Finite Group Databasebeta

← #6445751

Subgroups — #6445751

Order 512.

3036 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#3166C211yesmaximal, characteristic
3C2×#3078C211yesmaximal, characteristic
4#34953C211yesmaximal, characteristic
5#25849C214yesmaximal, complemented
6#25848C214yesmaximal, complemented
7#25842C218yesmaximal, complemented
8#23553C212yesmaximal
9#23552C212yesmaximal
10#21314C211yesmaximal, characteristic
11#21303C211yesmaximal, characteristic
12#20874C212yesmaximal
13#20874C212yesmaximal
14C4×#419C211yesmaximal, characteristic
15C2×#557—28no—
16C2×#557—28no—
17C2×#557—22no—
18C2×#557—22no—
19C2×#557C2214yes—
20C2×#557C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125—21no—
25C22×#125—21no—
26C22×#125C2211yescharacteristic
27C22×#125C2211yescharacteristic
28C22×C4×Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2211yescharacteristic
33#3166—28no—
34C2×#501C2212yes—
35C2×#501C2212yes—
36C2×#499C2214yes—
37#3078—28no—
38C2×#479C2214yes—
39C2×#477C2212yes—
40C2×#475C2212yes—
41C2×#428C2212yes—
42C2×#428C2212yes—
43C2×#428C2212yes—
44C2×#423C2211yescharacteristic
45C2×#419C2212yes—
46C2×#419C2212yes—
47C2×C4×M16C2211yescharacteristic
48#2584C2214yes—
49#2583—21no—
50#2583—21no—
51#2567C2212yes—
52#2546C2212yes—
53#2502—21no—
54#2434—21no—
55#2399—22no—
56C4×C4⋊Q8—22no—
57#2296C2212yes—
58#2296C2212yes—
59C4×#117C2211yescharacteristic
60#2059C2218yes—
61#2059C2218yes—
62#2058C2218yes—
63#2057C2218yes—
64#1989C2214yes—
65#1989C2214yes—
66#1985C2214yes—
67#1984C2214yes—
68#1865C2218yes—
69#1865C2218yes—
70#1863C2218yes—
71#1863C2218yes—
72#1763C2214yes—
73C4×#103C2214yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1728C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C23×Q8D411yescharacteristic
81C23×Q8C2311yescharacteristic
82C22×M16C2312yes—
83C22×M16C2312yes—
84#557—232no—
85#557—232no—
86#557—216no—
87#557—216no—
88#557—216no—
89#557—216no—
90#533—24no—
91#533—24no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2314yes—
102C2×#125—216no—
103C2×#125—216no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—22no—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—24no—
114C2×#125D418yes—
115C2×#125C2312yes—
116C2×#125C2314yes—
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C4×Q8—216no—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×C4×Q8—24no—
129C2×C4×Q8—22no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—22no—
133C2×C4×Q8—24no—
134C2×C4×Q8—24no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8D418yes—
138C2×C4×Q8D412yes—
139C2×C4×Q8D412yes—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2314yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2312yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C42—21no—
163C22×C42—21no—
164C22×C42—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168#501—28no—
169#501—28no—
170#499—216no—
171#479—216no—
172#477—28no—
173#475—28no—
174#428—28no—
175#428—28no—
176#428—28no—
177C2×#107C2314yes—
178C2×#107C2314yes—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182C2×#106C2314yes—
183#423—24no—
184C2×C4⋊C8C2312yes—
185C2×C4⋊C8C2312yes—
186#419—28no—
187#419Q8116yescomplemented
188C2×#103C2314yes—
189C2×#103C2314yes—
190C2×#103C2318yes—
191C4×M16—24no—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#398—22no—
195#398—22no—
196#398—24no—
197#395—24no—
198#395—22no—
199#395—22no—
200#391—22no—
201#391—22no—
202#389—24no—
203#389—24no—
204#389C2314yes—
205#389C2314yes—
206#389C2314yes—
207#389C2314yes—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#384—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—24no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219#376—21no—
220C43—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225#336C2318yes—
226C22×Q8—28no—
227C22×Q8—24no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—44no—
235C22×Q8—48no—
236C22×Q8C2×D414yes—
237C22×Q8C2×D412yes—
238C22×Q8C2×D412yes—
239C22×Q8C2×Q818yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—28no—
256C2×M16C2×Q818yes—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—232no—
260C4⋊Q8—232no—
261C4⋊Q8—216no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—48no—
269#125—232no—
270#125—232no—
271#125—232no—
272#125—232no—
273#125—216no—
274#125—28no—
275#125—28no—
276#125—28no—
277#125—28no—
278#125—24no—
279#125—24no—
280#125—24no—
281#125—24no—
282#125C2×D418yes—
283#125C2×D418yes—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C4×Q8—216no—
287C4×Q8—232no—
288C4×Q8—232no—
289C4×Q8—216no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8C2×D418yes—
307C4×Q8C2×D418yes—
308#117—24no—
309#117—24no—
310#117—24no—
311#117C2×Q818yes—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2414yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—28no—
366C2×C42—28no—
367C2×C42—28no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D412yes—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D414yes—
385C2×C42C2×D414yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—28no—
391#107—28no—
392#107—28no—
393#106—28no—
394#106—28no—
395#106—28no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398#103—28no—
399#103—28no—
400#103C2×Q8132yescomplemented
401#97—24no—
402C4×C8—24no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409C24C22×D411yescharacteristic
410C2×Q8—216no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—28no—
415C2×Q8—216no—
416C2×Q8—28no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—416no—
424C2×Q8—432no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—48no—
430C2×Q8—82no—
431C2×Q8C4⋊Q8116yescomplemented
432C2×Q8C22×D414yes—
433C2×Q8C22×Q818yes—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4#12014yes—
460C22×C4#12014yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×Q814yes—
473C22×C42−1+414yes—
474C22×C4C2511yescharacteristic
475M16—216no—
476M16—48no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—28no—
481C2×C8C22×Q818yes—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4C22×D414yes—
519C4⋊C4C22×D414yes—
520C4⋊C4C22×D414yes—
521C4⋊C4C22×Q818yes—
522C22⋊C4—28no—
523C22⋊C4—44no—
524C42—216no—
525C42—216no—
526C42—28no—
527C42—28no—
528C42—24no—
529C42—24no—
530C42—28no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—44no—
536C42—44no—
537C42—82no—
538C42#12014yes—
539C42#12014yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C42C22×D412yes—
544C23—22no—
545C23—22no—
546C23C2×#12012yes—
547C23D4×Q814yes—
548C23C23×D411yescharacteristic
549Q8—24no—
550Q8—216no—
551Q8—28no—
552Q8—48no—
553Q8—48no—
554Q8—432no—
555Q8—84no—
556Q8—84no—
557Q8C2×C4⋊Q818yes—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—216no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4C2×#12012yes—
597C2×C4C2×#12012yes—
598C2×C4C2×#12012yes—
599C2×C4C2×#12012yes—
600C2×C4C2×#12012yes—
601C2×C4C2×C4⋊Q814yes—
602C2×C4#53614yes—
603C2×C4D4×Q814yes—
604C2×C4D4×Q818yes—
605C2×C4#57514yes—
606C2×C4C23×D411yescharacteristic
607C2×C4C23×D411yescharacteristic
608C2×C4C23×Q812yes—
609C2×C4C2×2−1+412yes—
610C8—28no—
611C8—44no—
612C22—28no—
613C22—22no—
614C22—22no—
615C22#243414yes—
616C22C22×#12011yescharacteristic
617C22C2×D4×Q812yes—
618C22C2×#57512yes—
619C22#358612yes—
620C4—28no—
621C4—24no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—48no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—82no—
634C4—82no—
635C4#243414yes—
636C4C22×C4⋊Q811yescharacteristic
637C4C2×#53611yescharacteristic
638C4C2×D4×Q812yes—
639C2—24no—
640C2C2×#243411yescharacteristic
641C2#6035911yescharacteristic
642C2#6036811yescharacteristic
643C2#6136914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #25849C2semidirect
6. #25848C2semidirect
7. #25842C2semidirect
187. #419Q8semidirect
400. #103C2×Q8semidirect
431. C2×Q8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#31662
C2×#30782
#349532
#258492
#258482
#258422
#235532
#235522
#213142
#213032
#208742
C4×#4192