Finite Group Databasebeta

← #356962

Subgroups — #356962

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C211yesmaximal, characteristic
2C2×#1967C212yesmaximal
3C4⋊C4×M16C212yesmaximal, complemented
4C2×C4×#101C211yesmaximal, characteristic, complemented
5C2×C4×#100C211yesmaximal, characteristic
6#10963C212yesmaximal, complemented
7#10962C212yesmaximal
8#10847C211yesmaximal, characteristic, complemented
9#10763C211yesmaximal, characteristic, complemented
10#10762C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#423C2212yes—
16C22×#101C2212yes—
17C22×#101C2212yes—
18C22×#100C2212yes—
19C22×#100C2212yes—
20C2×C4×M16C2212yes—
21C2×C4×M16C2212yes—
22C2×C4×M16C2212yes—
23#2367—22no—
24#2367—22no—
25#2367—22no—
26#2367—22no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388C412yes—
30C2×#388C412yes—
31C2×#385C412yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C412yes—
35#1967C4116yescomplemented
36#1771—22no—
37#1771—22no—
38#1770—22no—
39C8×C4⋊C4—22no—
40C4×#101—24no—
41C4×#100—24no—
42C2×#343C2211yescharacteristic
43C2×#343C2211yescharacteristic
44C2×#343C2211yescharacteristic
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2211yescharacteristic
48#1728C2212yes—
49#1728C2212yes—
50C23×Q8C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4C2×C412yes—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C22×M16C2312yes—
56C22×M16C2312yes—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124C2×C418yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—42no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—42no—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C412yes—
82C22×C42C2×C412yes—
83C22×C42C2×C412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#423—24no—
88#423C2×C418yescomplemented
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#101—24no—
92C2×#101—24no—
93C2×#101—42no—
94C2×#101—42no—
95C2×#101C2×C418yescomplemented
96C2×#101C2×C418yescomplemented
97C2×#100—24no—
98C2×#100—24no—
99C2×#100—42no—
100C2×#100—42no—
101C2×#100C2×C418yes—
102C2×#100C2×C418yes—
103C4×M16—24no—
104C4×M16—24no—
105C4×M16—24no—
106C4×M16—42no—
107C4×M16—42no—
108C4×M16C2×C418yescomplemented
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115#391—22no—
116#391—22no—
117#390—22no—
118#390—22no—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#385—24no—
124#385—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138C2×#95C2×C411yescharacteristic
139C2×#95C2×C411yescharacteristic
140#343—24no—
141#343—22no—
142#343—22no—
143#343—22no—
144#343—42no—
145#343D414yes—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150C25C22×C412yes—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8C22×C414yes—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22×C414yes—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—82no—
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C2411yescharacteristic
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16C22×C414yes—
206C2×M16C22×C414yes—
207C2×M16C22×C414yes—
208C2×M16C22×C414yes—
209C2×M16C2×D414yes—
210C2×M16#4114yes—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215#124—28no—
216#124—28no—
217#124—28no—
218#124—28no—
219C4×Q8—44no—
220C4×Q8—44no—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×D4—44no—
224C4×D4—44no—
225C4×D4—44no—
226C4×D4—44no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C22⋊C412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4112yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4C4218yes—
258C2×C22⋊C4C4218yes—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—42no—
271C2×C42—82no—
272C2×C42C22×C414yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281C4⋊C8—28no—
282C4⋊C8—28no—
283#101—216no—
284#101—48no—
285#100—216no—
286#100—48no—
287#97—28no—
288#97—44no—
289#97—44no—
290C4×C8—28no—
291C4×C8—44no—
292C4×C8—44no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95C22⋊C411yescharacteristic
298#95C22⋊C411yescharacteristic
299#95C22⋊C411yescharacteristic
300#95C22⋊C411yescharacteristic
301#95C22⋊C412yes—
302#95C22⋊C412yes—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24C2×C4214yes—
314C24C23×C411yescharacteristic
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—42no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4C2×C4214yes—
391C22×C4C2×C4212yes—
392C22×C4C2×C4212yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4#11712yes—
403C22×C4#11712yes—
404C22×C4C4×D414yes—
405C22×C4C23×C412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+411yescharacteristic
415M16—24no—
416M16—24no—
417M16—24no—
418M16—24no—
419M16—24no—
420M16—24no—
421M16—24no—
422M16—24no—
423M16—24no—
424M16—42no—
425M16—42no—
426M16—42no—
427M16—42no—
428M16—42no—
429M16—42no—
430M16C4×D418yes—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—216no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—84no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—44no—
468C42—44no—
469C42—82no—
470C42—82no—
471C42C4×D414yes—
472C42C4×D414yes—
473C42C4×D414yes—
474C42C4×D414yes—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23C22×C4211yescharacteristic
503C23C22×C22⋊C411yescharacteristic
504C23C2×#11711yescharacteristic
505C23C2×C4×D411yescharacteristic
506C23C4×#4111yescharacteristic
507C23#51811yescharacteristic
508C23#52011yescharacteristic
509Q8—44no—
510Q8—44no—
511Q8—44no—
512Q8—44no—
513D4—44no—
514D4—44no—
515D4—44no—
516D4—44no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—88no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—88no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4C4×C22⋊C414yes—
571C2×C4C4×C22⋊C412yes—
572C2×C4C4×C22⋊C412yes—
573C2×C4#41312yes—
574C2×C4#41312yes—
575C2×C4C2×C4×D412yes—
576C2×C4C2×C4×D412yes—
577C2×C4C2×C4×D412yes—
578C2×C4C4×#4112yes—
579C2×C4#51812yes—
580C2×C4#52012yes—
581C2×C4D4212yes—
582C2×C4#54612yes—
583C2×C4#54711yescharacteristic
584C2×C4#54812yes—
585C2×C4#55012yes—
586C2×C4#55011yescharacteristic
587C2×C4#55312yes—
588C2×C4#55511yescharacteristic
589C2×C4#55611yescharacteristic
590C2×C4#55912yes—
591C8—28no—
592C8—44no—
593C8—44no—
594C8—44no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×C4×C22⋊C411yescharacteristic
619C22C42×D411yescharacteristic
620C22#227511yescharacteristic
621C22D4×C22⋊C411yescharacteristic
622C22#234111yescharacteristic
623C22#235411yescharacteristic
624C22#235711yescharacteristic
625C22C2×#41311yescharacteristic
626C22C2×#41311yescharacteristic
627C22#288811yescharacteristic
628C22#288911yescharacteristic
629C4—22no—
630C4—22no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—42no—
636C4—44no—
637C4—41no—
638C4—42no—
639C4—44no—
640C4—82no—
641C4#236112yes—
642C4#236312yes—
643C4#236612yes—
644C4#236712yes—
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—84no—
650C2#1382011yescharacteristic
651C2C4×#41311yescharacteristic
652C2#2039711yescharacteristic
653C2#2240811yescharacteristic
654C2#2241011yescharacteristic
655C2#2253711yescharacteristic
656C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×M16C2semidirect
4. C2×C4×#101C2semidirect
6. #10963C2semidirect
8. #10847C2semidirect
9. #10763C2semidirect
11. C4×#343C2semidirect
35. #1967C4semidirect
88. #423C2×C4semidirect
95. C2×#101C2×C4semidirect
96. C2×#101C2×C4semidirect
108. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
C2×#19672
C4⋊C4×M162
C2×C4×#1012
C2×C4×#1002
#109632
#109622
#108472
#107632
#107622
C4×#3432