Finite Group Databasebeta

← #185238

Subgroups — #185238

Order 512.

2300 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2#21849C211yesmaximal, characteristic
3#21849C211yesmaximal, characteristic
4C2×#1561C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1546C211yesmaximal, characteristic, complemented
7C2×#1544C211yesmaximal, characteristic, complemented
8#8345C214yesmaximal, complemented
9#8340C214yesmaximal, complemented
10#3503—24no—
11#3503—24no—
12#3503—24no—
13#3503—24no—
14C2×Q82—21no—
15C2×#557—21no—
16C2×#554—22no—
17C2×#554—21no—
18C2×D4×Q8—22no—
19C2×D4×Q8—21no—
20C2×#544C414yes—
21C2×#544C412yes—
22C2×#533C2212yes—
23C2×#533C2211yescharacteristic
24C22×C4⋊Q8C2212yes—
25C22×C4⋊Q8C2211yescharacteristic
26C2×#520C412yes—
27C2×C4×#41C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#423C2211yescharacteristic
32#1984—21no—
33#1984—21no—
34#1954—21no—
35#1954—21no—
36#1870—22no—
37#1870—22no—
38#1797C2214yes—
39#1797C2214yes—
40#1561—24no—
41#1549—24no—
42#1546—24no—
43#1544—24no—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2214yes—
47C2×#332C2214yes—
48C2×#332C2212yes—
49C2×#332C2212yes—
50C22×#41C2×C411yescharacteristic
51C23×Q8—22no—
52C23×Q8—21no—
53C22×M16—21no—
54C22×M16—21no—
55Q82—28no—
56Q82—28no—
57#557—28no—
58#557—28no—
59#554—28no—
60#554—28no—
61#554—216no—
62#554—216no—
63D4×Q8—216no—
64D4×Q8—216no—
65D4×Q8—28no—
66D4×Q8—28no—
67#544—216no—
68#544—216no—
69#544—28no—
70#544—28no—
71#533—24no—
72#533—24no—
73#533—24no—
74#533—24no—
75#533—22no—
76#533—22no—
77#533D414yes—
78#533D414yes—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2×C414yes—
92C2×C4⋊Q8C2×C412yes—
93C2×C4⋊Q8D414yes—
94C2×C4⋊Q8D414yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125—24no—
99C2×#125—22no—
100C2×#125C2×C414yes—
101C2×#125C2×C412yes—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109#520—28no—
110#520—28no—
111C4×#41—22no—
112C4×#41—22no—
113C4×#41—22no—
114C4×#41—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×D4—24no—
123C2×C4×D4—22no—
124C2×C4×D4D412yes—
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×#117C2×C414yes—
129C2×#117C2×C412yes—
130C2×#117C2×C411yescharacteristic
131C2×#117C2×C412yes—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2×C414yes—
135C22×C4⋊C4C2×C412yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#431—22no—
139#431—22no—
140#431—22no—
141#431—22no—
142#423—22no—
143#423—22no—
144#423—22no—
145#423—22no—
146C2×C4⋊C8—21no—
147C2×C4⋊C8—21no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8—24no—
156C2×C4×C8—21no—
157C2×C4×C8—21no—
158#336—24no—
159#336—24no—
160#332—24no—
161#332—24no—
162#332—28no—
163#332—28no—
164#332—24no—
165#332—24no—
166C2×#41—24no—
167C2×#41—24no—
168C2×#41—22no—
169C2×#41—22no—
170C22×Q8—28no—
171C22×Q8—28no—
172C22×Q8—24no—
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—21no—
179C22×Q8—82no—
180C22×Q8—82no—
181C22×Q8—84no—
182C22×Q8—84no—
183C22×Q8C22⋊C411yescharacteristic
184C22×D4C22⋊C412yes—
185C22×D4C22⋊C411yescharacteristic
186C23×C4—24no—
187C23×C4—22no—
188C23×C4C22×C412yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—82no—
197C2×M16—82no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C4⋊Q8—28no—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—28no—
206C4⋊Q8—232no—
207C4⋊Q8—216no—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8C22⋊C418yes—
222C4⋊Q8C2×D412yes—
223C4⋊Q8C2×D412yes—
224C4⋊Q8C2×D414yes—
225C4⋊Q8C2×D414yes—
226C4⋊Q8C2×D412yes—
227C4⋊Q8C2×D412yes—
228#125—232no—
229#125—216no—
230#125—28no—
231#125—28no—
232#125—24no—
233#125C22⋊C418yes—
234C22⋊Q8—28no—
235C22⋊Q8—28no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8—232no—
239C22⋊Q8—232no—
240C22⋊Q8—232no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C4×Q8—216no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×D4—232no—
259C4×D4—216no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268#117—22no—
269#117—22no—
270#117—24no—
271#117—28no—
272#117—28no—
273#117—24no—
274#117—24no—
275#117C22⋊C418yes—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22⋊C418yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C414yes—
301C2×C4⋊C4C22×C412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C22⋊C412yes—
305C2×C22⋊C4C22⋊C412yes—
306C2×C22⋊C4C22⋊C412yes—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42C22×C412yes—
314C2×C42C22×C411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2411yescharacteristic
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—42no—
333C4⋊C8—42no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336C22⋊C8—84no—
337C22⋊C8—84no—
338C4×C8—42no—
339C4×C8—42no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C412yes—
342#41—216no—
343#41—216no—
344C2×Q8—216no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—28no—
353C2×Q8—22no—
354C2×Q8—44no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—84no—
361C2×Q8—88no—
362C2×Q8—84no—
363C2×Q8—84no—
364C2×Q8—88no—
365C2×Q8—88no—
366C2×Q8—88no—
367C2×Q8—816no—
368C2×Q8#10114yes—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4#10114yes—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—84no—
399C22×C4—88no—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4#12012yes—
409C22×C4#12012yes—
410C22×C4#12012yes—
411C22×C4#12012yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415M16—84no—
416M16—84no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—44no—
423C2×C8—42no—
424C2×C8—44no—
425C2×C8—42no—
426C2×C8—42no—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4C2×C22⋊C414yes—
449C4⋊C4C2×C22⋊C418yes—
450C4⋊C4C2×C22⋊C414yes—
451C22⋊C4—216no—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C42—216no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—28no—
464C42C2×C22⋊C412yes—
465C42C2×C22⋊C412yes—
466C42#12012yes—
467C42#12012yes—
468C42#12012yes—
469C42#12012yes—
470C42C22×D411yescharacteristic
471C42C22×D411yescharacteristic
472C42C22×D411yescharacteristic
473C42C22×D411yescharacteristic
474C23—21no—
475C23—21no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23#37912yes—
483C23C2×#10114yes—
484C23C22×C22⋊C411yescharacteristic
485Q8—28no—
486Q8—44no—
487Q8—42no—
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—48no—
493Q8—44no—
494Q8—44no—
495Q8—88no—
496Q8—816no—
497D4—216no—
498D4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
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—48no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—88no—
532C2×C4—816no—
533C2×C4#37912yes—
534C2×C4#37912yes—
535C2×C4#37912yes—
536C2×C4C2×#10114yes—
537C2×C4C2×#10112yes—
538C2×C4C2×#10112yes—
539C2×C4C22×C22⋊C411yescharacteristic
540C2×C4C22×C22⋊C411yescharacteristic
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×#12011yescharacteristic
543C2×C4C2×#12011yescharacteristic
544C2×C4C2×#12011yescharacteristic
545C2×C4#57612yes—
546C2×C4#57712yes—
547C2×C4#57712yes—
548C2×C4#57812yes—
549C8—82no—
550C8—82no—
551C8—82no—
552C8—82no—
553C22—22no—
554C22—24no—
555C22—28no—
556C22—22no—
557C22—22no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22C2×#37911yescharacteristic
564C22C22×#10111yescharacteristic
565C22C22×#10111yescharacteristic
566C22#289712yes—
567C22#289812yes—
568C4—21no—
569C4—21no—
570C4—24no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—48no—
575C4—44no—
576C4—42no—
577C4—44no—
578C4#178414yes—
579C4#300811yescharacteristic
580C4#300911yescharacteristic
581C4#301011yescharacteristic
582C4#301111yescharacteristic
583C4#301212yes—
584C4#301312yes—
585C2—42no—
586C2—44no—
587C2#1777412yes—
588C2#1777712yes—
589C2C2×#178411yescharacteristic
590C2#2060211yescharacteristic
591C2#2060311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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×#1561C2semidirect
5. C2×#1549C2semidirect
6. C2×#1546C2semidirect
7. C2×#1544C2semidirect
8. #8345C2semidirect
9. #8340C2semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
#218492
C2×#15612
C2×#15492
C2×#15462
C2×#15442
#83452
#83402