Finite Group Databasebeta

← #185192

Subgroups — #185192

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 601 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×#1548C211yesmaximal, characteristic, complemented
6C2×#1542C211yesmaximal, characteristic, complemented
7C22×#333C211yesmaximal, characteristic, complemented
8#8343C214yesmaximal, complemented
9#8342C214yesmaximal, complemented
10#3503—24no—
11#3503—24no—
12#3503—24no—
13#3503—24no—
14C2×Q82—21no—
15C2×#557C412yes—
16C2×#554—22no—
17C2×#554—21no—
18C2×D4×Q8C414yes—
19C2×D4×Q8C412yes—
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#1548—24no—
42#1542—24no—
43C2×#333—24no—
44C2×#333C2214yes—
45C2×#333C2212yes—
46C2×#333C2214yes—
47C2×#333C2212yes—
48C2×#332C2212yes—
49C2×#332C2212yes—
50C22×#41D411yescharacteristic
51C23×Q8C2×C414yes—
52C23×Q8C2×C412yes—
53C22×M16—21no—
54C23×C8—21no—
55C23×C8—21no—
56Q82—28no—
57Q82—28no—
58#557—28no—
59#557—28no—
60#554—28no—
61#554—28no—
62#554—216no—
63#554—216no—
64D4×Q8—216no—
65D4×Q8—28no—
66D4×Q8—216no—
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⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
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⋊C4—22no—
136C22×C4⋊C4—21no—
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—28no—
162#333—24no—
163#333—28no—
164#333—24no—
165#332—24no—
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—21no—
179C22×Q8—88no—
180C22×Q8—84no—
181C22×Q8C22⋊C418yes—
182C22×Q8C22⋊C411yescharacteristic
183C22×Q8C22×C414yes—
184C22×Q8C22×C412yes—
185C22×D4C22⋊C412yes—
186C22×D4C22⋊C411yescharacteristic
187C23×C4—24no—
188C23×C4—22no—
189C23×C4C22×C412yes—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—82no—
207C4⋊Q8—28no—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—232no—
212C4⋊Q8—216no—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8—24no—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8C22⋊C418yes—
227C4⋊Q8C2×D412yes—
228C4⋊Q8C2×D412yes—
229C4⋊Q8C2×D414yes—
230C4⋊Q8C2×D414yes—
231C4⋊Q8C2×D412yes—
232C4⋊Q8C2×D412yes—
233#125—28no—
234#125—28no—
235#125—232no—
236#125—216no—
237#125—24no—
238#125—24no—
239C22⋊Q8—28no—
240C22⋊Q8—28no—
241C22⋊Q8—232no—
242C22⋊Q8—232no—
243C22⋊Q8—232no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—28no—
249C22⋊Q8C22⋊C418yes—
250C4×Q8—216no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8C22⋊C418yes—
263C4×D4—232no—
264C4×D4—216no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273#117—28no—
274#117—28no—
275#117—24no—
276#117—24no—
277#117—22no—
278#117—22no—
279#117—24no—
280#117—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
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—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C22⋊C411yescharacteristic
303C2×C4⋊C4C22⋊C412yes—
304C2×C4⋊C4C22⋊C411yescharacteristic
305C2×C4⋊C4C22⋊C412yes—
306C2×C4⋊C4C22×C412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4C22⋊C412yes—
310C2×C22⋊C4C22⋊C412yes—
311C2×C22⋊C4C22⋊C412yes—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42C22×C412yes—
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2411yescharacteristic
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340C22⋊C8—42no—
341C22⋊C8—48no—
342C22⋊C8—84no—
343C4×C8—42no—
344C24C2×C22⋊C412yes—
345C24C2×C22⋊C411yescharacteristic
346#41—28no—
347#41—28no—
348#41—28no—
349#41—28no—
350C2×Q8—216no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—28no—
357C2×Q8—22no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—84no—
363C2×Q8—88no—
364C2×Q8—88no—
365C2×Q8—816no—
366C2×Q8—816no—
367C2×Q8—88no—
368C2×Q8#10314yes—
369C2×Q8C2×C22⋊C418yes—
370C2×Q8C2×C22⋊C414yes—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4#10318yes—
376C2×D4#10314yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—84no—
397C22×C4—88no—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4#12012yes—
407C22×C4#12012yes—
408C22×C4#12012yes—
409C22×C4#12012yes—
410C22×C4C2×SD1614yes—
411C22×C4C2×SD1612yes—
412C22×C4C2×Q1614yes—
413C22×C4C2×Q1612yes—
414C22×C4#13712yes—
415C22×C4#13712yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419M16—84no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—44no—
430C2×C8—42no—
431C2×C8—84no—
432C2×C8—84no—
433C4⋊C4—216no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4C2×C22⋊C414yes—
457C22⋊C4—216no—
458C22⋊C4—28no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C42—216no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—28no—
470C42C2×C22⋊C412yes—
471C42C2×C22⋊C412yes—
472C42#12012yes—
473C42#12012yes—
474C42#12012yes—
475C42#12012yes—
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C42C22×D411yescharacteristic
479C42C22×D411yescharacteristic
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23#37912yes—
487C23C2×#10314yes—
488C23C2×#10312yes—
489C23#41912yes—
490C23C22×C22⋊C411yescharacteristic
491Q8—28no—
492Q8—44no—
493Q8—48no—
494Q8—44no—
495Q8—48no—
496Q8—42no—
497Q8—42no—
498Q8—44no—
499Q8—88no—
500Q8—816no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
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—28no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—88no—
533C2×C4—816no—
534C2×C4#37912yes—
535C2×C4#37912yes—
536C2×C4#37912yes—
537C2×C4C2×#10314yes—
538C2×C4C2×#10312yes—
539C2×C4#41912yes—
540C2×C4#44814yes—
541C2×C4#44814yes—
542C2×C4#45114yes—
543C2×C4#45114yes—
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C22×C22⋊C411yescharacteristic
546C2×C4C2×#12011yescharacteristic
547C2×C4C2×#12011yescharacteristic
548C2×C4C2×#12011yescharacteristic
549C2×C4C2×#12011yescharacteristic
550C2×C4C22×SD1611yescharacteristic
551C2×C4C22×Q1611yescharacteristic
552C2×C4C2×#13711yescharacteristic
553C2×C4C2×#13711yescharacteristic
554C2×C4#57712yes—
555C2×C4#57812yes—
556C8—82no—
557C8—82no—
558C8—82no—
559C8—82no—
560C22—24no—
561C22—24no—
562C22—22no—
563C22—22no—
564C22—28no—
565C22—24no—
566C22—44no—
567C22—42no—
568C22—42no—
569C22#178714yes—
570C22C2×#37911yescharacteristic
571C22#288812yes—
572C22C22×#10311yescharacteristic
573C22C2×#41911yescharacteristic
574C22#289812yes—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—24no—
579C4—22no—
580C4—42no—
581C4—48no—
582C4—44no—
583C4—42no—
584C4—44no—
585C4#178714yes—
586C4C2×#44811yescharacteristic
587C4C2×#44811yescharacteristic
588C4C2×#45111yescharacteristic
589C4C2×#45111yescharacteristic
590C4#301011yescharacteristic
591C4#301111yescharacteristic
592C4#301312yes—
593C2—24no—
594C2—44no—
595C2#1772212yes—
596C2#1776012yes—
597C2#2059211yescharacteristic
598C2C2×#178711yescharacteristic
599C2#2060311yescharacteristic
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×#1548C2semidirect
6. C2×#1542C2semidirect
7. C22×#333C2semidirect
8. #8343C2semidirect
9. #8342C2semidirect

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×#15482
C2×#15422
C22×#3332
#83432
#83422