Finite Group Databasebeta

← #185193

Subgroups — #185193

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4C2×#1550C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1542C211yesmaximal, characteristic, complemented
7C22×#332C211yesmaximal, characteristic, complemented
8#8343C214yesmaximal, complemented
9#8340C214yesmaximal, complemented
10#3503—24no—
11#3503—24no—
12#3503—24no—
13#3503—24no—
14C2×Q82C412yes—
15C2×#557—21no—
16C2×#554C412yes—
17C2×#554C414yes—
18C2×D4×Q8—22no—
19C2×D4×Q8—21no—
20C2×#544—22no—
21C2×#544—21no—
22C2×#533C2212yes—
23C2×#533C2211yescharacteristic
24C22×C4⋊Q8C2212yes—
25C22×C4⋊Q8C2211yescharacteristic
26C2×#520—21no—
27C2×C4×#41C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C2×#408—21no—
33#1984—21no—
34#1954—21no—
35#1869—22no—
36#1798—22no—
37#1797C2214yes—
38#1797C2214yes—
39#1729—21no—
40#1550—24no—
41#1549—24no—
42#1542—24no—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#332—24no—
46C2×#332C2214yes—
47C2×#332C2212yes—
48C2×#332C2214yes—
49C2×#332C2212yes—
50C22×#41D411yescharacteristic
51C23×Q8—22no—
52C23×Q8—21no—
53C22×M16—21no—
54C23×C8—21no—
55C23×C8—21no—
56Q82—28no—
57Q82—28no—
58#557—28no—
59#557—28no—
60#554—28no—
61#554—216no—
62#554—28no—
63#554—216no—
64D4×Q8—216no—
65D4×Q8—216no—
66D4×Q8—28no—
67D4×Q8—28no—
68#544—216no—
69#544—216no—
70#544—28no—
71#544—28no—
72#533—24no—
73#533—24no—
74#533—24no—
75#533—24no—
76#533—22no—
77#533—22no—
78#533D414yes—
79#533D414yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2×C414yes—
93C2×C4⋊Q8C2×C412yes—
94C2×C4⋊Q8D414yes—
95C2×C4⋊Q8D414yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—24no—
100C2×#125—22no—
101C2×#125—22no—
102C2×#125—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8C2×C414yes—
109C2×C22⋊Q8C2×C412yes—
110#520—28no—
111#520—28no—
112C4×#41—22no—
113C4×#41—22no—
114C4×#41—22no—
115C4×#41—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2×C414yes—
120C2×C4×Q8C2×C412yes—
121C2×C4×Q8C2×C411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×D4—24no—
124C2×C4×D4—22no—
125C2×C4×D4C2×C412yes—
126C2×C4×D4C2×C411yescharacteristic
127C2×C4×D4D412yes—
128C2×C4×D4D411yescharacteristic
129C2×#117—22no—
130C2×#117—21no—
131C2×#117D412yes—
132C2×#117D411yescharacteristic
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2×C414yes—
136C22×C4⋊C4C2×C412yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#431—22no—
140#431—22no—
141#423—22no—
142#423—22no—
143C2×C4⋊C8—22no—
144C2×C4⋊C8—22no—
145C2×C4⋊C8—22no—
146C2×C4⋊C8—22no—
147C2×C4⋊C8—21no—
148C2×C4⋊C8—21no—
149C2×C4⋊C8—21no—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154#408—44no—
155C2×C22⋊C8—24no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8—24no—
158C2×C4×C8—21no—
159#336—24no—
160#336—24no—
161#333—24no—
162#333—24no—
163#332—28no—
164#332—24no—
165#332—28no—
166#332—24no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41SD1614yes—
172C2×#41Q1614yes—
173C22×Q8—28no—
174C22×Q8—28no—
175C22×Q8—24no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—21no—
181C22×Q8—82no—
182C22×Q8—82no—
183C22×Q8—84no—
184C22×Q8—84no—
185C22×Q8C22⋊C411yescharacteristic
186C22×Q8C22×C412yes—
187C22×D4C22⋊C412yes—
188C22×D4C22⋊C411yescharacteristic
189C23×C4—24no—
190C23×C4—22no—
191C23×C4C22×C412yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C2×M16—42no—
198C2×M16—42no—
199C2×M16—42no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—82no—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—28no—
212C4⋊Q8—28no—
213C4⋊Q8—232no—
214C4⋊Q8—216no—
215C4⋊Q8—28no—
216C4⋊Q8—28no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8C22⋊C418yes—
229C4⋊Q8C2×D412yes—
230C4⋊Q8C2×D412yes—
231C4⋊Q8C2×D412yes—
232C4⋊Q8C2×D412yes—
233C4⋊Q8C2×D414yes—
234C4⋊Q8C2×D414yes—
235#125—28no—
236#125—28no—
237#125—232no—
238#125—216no—
239#125—24no—
240#125—24no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—232no—
244C22⋊Q8—232no—
245C22⋊Q8—232no—
246C22⋊Q8—28no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—28no—
251C22⋊Q8C22⋊C418yes—
252C4×Q8—216no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×Q8—28no—
256C4×Q8—28no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8C22⋊C418yes—
265C4×D4—232no—
266C4×D4—216no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275#117—28no—
276#117—28no—
277#117—24no—
278#117—24no—
279#117—22no—
280#117—22no—
281#117—24no—
282#117—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4C22⋊C411yescharacteristic
303C2×C4⋊C4C22⋊C412yes—
304C2×C4⋊C4C22⋊C411yescharacteristic
305C2×C4⋊C4C22⋊C412yes—
306C2×C4⋊C4C22⋊C418yes—
307C2×C4⋊C4C22×C412yes—
308C2×C4⋊C4C22×C414yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4C22⋊C412yes—
312C2×C22⋊C4C22⋊C412yes—
313C2×C22⋊C4C22⋊C412yes—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42C22×C412yes—
321C2×C42C22×C411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2411yescharacteristic
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—48no—
344C22⋊C8—84no—
345C4×C8—42no—
346C24C2×C22⋊C412yes—
347C24C2×C22⋊C411yescharacteristic
348#41—28no—
349#41—28no—
350#41—28no—
351#41—28no—
352C2×Q8—216no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—28no—
360C2×Q8—22no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—84no—
367C2×Q8—88no—
368C2×Q8—84no—
369C2×Q8—84no—
370C2×Q8—88no—
371C2×Q8—88no—
372C2×Q8—88no—
373C2×Q8—816no—
374C2×Q8#10314yes—
375C2×Q8C2×C22⋊C414yes—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4#10318yes—
381C2×D4#10314yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—84no—
402C22×C4—88no—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4#12012yes—
414C22×C4#12012yes—
415C22×C4C2×SD1614yes—
416C22×C4C2×SD1612yes—
417C22×C4C2×Q1614yes—
418C22×C4C2×Q1612yes—
419C22×C4#13712yes—
420C22×C4#13712yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424M16—84no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—44no—
435C2×C8—42no—
436C2×C8—84no—
437C2×C8—84no—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4C2×C22⋊C414yes—
461C4⋊C4C2×C22⋊C418yes—
462C22⋊C4—216no—
463C22⋊C4—28no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C42—216no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—28no—
475C42C2×C22⋊C412yes—
476C42C2×C22⋊C412yes—
477C42#12012yes—
478C42#12012yes—
479C42#12012yes—
480C42#12012yes—
481C42C22×D411yescharacteristic
482C42C22×D411yescharacteristic
483C42C22×D411yescharacteristic
484C42C22×D411yescharacteristic
485C23—22no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23#37912yes—
492C23C2×#10314yes—
493C23C2×#10312yes—
494C23#41912yes—
495C23C22×C22⋊C411yescharacteristic
496Q8—28no—
497Q8—44no—
498Q8—44no—
499Q8—44no—
500Q8—42no—
501Q8—44no—
502Q8—42no—
503Q8—48no—
504Q8—44no—
505Q8—88no—
506Q8—816no—
507D4—216no—
508D4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—88no—
539C2×C4—816no—
540C2×C4#37912yes—
541C2×C4#37912yes—
542C2×C4#37912yes—
543C2×C4C2×#10314yes—
544C2×C4C2×#10312yes—
545C2×C4#41912yes—
546C2×C4#44814yes—
547C2×C4#44814yes—
548C2×C4#45114yes—
549C2×C4#45114yes—
550C2×C4C22×C22⋊C411yescharacteristic
551C2×C4C22×C22⋊C411yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×#12011yescharacteristic
555C2×C4C2×#12011yescharacteristic
556C2×C4C22×SD1611yescharacteristic
557C2×C4C22×Q1611yescharacteristic
558C2×C4C2×#13711yescharacteristic
559C2×C4C2×#13711yescharacteristic
560C2×C4#57612yes—
561C2×C4#57712yes—
562C8—82no—
563C8—82no—
564C8—82no—
565C8—82no—
566C22—24no—
567C22—24no—
568C22—22no—
569C22—22no—
570C22—28no—
571C22—24no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22#178714yes—
576C22C2×#37911yescharacteristic
577C22#288912yes—
578C22C22×#10311yescharacteristic
579C22C2×#41911yescharacteristic
580C22#289712yes—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—24no—
585C4—22no—
586C4—42no—
587C4—44no—
588C4—42no—
589C4—48no—
590C4—44no—
591C4#178714yes—
592C4C2×#44811yescharacteristic
593C4C2×#44811yescharacteristic
594C4C2×#45111yescharacteristic
595C4C2×#45111yescharacteristic
596C4#300811yescharacteristic
597C4#300911yescharacteristic
598C4#301212yes—
599C2—24no—
600C2—44no—
601C2#1772312yes—
602C2#1776112yes—
603C2#2059311yescharacteristic
604C2C2×#178711yescharacteristic
605C2#2060211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1550C2semidirect
5. C2×#1549C2semidirect
6. C2×#1542C2semidirect
7. C22×#332C2semidirect
8. #8343C2semidirect
9. #8340C2semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
#218472
#206292
C2×#15502
C2×#15492
C2×#15422
C22×#3322
#83432
#83402