Finite Group Databasebeta

← #364795

Subgroups — #364795

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2962C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×C4×#101C211yesmaximal, characteristic
6#11216C212yesmaximal
7#11062C212yesmaximal
8#10942C212yesmaximal
9#10896C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10853C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078C418yescomplemented
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19#2962—24no—
20#2962—24no—
21C2×#446—21no—
22C2×#446—22no—
23C2×C8×Q8—21no—
24C2×#428C2212yes—
25C2×#424C2212yes—
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C2×#419C2212yes—
29C22×#103C2212yes—
30C22×#101C2211yescharacteristic
31C22×#101C2211yescharacteristic
32C2×#405C2212yes—
33C2×C4×M16C2211yescharacteristic
34C2×C4×M16C2211yescharacteristic
35C2×#384C2211yescharacteristic
36#1992—22no—
37#1990—21no—
38#1990—21no—
39#1803—21no—
40#1803—21no—
41#1803—22no—
42C4×#101—24no—
43C2×#343C2212yes—
44C2×#343C2211yescharacteristic
45C2×#343C2211yescharacteristic
46C2×#343C2212yes—
47C2×#341C2212yes—
48C2×#341C2212yes—
49#1736C2212yes—
50#1736C2212yes—
51C2×#340C2212yes—
52C23×Q8C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C23×C8C2312yes—
59#533—22no—
60#533—22no—
61#533C2×C414yes—
62#533D414yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×#125D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—22no—
77C2×C4×Q8C2×C414yes—
78C2×#117C2312yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#479—216no—
87#477—28no—
88#475—28no—
89#446—216no—
90#446—28no—
91C8×Q8—28no—
92#428—24no—
93#428C2×C418yes—
94C2×#107—22no—
95C2×#106—22no—
96#424—24no—
97#424—24no—
98#423—22no—
99#423C2×C414yes—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8D412yes—
109#419—24no—
110#419C2×C418yes—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—24no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—42no—
117C2×#101—22no—
118C2×#101—22no—
119C2×#101—22no—
120C2×#101—22no—
121C2×#101—22no—
122C2×#101Q814yescomplemented
123#405—24no—
124#405—24no—
125C4×M16—22no—
126C4×M16—22no—
127C4×M16—22no—
128C4×M16D414yes—
129C2×#97—22no—
130C2×#97—21no—
131C2×#97—21no—
132C2×C4×C8—22no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135#384—21no—
136#384—21no—
137#384—21no—
138#384—21no—
139#343—22no—
140#343—22no—
141#343—24no—
142#343—42no—
143#341—24no—
144#341—42no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8C22×C412yes—
165C22×Q8C2×Q812yes—
166C22×Q8#4112yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4C2411yescharacteristic
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—22no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16C22×C414yes—
197C2×M16C2×D412yes—
198C2×M16C2×D412yes—
199C2×M16C2×Q812yes—
200C2×M16C2×Q812yes—
201C2×M16#4112yes—
202C2×M16#4112yes—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8C22⋊C414yes—
219C4⋊Q8C22⋊C414yes—
220#125—24no—
221#125C22⋊C418yes—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×Q8—42no—
238#117—22no—
239#117—22no—
240#117C22×C414yes—
241#117C2×D414yes—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C22⋊C4—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—42no—
276C2×C42C22×C412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42#4112yes—
284#107—28no—
285#106—28no—
286C4⋊C8—28no—
287C4⋊C8—28no—
288C4⋊C8—24no—
289C4⋊C8—24no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—24no—
293#103—28no—
294#103—28no—
295#103—44no—
296#103—44no—
297#101—28no—
298#101—28no—
299#97—28no—
300#97—24no—
301#97—42no—
302C4×C8—28no—
303C4×C8—24no—
304C4×C8—42no—
305C24C2×#4111yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8C4×Q814yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C22×C42−1+411yescharacteristic
385M16—22no—
386M16—22no—
387M16—22no—
388M16—24no—
389M16—24no—
390M16—22no—
391M16—22no—
392M16—22no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—41no—
397M16—41no—
398M16—42no—
399M16—41no—
400M16—42no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16C22⋊Q814yes—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4C2×C22⋊C414yes—
451C4⋊C4C2×C22⋊C414yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C42—24no—
455C42—24no—
456C42—44no—
457C42—44no—
458C42C2×C22⋊C412yes—
459C42C2×C22⋊C412yes—
460C42C4×D412yes—
461C42C4×D412yes—
462C42#12012yes—
463C42#12012yes—
464C42#12112yes—
465C42#12112yes—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23C2×C4×D411yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×#12311yescharacteristic
473C23#52911yescharacteristic
474C23#53311yescharacteristic
475C23#54111yescharacteristic
476C23#54211yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486Q8—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—162no—
519C2×C4—162no—
520C2×C4#38612yes—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4#47112yes—
525C2×C4#47312yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×Q811yescharacteristic
529C2×C4C4×#4111yescharacteristic
530C2×C4#51911yescharacteristic
531C2×C4#52011yescharacteristic
532C2×C4C2×#12011yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#53711yescharacteristic
538C2×C4#54611yescharacteristic
539C2×C4#54711yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4#54811yescharacteristic
542C2×C4#54811yescharacteristic
543C2×C4#54811yescharacteristic
544C2×C4#54811yescharacteristic
545C2×C4D4×Q811yescharacteristic
546C2×C4D4×Q811yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4#55011yescharacteristic
549C2×C4#55011yescharacteristic
550C2×C4#55211yescharacteristic
551C2×C4#55211yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55611yescharacteristic
554C8—24no—
555C8—44no—
556C8—42no—
557C8—42no—
558C8—42no—
559C8—42no—
560C8—42no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22C2×#38611yescharacteristic
574C22C4×C22⋊Q811yescharacteristic
575C22#232311yescharacteristic
576C22#242111yescharacteristic
577C22#242311yescharacteristic
578C22#242611yescharacteristic
579C22#242911yescharacteristic
580C22C2×#47111yescharacteristic
581C22C2×#47311yescharacteristic
582C22#306811yescharacteristic
583C22#306911yescharacteristic
584C4—22no—
585C4—22no—
586C4—22no—
587C4—44no—
588C4—44no—
589C4—44no—
590C4—82no—
591C4—162no—
592C4#190814yes—
593C4Q8×C22⋊C411yescharacteristic
594C4#234111yescharacteristic
595C4#236011yescharacteristic
596C4#236211yescharacteristic
597C4#240711yescharacteristic
598C4#240911yescharacteristic
599C4#253611yescharacteristic
600C4#253811yescharacteristic
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2#1396911yescharacteristic
606C2C2×#190811yescharacteristic
607C2#2100411yescharacteristic
608C2#2531411yescharacteristic
609C2#2531511yescharacteristic
610C2#2549111yescharacteristic
611C2#2549211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #3078C4semidirect
122. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29622
#213222
C2×#18032
C2×C4×#1012
#112162
#110622
#109422
#108962
#108572
#108532