Finite Group Databasebeta

← #188359

Subgroups — #188359

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#520C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#21349C211yesmaximal, characteristic, complemented
4C2×#1477C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8566C212yesmaximal, complemented
7#8565C212yesmaximal, complemented
8#8564C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7892C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12C2×#520C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2276C412yes—
20#2275—22no—
21#2275C412yes—
22#2274—21no—
23#2274—21no—
24#2274C411yescharacteristic
25#2274C411yescharacteristic
26C42×Q8C2212yes—
27C42×D4—22no—
28C42×D4C2212yes—
29C4×#117—21no—
30C4×#117C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1939—21no—
34#1939—21no—
35#1939—21no—
36#1939—21no—
37#1850—22no—
38#1850—22no—
39#1842C2212yes—
40#1831C2212yes—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1477—24no—
44#1476—24no—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41C2×C411yescharacteristic
56C23×C8—21no—
57C23×C8—21no—
58#520—24no—
59#520—24no—
60#520—24no—
61#520C818yescomplemented
62C4×#41—22no—
63C4×#41—22no—
64C4×#41D414yes—
65C4×#41Q814yes—
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yes—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117C2×C411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C42—21no—
89C22×C42—21no—
90C22×C42—22no—
91C22×C42C2311yescharacteristic
92#431—22no—
93#431—22no—
94#431—22no—
95#431—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2312yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4D412yes—
147C4×C22⋊C4D412yes—
148#376—22no—
149#376C2×C412yes—
150C2×#95—21no—
151C2×#95—22no—
152C43—22no—
153C43C2312yes—
154#336—22no—
155#336—22no—
156#336—22no—
157#336—22no—
158Q8⋊C8—22no—
159Q8⋊C8—22no—
160Q8⋊C8—24no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—24no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41C4⋊C414yes—
168C2×#41C2×C814yes—
169C2×#41M1614yes—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—81no—
194C22×C8—81no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—24no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8C4⋊C414yes—
202C4×Q8C2×C818yescomplemented
203C4×Q8C2×D412yes—
204C4×Q8C2×Q812yes—
205C4×D4—216no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—216no—
213C4×D4—22no—
214C4×D4—24no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4C4⋊C414yes—
218C4×D4C2×C818yescomplemented
219C4×D4C2×D412yes—
220C4×D4C2×Q812yes—
221#117—28no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—28no—
226#117—24no—
227#117—24no—
228#117C4⋊C414yes—
229#117C2×C814yes—
230#117M1614yes—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C2×C814yes—
246C2×C4⋊C4M1614yes—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4C22⋊C412yes—
257C2×C22⋊C4C22⋊C412yes—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—82no—
272C2×C42—82no—
273C2×C42—81no—
274C2×C42—81no—
275C2×C42C22×C412yes—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×Q812yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288C2×C42C2411yescharacteristic
289C4⋊C8—22no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—22no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—44no—
296C4⋊C8—44no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—82no—
300C4⋊C8—82no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—22no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—22no—
309C4×C8—42no—
310C4×C8—42no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95C22⋊C412yes—
318#95C22⋊C412yes—
319#95C22⋊C412yes—
320#95C22⋊C412yes—
321C24—21no—
322C24C2×C22⋊C411yescharacteristic
323#41—28no—
324#41—28no—
325#41—28no—
326#41C4⋊C8116yescomplemented
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8C2×C4⋊C412yes—
331C2×Q8C22×C812yes—
332C2×Q8C2×M1612yes—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4C2×C4⋊C412yes—
340C2×D4C22×C812yes—
341C2×D4C2×M1612yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—82no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C4⋊C412yes—
385C22×C4#11712yes—
386C22×C4#12112yes—
387C22×C4#12112yes—
388C22×C4C22⋊Q812yes—
389C22×C4C22⋊Q812yes—
390C22×C4C22×C812yes—
391C22×C4C2×M1612yes—
392C22×C4C2×M1612yes—
393C22×C4C2×M1612yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—48no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—84no—
417C4⋊C4—28no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4C22⋊C814yes—
431C4⋊C4C22⋊C814yes—
432C4⋊C4C2×C4⋊C412yes—
433C4⋊C4C2×C4⋊C412yes—
434C4⋊C4C22×C814yes—
435C4⋊C4C2×M1614yes—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—24no—
442C22⋊C4—28no—
443C22⋊C4C22⋊C818yes—
444C42—28no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—24no—
454C42—24no—
455C42—28no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—82no—
461C42—82no—
462C42—82no—
463C42—82no—
464C42—82no—
465C42C2×C4⋊C412yes—
466C42#11712yes—
467C42#12112yes—
468C42#12112yes—
469C42C22⋊Q812yes—
470C42C22⋊Q812yes—
471C42C22×C814yes—
472C42C2×M1614yes—
473C42C22×D411yescharacteristic
474C42C22×Q811yescharacteristic
475C42C2×#4111yescharacteristic
476C42C2×#4111yescharacteristic
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23#38012yes—
489C23C2×C22⋊C812yes—
490C23#40712yes—
491C23C22×C22⋊C411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—22no—
495Q8C2×C4⋊C814yes—
496D4—216no—
497D4—22no—
498D4—22no—
499D4—22no—
500D4C2×C4⋊C814yes—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—216no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—88no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—88no—
548C2×C4#38012yes—
549C2×C4#38012yes—
550C2×C4#38012yes—
551C2×C4C2×C22⋊C814yes—
552C2×C4C2×C22⋊C814yes—
553C2×C4C2×C22⋊C812yes—
554C2×C4#40712yes—
555C2×C4C2×C4⋊C814yes—
556C2×C4#42314yes—
557C2×C4#43114yes—
558C2×C4#43214yes—
559C2×C4C22×C4⋊C411yescharacteristic
560C2×C4C2×#11711yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×C22⋊Q811yescharacteristic
564C2×C4C2×C22⋊Q811yescharacteristic
565C2×C4C23×C811yescharacteristic
566C2×C4C22×M1611yescharacteristic
567C2×C4C22×M1611yescharacteristic
568C2×C4C22×M1611yescharacteristic
569C2×C4#57611yescharacteristic
570C2×C4#57711yescharacteristic
571C2×C4#57711yescharacteristic
572C2×C4#57811yescharacteristic
573C8—42no—
574C8—42no—
575C8—82no—
576C8—81no—
577C8—82no—
578C8—81no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22#179714yes—
593C22C2×#38011yescharacteristic
594C22C22×C22⋊C811yescharacteristic
595C22C2×#40711yescharacteristic
596C22#289711yescharacteristic
597C22#289811yescharacteristic
598C22#290111yescharacteristic
599C22#290111yescharacteristic
600C4—24no—
601C4—24no—
602C4—24no—
603C4—22no—
604C4—21no—
605C4—22no—
606C4—21no—
607C4—44no—
608C4—42no—
609C4—44no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4#179714yes—
614C4C22×C4⋊C811yescharacteristic
615C4C2×#42311yescharacteristic
616C4C2×#43111yescharacteristic
617C4C2×#43211yescharacteristic
618C4#304511yescharacteristic
619C4#304611yescharacteristic
620C4#308311yescharacteristic
621C4#308411yescharacteristic
622C2—24no—
623C2—44no—
624C2#1743811yescharacteristic
625C2#1743911yescharacteristic
626C2#1745311yescharacteristic
627C2#1745411yescharacteristic
628C2C2×#179711yescharacteristic
629C2#2065211yescharacteristic
630C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21972C2semidirect
3. #21349C2semidirect
6. #8566C2semidirect
7. #8565C2semidirect
8. #8564C2semidirect
9. #8563C2semidirect
10. #7892C2semidirect
11. C2×#1279C2semidirect
61. #520C8semidirect
202. C4×Q8C2×C8semidirect
218. C4×D4C2×C8semidirect
326. #41C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5202
#219722
#213492
C2×#14772
C2×#14762
#85662
#85652
#85642
#85632
#78922
C2×#12792