Finite Group Databasebeta

← #879801

Subgroups — #879801

Order 512.

2810 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C212yesmaximal, complemented
2C2×#2429C212yesmaximal, complemented
3#16863C214yesmaximal, complemented
4#15666C214yesmaximal
5#15626C214yesmaximal, complemented
6#15236C212yesmaximal, complemented
7#15177C212yesmaximal
8#15131C212yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14707C211yesmaximal, characteristic, complemented
11#14565C214yesmaximal, complemented
12#14017C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19#2429—216no—
20#2421—216no—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#397C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C43C2211yescharacteristic
45#1444C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C2×C4⋊Q8—42no—
73C2×C4⋊Q8—42no—
74C2×C4⋊Q8D418yescomplemented
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—44no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8—44no—
81C2×C4×Q8—28no—
82C2×C4×Q8—28no—
83C2×C4×Q8—44no—
84C2×C4×D4—24no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#400—216no—
116#398—216no—
117#397—216no—
118#396—216no—
119#396—216no—
120#391—28no—
121#391—28no—
122#391—28no—
123#390—28no—
124#389—28no—
125#389—216no—
126#389—28no—
127#388—216no—
128#388—216no—
129#387—216no—
130#384—42no—
131#382—42no—
132#382D418yescomplemented
133#380—28no—
134#380—44no—
135C4×C4⋊C4—42no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—28no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C43—42no—
161C25C2412yes—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8C2×D418yes—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—41no—
173C22×D4—41no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C4⋊Q8—416no—
194C22⋊Q8—432no—
195C22⋊Q8—432no—
196C4×Q8—432no—
197C4×D4—416no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D418yes—
233C2×C4⋊C4C2×D4116yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4#4118yes—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42#4114yes—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—41no—
300#95—41no—
301#95—44no—
302#95—44no—
303#95C2×D4116yes—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C2×#4114yes—
319C24C2511yescharacteristic
320C2×Q8—216no—
321C2×Q8—216no—
322C2×Q8—216no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×D4—28no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—44no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4C22×D414yes—
403C22×C4C22×D418yes—
404C22×C4C22×D418yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4112yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+414yes—
417C22×C42−1+414yes—
418C22×C42−1+412yes—
419C4⋊C4—216no—
420C4⋊C4—416no—
421C4⋊C4—48no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—44no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—48no—
430C4⋊C4—88no—
431C22⋊C4—216no—
432C22⋊C4—48no—
433C22⋊C4—416no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—48no—
439C22⋊C4—88no—
440C42—28no—
441C42—28no—
442C42—28no—
443C42—48no—
444C42—416no—
445C42—48no—
446C42—48no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C23×D411yescharacteristic
483C23C23×D411yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4112yes—
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×2−1+411yescharacteristic
490C23C2×2−1+412yes—
491C23C2×2−1+412yes—
492C23#58512yes—
493Q8—432no—
494D4—416no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—816no—
567C2×C4#53414yes—
568C2×C4#53714yes—
569C2×C4#53814yes—
570C2×C4#54114yes—
571C2×C4D4214yes—
572C2×C4#54714yes—
573C2×C4#54818yes—
574C2×C4#55014yes—
575C2×C4#56214yes—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22#344411yescharacteristic
592C22C2×#53411yescharacteristic
593C22C2×#53711yescharacteristic
594C22C2×#53811yescharacteristic
595C22C2×#54111yescharacteristic
596C22C2×D4211yescharacteristic
597C22C2×#54711yescharacteristic
598C22C2×#54812yes—
599C22C2×#55011yescharacteristic
600C22D4×#4112yes—
601C22#349712yes—
602C22#349912yes—
603C22#350911yescharacteristic
604C22#350912yes—
605C22#351211yescharacteristic
606C22#351412yes—
607C22#352012yes—
608C22#352012yes—
609C22C2×#56211yescharacteristic
610C22#354112yes—
611C22#354712yes—
612C22#355212yes—
613C22#355512yes—
614C4—24no—
615C4—28no—
616C4—48no—
617C4—44no—
618C4—42no—
619C4—44no—
620C4—48no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4—84no—
625C4#268218yes—
626C2—84no—
627C2#3508611yescharacteristic
628C2#3613611yescharacteristic
629C2#3615111yescharacteristic
630C2#3739311yescharacteristic
631C2#3916612yes—
632C2#3937911yescharacteristic
633C2#3938511yescharacteristic
634C2#3995912yes—
635C2#4605512yes—
636C2C2×#268211yescharacteristic
637C2#5965412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2421C2semidirect
2. C2×#2429C2semidirect
3. #16863C2semidirect
5. #15626C2semidirect
6. #15236C2semidirect
8. #15131C2semidirect
9. #14714C2semidirect
10. #14707C2semidirect
11. #14565C2semidirect
12. #14017C2semidirect
74. C2×C4⋊Q8D4semidirect
132. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24212
C2×#24292
#168632
#156662
#156262
#152362
#151772
#151312
#147142
#147072
#145652
#140172
C4×#3902