Finite Group Databasebeta

← #2736605

Subgroups — #2736605

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C212yesmaximal
2C2×#2440C212yesmaximal
3#17120C212yesmaximal, complemented
4#17099C214yesmaximal, complemented
5#17085C212yesmaximal, complemented
6#16675C212yesmaximal
7#16262C214yesmaximal
8#15150C211yesmaximal, characteristic
9#15146C212yesmaximal, complemented
10#15143C212yesmaximal, complemented
11#15139C212yesmaximal, complemented
12#15132C212yesmaximal
13#15128C212yesmaximal, complemented
14#13904C211yesmaximal, characteristic
15C4×#397C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2214yes—
18C23×C4⋊C4C2211yescharacteristic
19#2817—216no—
20#2440—216no—
21C2×#400C2214yes—
22C2×#400C2214yes—
23C2×#398C2214yes—
24C2×#397C2214yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2211yescharacteristic
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#385C2214yes—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2214yes—
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1443C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2211yescharacteristic
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2212yes—
67#1439C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×Q8C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—44no—
82C2×C4⋊Q8—44no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—48no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#400—216no—
115#400—216no—
116#398—216no—
117#397—28no—
118#397—216no—
119#397—44no—
120#397D418yes—
121#395—216no—
122#395—216no—
123#395—216no—
124#395—48no—
125#389—28no—
126#389—28no—
127#389—44no—
128#385—216no—
129#384—28no—
130#384—42no—
131#381—28no—
132#381—44no—
133#380—28no—
134#380—42no—
135C4×C4⋊C4—42no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—42no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—28no—
171C22×Q8—28no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—44no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C2×D414yes—
188C23×C4C2×D418yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C4⋊Q8—432no—
204C22⋊Q8—464no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—416no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C2×D418yes—
267C2×C22⋊C4C2×D414yes—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—41no—
298#95—41no—
299#95—41no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95C2×D418yes—
308#95C2×D414yes—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—21no—
317C24C22×D412yes—
318C24C22×D412yes—
319C24H(5,𝔽2)12yes—
320C24C2511yescharacteristic
321C2×Q8—416no—
322C2×Q8—416no—
323C2×Q8—88no—
324C2×Q8—88no—
325C2×Q8—88no—
326C2×Q8—88no—
327C2×Q8—88no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D418yes—
418C22×C4C22×D412yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+414yes—
423C22×C42−1+414yes—
424C22×C42−1+414yes—
425C22×C42−1+414yes—
426C4⋊C4—216no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—84no—
441C22⋊C4—216no—
442C22⋊C4—216no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—416no—
446C22⋊C4—84no—
447C42—48no—
448C42—48no—
449C42—44no—
450C42—44no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23D4214yes—
474C23C23×D411yescharacteristic
475C23C23×D411yescharacteristic
476C23C2×H(5,𝔽2)11yescharacteristic
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×2−1+412yes—
481C23C2×2−1+412yes—
482C23C2×2−1+412yes—
483C23C2×2−1+412yes—
484C23#58511yescharacteristic
485Q8—832no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—416no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—816no—
569C2×C4#53414yes—
570C2×C4#53618yes—
571C2×C4#56314yes—
572C2×C4#56314yes—
573C2×C4#56318yes—
574C22—28no—
575C22—24no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22C2×#53411yescharacteristic
586C22C2×#53612yes—
587C22C2×D4211yescharacteristic
588C22#348612yes—
589C22#349611yescharacteristic
590C22#349611yescharacteristic
591C22#349811yescharacteristic
592C22#349912yes—
593C22#349912yes—
594C22C2×#56311yescharacteristic
595C22C2×#56312yes—
596C22C2×#56311yescharacteristic
597C22#353711yescharacteristic
598C22#353811yescharacteristic
599C22#354412yes—
600C22#354412yes—
601C22#354412yes—
602C22#355011yescharacteristic
603C22#355212yes—
604C22#355212yes—
605C22#355512yes—
606C22#355811yescharacteristic
607C22#356312yes—
608C4—42no—
609C4—42no—
610C4—44no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—48no—
615C4—44no—
616C4—44no—
617C4—84no—
618C4—84no—
619C4—84no—
620C4—81no—
621C4#281718yescomplemented
622C2—44no—
623C2C2×#281711yescharacteristic
624C2#3933712yes—
625C2#3935811yescharacteristic
626C2#4243612yes—
627C2#4280711yescharacteristic
628C2#4548811yescharacteristic
629C2#4599312yes—
630C2#4614711yescharacteristic
631C2#5296811yescharacteristic
632C2#5303111yescharacteristic
633C2#5303812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17120C2semidirect
4. #17099C2semidirect
5. #17085C2semidirect
9. #15146C2semidirect
10. #15143C2semidirect
11. #15139C2semidirect
13. #15128C2semidirect
621. C4#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#24402
#171202
#170992
#170852
#166752
#162622
#151502
#151462
#151432
#151392
#151322
#151282
#139042
C4×#3972