteristic
469C23#54611yescharacteristic
470C23#54611yescharacteristic
471C23#54811yescharacteristic
472C23#54811yescharacteristic
473C23#57514yes—
474C23#57714yes—
475Q8—24no—
476Q8—24no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483D4—24no—
484D4—24no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—816no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4#39014yes—
531C2×C4#39014yes—
532C2×C4C22×C22⋊C411yescharacteristic
533C2×C4C2×#11711yescharacteristic
534C2×C4C2×#11711yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4#52011yescharacteristic
537C2×C4#52011yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53711yescharacteristic
547C2×C4#53711yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4#54811yescharacteristic
551C2×C4#54811yescharacteristic
552C2×C4#55211yescharacteristic
553C2×C4#55211yescharacteristic
554C2×C4#55211yescharacteristic
555C2×C4#55211yescharacteristic
556C2×C4#55311yescharacteristic
557C2×C4#55311yescharacteristic
558C2×C4#55311yescharacteristic
559C2×C4#55311yescharacteristic
560C8—24no—
561C8—24no—
562C8—44no—
563C22—44no—
564C22—48no—
565C22—84no—
566C22—82no—
567C22—84no—
568C22—82no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22C2×#39011yescharacteristic
573C22#235211yescharacteristic
574C22#235211yescharacteristic
575C22#253511yescharacteristic
576C22#253611yescharacteristic
577C22#254011yescharacteristic
578C22#254011yescharacteristic
579C22#294312yes—
580C22#294912yes—
581C22#305412yes—
582C22#305712yes—
583C22#315012yes—
584C22#315412yes—
585C22#332314yes—
586C22#332614yes—
587C22#337114yes—
588C22#337614yes—
589C4—24no—
590C4—24no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—162no—
596C4—162no—
597C4#228011yescharacteristic
598C4#231111yescharacteristic
599C4#235211yescharacteristic
600C4#235211yescharacteristic
601C4#253511yescharacteristic
602C4#253611yescharacteristic
603C4#256211yescharacteristic
604C4#256211yescharacteristic
605C2—84no—
606C2#1390511yescharacteristic
607C2#2122811yescharacteristic
608C2#2123111yescharacteristic
609C2#2276012yes—
610C2#2276512yes—
611C2#2814712yes—
612C2#2819212yes—
613C2#2841012yes—
614C2#2841212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1835C2semidirect
3. C2×#1805C2semidirect
5. C2×#1770C2semidirect
6. #13942C2semidirect
8. #11378C2semidirect
10. #10928C2semidirect
17. C2×C4×SD16C4semidirect
265. C4×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#18352
C2×#18052
C2×#18042
C2×#17702
#139422
#139412
#113782
#110232
#109282
#109272