Finite Group Databasebeta

← #220151

Subgroups — #220151

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3168C211yesmaximal, characteristic
2#22531C211yesmaximal, characteristic, complemented
3#22474C211yesmaximal, characteristic, complemented
4C2×#1592C211yesmaximal, characteristic, complemented
5C2×SD16⋊C8C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9660C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic, complemented
9#8950C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8946C211yesmaximal, characteristic, complemented
12#8596C211yesmaximal, characteristic, complemented
13#8315C211yesmaximal, characteristic, complemented
14#8315C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3168—24no—
19#3168—24no—
20C2×#499C411yescharacteristic
21C2×#495C411yescharacteristic
22C2×#488C411yescharacteristic
23C2×#488C411yescharacteristic
24C2×#467C411yescharacteristic
25C2×#466C411yescharacteristic
26C2×#465C411yescharacteristic
27C2×#465C411yescharacteristic
28C2×#446C2211yescharacteristic
29C2×C8×Q8C2211yescharacteristic
30C2×C4×Q16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×C4×SD16C2211yescharacteristic
33C2×C4×D8C2211yescharacteristic
34C2×#436C2211yescharacteristic
35C2×C8×D4C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#1984C2211yescharacteristic
38#1984C2211yescharacteristic
39#1955—21no—
40#1954—21no—
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1870—21no—
44#1869—21no—
45#1818—21no—
46#1798—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1750—21no—
50C4×C22⋊C8C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52#1729—21no—
53#1592—24no—
54SD16⋊C8—24no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×Q16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×D8C2×C411yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C23×C8C2×C411yescharacteristic
78C23×C8C2×C411yescharacteristic
79#529—22no—
80#529—22no—
81#529—22no—
82#529—22no—
83#529—41no—
84#529—41no—
85#529—41no—
86#529—41no—
87C2×#126—21no—
88C2×#126—21no—
89C2×#125C2×C411yescharacteristic
90C2×#125C2×C411yescharacteristic
91C2×#124C2×C411yescharacteristic
92C2×#124C2×C411yescharacteristic
93C2×#123—21no—
94C2×#123—21no—
95C2×C22⋊Q8C2×C411yescharacteristic
96C2×C22⋊Q8C2×C411yescharacteristic
97C2×#121C2×C411yescharacteristic
98C2×#121C2×C411yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×#117—21no—
106C2×#117—21no—
107C22×C42C2311yescharacteristic
108#499—24no—
109#495—24no—
110#488—24no—
111#488—24no—
112#467—24no—
113#466—24no—
114#465—24no—
115#465—24no—
116#446—22no—
117#446—41no—
118C8×Q8—22no—
119C8×Q8—41no—
120C4×Q16—22no—
121C4×Q16—22no—
122C4×SD16—22no—
123C4×SD16—22no—
124C4×SD16D414yes—
125C4×SD16D414yes—
126C4×D8—22no—
127C4×D8—22no—
128#436—22no—
129#436—41no—
130C8×D4—22no—
131C8×D4—41no—
132C2×#107C2×C411yescharacteristic
133C2×#107C2×C411yescharacteristic
134C2×#106C2×C411yescharacteristic
135C2×#106C2×C411yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#102C2×C411yescharacteristic
147C2×#102C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×C22⋊C8—21no—
151C2×C22⋊C8—21no—
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8—21no—
158C2×C22⋊C8—21no—
159C2×C22⋊C8D411yescharacteristic
160C2×C22⋊C8D411yescharacteristic
161C2×#97—21no—
162C2×C4×C8—22no—
163C2×C4×C8—22no—
164C2×C4×C8—21no—
165C2×C4×C8C2×C411yescharacteristic
166C2×C4×C8C2×C411yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336D411yescharacteristic
181#336D411yescharacteristic
182#335—22no—
183#335—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—41no—
187Q8⋊C8—41no—
188D4⋊C8—22no—
189D4⋊C8—22no—
190D4⋊C8—41no—
191D4⋊C8—41no—
192#322—22no—
193#322—22no—
194C22×Q8C22×C411yescharacteristic
195C22×Q8C22×C411yescharacteristic
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C22×C411yescharacteristic
199C22×D4C22×C411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4#4111yescharacteristic
205C2×Q16—22no—
206C2×Q16—22no—
207C2×Q16—41no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×SD16—22no—
211C2×SD16—22no—
212C2×SD16—22no—
213C2×SD16C22⋊C414yes—
214C2×D8—22no—
215C2×D8—22no—
216C2×D8—41no—
217C2×M16—42no—
218C2×M16—42no—
219C2×M16—81no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—41no—
232C22×C8—42no—
233C22×C8—42no—
234C22×C8—81no—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C22×C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243#126—42no—
244#126—42no—
245#126—42no—
246#126—42no—
247#125—22no—
248#125—22no—
249#125—41no—
250#125—41no—
251#124—22no—
252#124—22no—
253#124—41no—
254#124—41no—
255#123—42no—
256#123—42no—
257#123—42no—
258#123—42no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—41no—
262C22⋊Q8—41no—
263#121—22no—
264#121—22no—
265#121—41no—
266#121—41no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8C2×D412yes—
274C4×Q8C2×D412yes—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4C2×D412yes—
286C4×D4C2×D412yes—
287#117—42no—
288#117—42no—
289#117—42no—
290#117—42no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42#4111yescharacteristic
322C2×C42C2411yescharacteristic
323#107—22no—
324#107—22no—
325#106—22no—
326#106C22⋊C414yes—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337#103—22no—
338#103—22no—
339#103—22no—
340#103C22⋊C414yes—
341#102—22no—
342#102—22no—
343#102—22no—
344#102C22⋊C414yes—
345C22⋊C8—41no—
346C22⋊C8—41no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—81no—
353C22⋊C8—81no—
354C22⋊C8—81no—
355C22⋊C8—81no—
356#97—42no—
357C4×C8—22no—
358C4×C8—22no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—42no—
370C4×C8C2×D412yes—
371C4×C8C2×D412yes—
372C24—21no—
373C24—21no—
374C24C4×D411yescharacteristic
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8C2×C22⋊C412yes—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4C2×C22⋊C412yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4111yescharacteristic
445Q16—44no—
446SD16—28no—
447SD16—44no—
448D8—44no—
449M16—84no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—84no—
490C2×C8C2×C22⋊C412yes—
491C2×C8C2×C22⋊C412yes—
492C2×C8#13112yes—
493C2×C8#13112yes—
494C2×C8#13112yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4C2×C22⋊C412yes—
512C4⋊C4C2×C22⋊C412yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—41no—
534C42—41no—
535C42C22×D411yescharacteristic
536C42C22×D411yescharacteristic
537C42H(5,𝔽2)11yescharacteristic
538C42H(5,𝔽2)11yescharacteristic
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#44412yes—
560C23C2×C4×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578D4—84no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4C22×C22⋊C411yescharacteristic
627C2×C4#51811yescharacteristic
628C2×C4D4211yescharacteristic
629C2×C4D4211yescharacteristic
630C2×C4#54611yescharacteristic
631C2×C4#54611yescharacteristic
632C2×C4C2×#13111yescharacteristic
633C2×C4C2×#13111yescharacteristic
634C2×C4C2×#13111yescharacteristic
635C2×C4#56811yescharacteristic
636C2×C4#57511yescharacteristic
637C2×C4#57511yescharacteristic
638C2×C4#57611yescharacteristic
639C2×C4#57811yescharacteristic
640C8—22no—
641C8—22no—
642C8—41no—
643C8—41no—
644C8—41no—
645C8—41no—
646C8—41no—
647C8—81no—
648C8—81no—
649C8—81no—
650C8—81no—
651C8—81no—
652C8—81no—
653C8#40814yes—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22D4×C22⋊C411yescharacteristic
676C22#293111yescharacteristic
677C22#293511yescharacteristic
678C22#294311yescharacteristic
679C22#295111yescharacteristic
680C22C2×#44411yescharacteristic
681C22#295711yescharacteristic
682C4—21no—
683C4—21no—
684C4—41no—
685C4—42no—
686C4—41no—
687C4—42no—
688C4—41no—
689C4—42no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4C2×#40811yescharacteristic
694C4#288011yescharacteristic
695C4#297911yescharacteristic
696C4#298111yescharacteristic
697C4#326811yescharacteristic
698C4#326811yescharacteristic
699C4#326911yescharacteristic
700C4#326911yescharacteristic
701C2—42no—
702C2—82no—
703C2—161no—
704C2#1796311yescharacteristic
705C2#1797311yescharacteristic
706C2#1807411yescharacteristic
707C2#1809211yescharacteristic
708C2#2240211yescharacteristic
709C2#2243911yescharacteristic
710C2#2245211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#466C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22531C2semidirect
3. #22474C2semidirect
4. C2×#1592C2semidirect
5. C2×SD16⋊C8C2semidirect
6. #9660C2semidirect
7. #9660C2semidirect
8. #8951C2semidirect
9. #8950C2semidirect
10. #8947C2semidirect
11. #8946C2semidirect
12. #8596C2semidirect
13. #8315C2semidirect
14. #8315C2semidirect
15. #8144C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31682
#225312
#224742
C2×#15922
C2×SD16⋊C82
#96602
#89512
#89502
#89472
#89462
#85962
#83152
#81442