Finite Group Databasebeta

← #108335

Subgroups — #108335

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C22×#332C211yesmaximal, characteristic
4#10849C211yesmaximal, characteristic
5#10849C211yesmaximal, characteristic
6C2×#1279C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8—21no—
10C22×C4⋊C8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13#2365—22no—
14#2365—22no—
15#2365—22no—
16#2365C414yescomplemented
17C2×#391—21no—
18C2×#389—21no—
19C2×#387—22no—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990—22no—
25#1990C414yescomplemented
26C2×#340—22no—
27C2×#340—21no—
28C2×#340—21no—
29C2×#340—21no—
30C2×#340—21no—
31C2×#332—22no—
32C2×#332—22no—
33C2×#332C414yescomplemented
34#1279—22no—
35#1279—22no—
36C23×Q8—21no—
37C23×C8—21no—
38C23×C8—21no—
39C23×C8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8D412yes—
50C2×C4⋊Q8D412yes—
51C2×C4⋊Q8Q812yes—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—42no—
55C2×C4×Q8—42no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C412yes—
83C2×C4⋊C8C2×C412yes—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2×C412yes—
90#391—24no—
91#391—24no—
92#389—24no—
93#389—24no—
94#387—28no—
95#387—28no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384C2×C412yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114#340—28no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#332—28no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—41no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8—41no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8—41no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8C4214yes—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C4⋊C414yes—
179C4⋊Q8C4⋊C414yes—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8—48no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C412yes—
234C2×C4⋊C4C4⋊C414yes—
235C2×C4⋊C4D812yes—
236C2×C4⋊C4SD1612yes—
237C2×C4⋊C4SD1612yes—
238C2×C4⋊C4Q1612yes—
239C2×C42—28no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×Q811yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—41no—
277C4×C8—41no—
278#95—28no—
279#95—28no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284C24#12011yescharacteristic
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C4⋊C411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4C22⋊Q811yescharacteristic
374C22×C4C22⋊Q811yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12311yescharacteristic
377C22×C4C2×D811yescharacteristic
378C22×C4C2×SD1611yescharacteristic
379C22×C4C2×SD1611yescharacteristic
380C22×C4C2×Q1611yescharacteristic
381C22×C4#13511yescharacteristic
382C22×C4#13511yescharacteristic
383C22×C4#13511yescharacteristic
384C22×C4#13511yescharacteristic
385C22×C4AGL(1,ℤ/8)11yescharacteristic
386C22×C4AGL(1,ℤ/8)11yescharacteristic
387C22×C4AGL(1,ℤ/8)11yescharacteristic
388C22×C4AGL(1,ℤ/8)11yescharacteristic
389C22×C4#13711yescharacteristic
390C22×C4#13711yescharacteristic
391C22×C4#13711yescharacteristic
392C22×C4#13711yescharacteristic
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—81no—
407C2×C8—81no—
408C2×C8—81no—
409C2×C8—81no—
410C2×C8—81no—
411C2×C8—81no—
412C2×C8—81no—
413C2×C8—81no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—416no—
432C4⋊C4—44no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—82no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4#9518yes—
447C4⋊C4#10214yes—
448C4⋊C4#10314yes—
449C42—416no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42C2×C4211yescharacteristic
459C42C2×C22⋊C411yescharacteristic
460C42C2×C4⋊C411yescharacteristic
461C42C2×C4⋊C411yescharacteristic
462C42#11711yescharacteristic
463C42#11711yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42C22⋊Q811yescharacteristic
471C42C22⋊Q811yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C23#37911yescharacteristic
475C23#38511yescharacteristic
476C23#38611yescharacteristic
477C23#44711yescharacteristic
478C23#44811yescharacteristic
479C23#44911yescharacteristic
480C23#44911yescharacteristic
481C23#45011yescharacteristic
482C23#45111yescharacteristic
483C23#45211yescharacteristic
484C23#45211yescharacteristic
485C23#45411yescharacteristic
486C23#45411yescharacteristic
487C23#45611yescharacteristic
488C23#45611yescharacteristic
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—44no—
495Q8—44no—
496Q8—44no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—161no—
548C2×C4—161no—
549C2×C4—161no—
550C2×C4—161no—
551C2×C4C2×#9511yescharacteristic
552C2×C4C4×C22⋊C411yescharacteristic
553C2×C4C4×C22⋊C411yescharacteristic
554C2×C4#38011yescharacteristic
555C2×C4#38011yescharacteristic
556C2×C4#38111yescharacteristic
557C2×C4#38511yescharacteristic
558C2×C4#38511yescharacteristic
559C2×C4#38511yescharacteristic
560C2×C4#38611yescharacteristic
561C2×C4#38611yescharacteristic
562C2×C4#38611yescharacteristic
563C2×C4C2×#10211yescharacteristic
564C2×C4C2×#10311yescharacteristic
565C2×C4#41611yescharacteristic
566C2×C4#41611yescharacteristic
567C2×C4#41711yescharacteristic
568C2×C4#41711yescharacteristic
569C2×C4#41811yescharacteristic
570C2×C4#41911yescharacteristic
571C2×C4C4×D811yescharacteristic
572C2×C4C4×D811yescharacteristic
573C2×C4C4×SD1611yescharacteristic
574C2×C4C4×SD1611yescharacteristic
575C2×C4C4×SD1611yescharacteristic
576C2×C4C4×SD1611yescharacteristic
577C2×C4C4×Q1611yescharacteristic
578C2×C4C4×Q1611yescharacteristic
579C2×C4#44011yescharacteristic
580C2×C4#44011yescharacteristic
581C2×C4#44011yescharacteristic
582C2×C4#44011yescharacteristic
583C2×C4#44111yescharacteristic
584C2×C4#44111yescharacteristic
585C2×C4#44211yescharacteristic
586C2×C4#44211yescharacteristic
587C2×C4#44711yescharacteristic
588C2×C4#44811yescharacteristic
589C2×C4#44911yescharacteristic
590C2×C4#44911yescharacteristic
591C2×C4#45011yescharacteristic
592C2×C4#45111yescharacteristic
593C2×C4#45211yescharacteristic
594C2×C4#45211yescharacteristic
595C2×C4#45911yescharacteristic
596C2×C4#46011yescharacteristic
597C2×C4#46111yescharacteristic
598C2×C4#46211yescharacteristic
599C2×C4#46311yescharacteristic
600C2×C4#46311yescharacteristic
601C2×C4#46411yescharacteristic
602C2×C4#46411yescharacteristic
603C2×C4#47411yescharacteristic
604C2×C4#47511yescharacteristic
605C2×C4#47611yescharacteristic
606C2×C4#47711yescharacteristic
607C2×C4#47811yescharacteristic
608C2×C4#47811yescharacteristic
609C2×C4#47911yescharacteristic
610C2×C4#47911yescharacteristic
611C8—44no—
612C8—82no—
613C8—82no—
614C22C22⋊#9511yescharacteristic
615C22#162511yescharacteristic
616C22#162611yescharacteristic
617C22#162711yescharacteristic
618C22#162811yescharacteristic
619C22#163511yescharacteristic
620C22#163611yescharacteristic
621C22#163711yescharacteristic
622C22#163811yescharacteristic
623C22#164711yescharacteristic
624C22#164811yescharacteristic
625C22#164911yescharacteristic
626C22#165011yescharacteristic
627C22#165111yescharacteristic
628C22#165211yescharacteristic
629C22#165311yescharacteristic
630C22#165411yescharacteristic
631C22#178611yescharacteristic
632C22#178711yescharacteristic
633C22#178811yescharacteristic
634C22#178811yescharacteristic
635C22#179311yescharacteristic
636C22#179311yescharacteristic
637C22#185711yescharacteristic
638C22#185711yescharacteristic
639C22#186211yescharacteristic
640C22#186311yescharacteristic
641C22#186411yescharacteristic
642C22#186511yescharacteristic
643C22#187711yescharacteristic
644C22#187811yescharacteristic
645C22#187911yescharacteristic
646C22#188011yescharacteristic
647C22#188611yescharacteristic
648C22#188811yescharacteristic
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—48no—
654C4—44no—
655C4—44no—
656C4—82no—
657C4—82no—
658C4—162no—
659C4C4×#10211yescharacteristic
660C4C4×#10311yescharacteristic
661C4#176211yescharacteristic
662C4#176311yescharacteristic
663C4#180211yescharacteristic
664C4#180311yescharacteristic
665C4#180411yescharacteristic
666C4#180511yescharacteristic
667C4#186211yescharacteristic
668C4#186311yescharacteristic
669C4#186411yescharacteristic
670C4#186511yescharacteristic
671C4#187711yescharacteristic
672C4#187811yescharacteristic
673C4#187911yescharacteristic
674C4#188011yescharacteristic
675C2#844611yescharacteristic
676C2#844711yescharacteristic
677C2#844811yescharacteristic
678C2#844911yescharacteristic
679C2#892211yescharacteristic
680C2#892311yescharacteristic
681C2#892511yescharacteristic
682C2#892611yescharacteristic
683C2#905811yescharacteristic
684C2#905911yescharacteristic
685C2#906211yescharacteristic
686C2#906511yescharacteristic
687C2#1075011yescharacteristic
688C2#1078811yescharacteristic
689C2#1078911yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2365C4semidirect
25. #1990C4semidirect
33. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#19902
C22×#3322
#108492
C2×#12792