Finite Group Databasebeta

← #6343116

Subgroups — #6343116

Order 512.

2919 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#41C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic
3#60125C211yesmaximal, characteristic
4#17514C2116yesmaximal, complemented
5#17491C214yesmaximal, complemented
6#17486C212yesmaximal, complemented
7#17449C214yesmaximal, complemented
8C2×#1487C212yesmaximal, complemented
9Q8×#41—24no—
10Q8×#41—24no—
11Q8×#41C418yes—
12Q8×#41C2218yes—
13C2×Q82C411yescharacteristic
14C2×#557—21no—
15C2×#557C411yescharacteristic
16C2×#554—22no—
17C2×#554—21no—
18C2×#554C412yes—
19C2×#554C411yescharacteristic
20C2×D4×Q8C411yescharacteristic
21C2×D4×Q8C412yes—
22C2×#533—21no—
23C2×#533—22no—
24C2×#533C2211yescharacteristic
25C2×#533C2212yes—
26C2×C4×#41—21no—
27C2×C4×#41C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2212yes—
30#2989—24no—
31#2983—21no—
32#2975—21no—
33#2973—22no—
34#2965C2218yes—
35#2925—24no—
36#2925C2218yes—
37#2921—22no—
38#2920C2214yes—
39#2920C2214yes—
40C2×#431C2212yes—
41C2×#423C2212yes—
42#2879—21no—
43#2873—21no—
44#1497C22116yescomplemented
45#1493C2218yescomplemented
46#1491C22116yescomplemented
47#1487C2214yes—
48#1487C2214yes—
49#1487C2218yes—
50#1487C2218yescomplemented
51#1485C22116yescomplemented
52#1481C2218yescomplemented
53#1479C22116yescomplemented
54C2×Q8⋊C8C2218yescomplemented
55C22×#41D411yescharacteristic
56C23×Q8C2×C411yescharacteristic
57C23×Q8C2×C412yes—
58C22×M16—22no—
59Q82—24no—
60Q82C2×C418yes—
61#557—28no—
62#557—24no—
63#557—28no—
64#557C2×C418yes—
65#554—216no—
66#554—28no—
67#554—24no—
68#554—28no—
69#554—216no—
70#554—28no—
71#554C2×C4116yes—
72#554C2×C418yes—
73D4×Q8—28no—
74D4×Q8—24no—
75D4×Q8C2×C418yes—
76D4×Q8C2×C4116yes—
77#533—216no—
78#533—28no—
79#533—28no—
80#533—216no—
81#533—22no—
82#533—22no—
83#533—24no—
84#533—24no—
85#533C2×C414yes—
86#533C2×C418yes—
87#533C2314yes—
88#533C2318yes—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8C2×C411yescharacteristic
94C2×C4⋊Q8C2×C412yes—
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×C4⋊Q8C2×C412yes—
97C2×#125—22no—
98C2×#125—21no—
99C2×#125—22no—
100C2×#125—21no—
101C2×#125—21no—
102C2×#125—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2×C412yes—
112C2×C22⋊Q8C2×C412yes—
113C2×C22⋊Q8C2×C412yes—
114C4×#41—28no—
115C4×#41—28no—
116C4×#41C2×C414yes—
117C4×#41D414yes—
118C4×#41D414yes—
119C4×#41C2314yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—22no—
126C2×C4×Q8—22no—
127C2×C4×Q8—24no—
128C2×C4×Q8—22no—
129C2×C4×Q8—24no—
130C2×C4×Q8—22no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2×C411yescharacteristic
133C2×C4×Q8C2×C412yes—
134C2×C4×Q8C2×C411yescharacteristic
135C2×C4×Q8C2×C412yes—
136C2×C4×Q8C2×C414yes—
137C2×C4×Q8C2×C418yes—
138C2×C4×Q8C2×C411yescharacteristic
139C2×C4×Q8D411yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2314yes—
144C2×C4×Q8C2318yes—
145C2×C4×D4—22no—
146C2×C4×D4—21no—
147C2×C4×D4—22no—
148C2×C4×D4—21no—
149C2×C4×D4C2×C412yes—
150C2×C4×D4C2×C411yescharacteristic
151C2×C4×D4D412yes—
152C2×C4×D4D411yescharacteristic
153C2×#117—22no—
154C2×#117—21no—
155C2×#117D411yescharacteristic
156C2×#117D412yes—
157C22×C4⋊C4—21no—
158C22×C4⋊C4—22no—
159C22×C4⋊C4C2×C411yescharacteristic
160C22×C4⋊C4C2×C412yes—
161C22×C42—22no—
162C22×C42—21no—
163C22×C42C2311yescharacteristic
164C22×C42C2312yes—
165#446—28no—
166#436—28no—
167#435—216no—
168#433—216no—
169#432—24no—
170#432—24no—
171#432—24no—
172#432C2318yes—
173#431—22no—
174#431—22no—
175#431—24no—
176#431C2318yes—
177#431C2314yes—
178#431C2314yes—
179#431C2318yes—
180#423—22no—
181#423—22no—
182#423—24no—
183#423C2314yes—
184#423C2314yes—
185C2×C4⋊C8—22no—
186C2×C4⋊C8—22no—
187C2×C4⋊C8C2314yes—
188#408—28no—
189#407—24no—
190#407—24no—
191C2×C22⋊C8—22no—
192#405—28no—
193C4×M16—24no—
194C4×M16C2318yes—
195C2×C4×C8C2314yes—
196Q8⋊C8C23116yes—
197Q8⋊C8C23116yes—
198Q8⋊C8C23132yes—
199C2×#41—28no—
200C2×#41C22⋊C414yes—
201C2×#41C2×D414yes—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—21no—
208C22×Q8—44no—
209C22×Q8—48no—
210C22×Q8C22⋊C411yescharacteristic
211C22×Q8C22×C411yescharacteristic
212C22×Q8C22×C414yes—
213C22×Q8C22×C418yes—
214C22×Q8C22×C411yescharacteristic
215C22×Q8C22×C412yes—
216C22×D4C22⋊C412yes—
217C22×D4C22⋊C411yescharacteristic
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4C22×C412yes—
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C2×M16—24no—
229C2×M16—24no—
230C2×M16—28no—
231C2×M16—48no—
232C22×C8—24no—
233C22×C8—24no—
234C4⋊Q8—216no—
235C4⋊Q8—28no—
236C4⋊Q8—216no—
237C4⋊Q8—28no—
238C4⋊Q8—24no—
239C4⋊Q8—22no—
240C4⋊Q8—24no—
241C4⋊Q8—22no—
242C4⋊Q8C22×C414yes—
243C4⋊Q8C22×C418yes—
244C4⋊Q8C22×C414yes—
245C4⋊Q8C22×C418yes—
246#125—216no—
247#125—28no—
248#125—216no—
249#125—28no—
250#125—28no—
251#125—24no—
252#125—24no—
253#125—28no—
254C22⋊Q8—28no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—28no—
261C22⋊Q8—216no—
262C22⋊Q8—216no—
263C22⋊Q8—216no—
264C22⋊Q8—216no—
265C22⋊Q8—216no—
266C22⋊Q8—216no—
267C22⋊Q8C22×C418yes—
268C22⋊Q8C22×C418yes—
269C22⋊Q8C22×C418yes—
270C4×Q8—28no—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—28no—
279C4×Q8—24no—
280C4×Q8—28no—
281C4×Q8—24no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—22no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—21no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8C22⋊C414yes—
295C4×Q8C22⋊C414yes—
296C4×Q8C22×C412yes—
297C4×Q8C22×C412yes—
298C4×Q8C22×C414yes—
299C4×Q8C22×C414yes—
300C4×Q8C22×C418yes—
301C4×Q8C22×C414yes—
302C4×Q8C22×C418yes—
303C4×Q8C22×C414yes—
304C4×Q8C2×D414yes—
305C4×Q8C2412yes—
306C4×Q8C2412yes—
307C4×Q8C2414yes—
308C4×D4—216no—
309C4×D4—28no—
310C4×D4—216no—
311C4×D4—28no—
312C4×D4C22⋊C418yes—
313C4×D4C22⋊C414yes—
314C4×D4C22⋊C418yes—
315C4×D4C22⋊C414yes—
316C4×D4C22×C418yes—
317C4×D4C22×C414yes—
318C4×D4C2×D418yes—
319C4×D4C2×D414yes—
320#117—28no—
321#117—24no—
322#117—28no—
323#117—24no—
324#117C22⋊C418yes—
325#117C22⋊C414yes—
326#117C2×D414yes—
327#117C2×D418yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4C22⋊C412yes—
353C2×C4⋊C4C22⋊C411yescharacteristic
354C2×C4⋊C4C22⋊C412yes—
355C2×C4⋊C4C22⋊C411yescharacteristic
356C2×C4⋊C4C22×C411yescharacteristic
357C2×C4⋊C4C22×C411yescharacteristic
358C2×C4⋊C4C22×C412yes—
359C2×C4⋊C4C22×C414yes—
360C2×C4⋊C4C22×C418yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4C22⋊C412yes—
365C2×C22⋊C4C22⋊C412yes—
366C2×C22⋊C4C22⋊C412yes—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—28no—
372C2×C42—24no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—28no—
377C2×C42—24no—
378C2×C42C22×C412yes—
379C2×C42C22×C411yescharacteristic
380C2×C42C22×C414yes—
381C2×C42C22×C412yes—
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2×D414yes—
389C2×C42C2×D414yes—
390C2×C42C2411yescharacteristic
391C2×C42C2412yes—
392C2×C42C2414yes—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—28no—
399C4⋊C8C2418yes—
400C22⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—28no—
403C22⋊C8—28no—
404C22⋊C8—28no—
405#97—28no—
406#97—28no—
407C4×C8—24no—
408C4×C8—28no—
409C4×C8C2418yes—
410C24C2×C22⋊C412yes—
411C24C2×C22⋊C411yescharacteristic
412#41—28no—
413#41—28no—
414#41—216no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—24no—
418C2×Q8—28no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—28no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—28no—
425C2×Q8—216no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—416no—
434C2×Q8—82no—
435C2×Q8—84no—
436C2×Q8C2×C22⋊C414yes—
437C2×Q8C23×C412yes—
438C2×Q8C23×C414yes—
439C2×Q8C23×C412yes—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4C2×C22⋊C418yes—
443C2×D4C2×C22⋊C414yes—
444C22×C4—21no—
445C22×C4—28no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—81no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C414yes—
465C22×C4C2×C22⋊C412yes—
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C414yes—
468C22×C4C2×C22⋊C412yes—
469C22×C4C2×C22⋊C412yes—
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C414yes—
472C22×C4C2×C22⋊C412yes—
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C412yes—
475C22×C4#13718yes—
476C22×C4C23×C411yescharacteristic
477C22×C4C23×C414yes—
478C22×C4C23×C412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485M16—48no—
486M16—48no—
487M16—88no—
488C2×C8—28no—
489C2×C8—28no—
490C2×C8—48no—
491C2×C8—84no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4C2×C22⋊C418yes—
517C4⋊C4C2×C22⋊C414yes—
518C4⋊C4C2×C22⋊C418yes—
519C4⋊C4C2×C22⋊C414yes—
520C4⋊C4C23×C412yes—
521C4⋊C4C23×C412yes—
522C4⋊C4C23×C414yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4C2×C22⋊C418yes—
527C22⋊C4C2×C22⋊C418yes—
528C22⋊C4C2×C22⋊C418yes—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—28no—
533C42—24no—
534C42—28no—
535C42—24no—
536C42—22no—
537C42—24no—
538C42—22no—
539C42C2×C22⋊C414yes—
540C42C2×C22⋊C412yes—
541C42C2×C22⋊C414yes—
542C42C2×C22⋊C412yes—
543C42C2×C22⋊C414yes—
544C42C2×C22⋊C412yes—
545C42C23×C411yescharacteristic
546C42C23×C414yes—
547C42C23×C412yes—
548C42C22×D412yes—
549C42C22×D414yes—
550C42C22×D411yescharacteristic
551C42C22×D414yes—
552C42C22×D411yescharacteristic
553C42C22×D412yes—
554C42C2511yescharacteristic
555C23—24no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23#41914yes—
561C23C22×C22⋊C414yes—
562C23C22×C22⋊C412yes—
563C23C22×C22⋊C411yescharacteristic
564Q8—24no—
565Q8—28no—
566Q8—22no—
567Q8—24no—
568Q8—28no—
569Q8—48no—
570Q8—416no—
571Q8—42no—
572Q8—44no—
573Q8—84no—
574Q8—88no—
575Q8#56814yes—
576D4—216no—
577D4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4#41914yes—
612C2×C4#41918yes—
613C2×C4C22×C22⋊C414yes—
614C2×C4C22×C22⋊C411yescharacteristic
615C2×C4C22×C22⋊C411yescharacteristic
616C2×C4C22×C22⋊C414yes—
617C2×C4C22×C22⋊C412yes—
618C2×C4C22×C22⋊C412yes—
619C2×C4C22×C22⋊C414yes—
620C2×C4C22×C22⋊C411yescharacteristic
621C2×C4C22×C22⋊C412yes—
622C2×C4C2×#13714yes—
623C2×C4C2×#13714yes—
624C2×C4C2×#13714yes—
625C2×C4C24×C411yescharacteristic
626C2×C4C23×D411yescharacteristic
627C2×C4C23×D411yescharacteristic
628C8—48no—
629C8—48no—
630C22—21no—
631C22—22no—
632C22—21no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—48no—
639C22C2×#41912yes—
640C22C2×#41914yes—
641C22C23×C22⋊C411yescharacteristic
642C4—24no—
643C4—21no—
644C4—22no—
645C4—24no—
646C4—24no—
647C4—21no—
648C4—21no—
649C4—22no—
650C4—24no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4C2×#41912yes—
657C4C2×#41914yes—
658C4C2×#56812yes—
659C4C22×#13712yes—
660C2—22no—
661C2—24no—
662C2—44no—
663C2#6006411yescharacteristic
664C2C22×#41911yescharacteristic
665C2#6009311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17514C2semidirect
5. #17491C2semidirect
6. #17486C2semidirect
7. #17449C2semidirect
8. C2×#1487C2semidirect
44. #1497C22semidirect
45. #1493C22semidirect
46. #1491C22semidirect
50. #1487C22semidirect
51. #1485C22semidirect
52. #1481C22semidirect
53. #1479C22semidirect
54. C2×Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#412
#602772
#601252
#175142
#174912
#174862
#174492
C2×#14872