Finite Group Databasebeta

← #1393184

Subgroups — #1393184

Order 512.

5388 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#548C211yesmaximal, characteristic
2C2×#2449C213yesmaximal, complemented
3C2×#2616C211yesmaximal, characteristic, complemented
4C22×#395C211yesmaximal, characteristic, complemented
5#16510C213yesmaximal, complemented
6#16505C213yesmaximal, complemented
7#16264C213yesmaximal, complemented
8#16254C213yesmaximal, complemented
9#15158C211yesmaximal, characteristic, complemented
10#14889C211yesmaximal, characteristic, complemented
11#14874C211yesmaximal, characteristic, complemented
12#14747C213yesmaximal, complemented
13#14719C213yesmaximal, complemented
14#14302C211yesmaximal, characteristic, complemented
15#13899C213yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C2×#548—224no—
19C22×#124C2213yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2213yes—
22C22×C22⋊Q8C2213yes—
23C22×#121C2213yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2213yes—
26C23×C4⋊C4C2213yes—
27#2616—28no—
28#2449—224no—
29C2×#399C2213yes—
30C2×#398C2213yes—
31C2×#396C2213yes—
32C2×#395—28no—
33C2×#395C2211yescharacteristic
34C2×#395C2211yescharacteristic
35C2×#395C2213yes—
36C2×#395C2213yes—
37C2×#395C2213yes—
38C2×#395C2213yes—
39C2×#395C2211yescharacteristic
40C2×#394C2213yes—
41C2×#394C2213yes—
42C2×#394C2213yes—
43C2×#389C2213yes—
44C2×#387C2213yes—
45C2×#386C2213yes—
46C2×#386C2213yes—
47C2×#386C2216yes—
48C2×#385C2213yes—
49C2×#385C2213yes—
50C2×#385C2213yes—
51C2×#380C2213yes—
52C22×#95C2213yes—
53#1446C2211yescharacteristic
54#1446C2213yes—
55#1442C2213yes—
56#1441C2213yes—
57#1441C2213yes—
58#1441C2216yes—
59#1441C2213yes—
60#1440C2216yes—
61#1439C2216yes—
62#1439C2216yes—
63#1439C2216yes—
64#1439C2216yes—
65#1438C2213yes—
66C4⋊#95C2213yes—
67#1434C2213yes—
68C22⋊#95C2213yes—
69C22⋊#95C2213yes—
70C22×#41—24no—
71C22×#41—212no—
72C22×#41—46no—
73C23×Q8—24no—
74C23×Q8—212no—
75C23×Q8—46no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2313yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2313yes—
80C23×D4C2313yes—
81C24×C4C2313yes—
82C24×C4C2311yescharacteristic
83C24×C4C2313yes—
84C24×C4C2313yes—
85#548—2128no—
86C2×#124—224no—
87C2×#124—212no—
88C2×C22⋊Q8—212no—
89C2×C22⋊Q8—212no—
90C2×C22⋊Q8—224no—
91C2×C22⋊Q8—212no—
92C2×C22⋊Q8—212no—
93C2×C22⋊Q8—46no—
94C2×#121—224no—
95C2×#121—212no—
96C2×C4×Q8—212no—
97C2×C4×D4—224no—
98C2×C4×D4D4124yes—
99C22×C4⋊C4—212no—
100C22×C4⋊C4—212no—
101C22×C4⋊C4—46no—
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2313yes—
108C22×C4⋊C4C2313yes—
109C22×C4⋊C4C2313yes—
110C22×C4⋊C4C2313yes—
111C22×C4⋊C4C2313yes—
112C22×C4⋊C4C2313yes—
113C22×C4⋊C4C2313yes—
114C22×C4⋊C4C2313yes—
115C22×C22⋊C4C2316yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2313yes—
119C22×C22⋊C4C2313yes—
120C22×C22⋊C4C2313yes—
121C22×C22⋊C4C2313yes—
122C22×C22⋊C4C2313yes—
123C22×C22⋊C4C2316yes—
124C22×C22⋊C4C2316yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C42C2313yes—
127#399—212no—
128#398—212no—
129#396—212no—
130#395—24no—
131#395—24no—
132#395—212no—
133#395—212no—
134#395—212no—
135#395—46no—
136#395D418yescomplemented
137#394—212no—
138#394—212no—
139#394—46no—
140#389—212no—
141#387—212no—
142#386—212no—
143#386—212no—
144#386—412no—
145#385—212no—
146#385—212no—
147#385—212no—
148#380—212no—
149C2×#95—212no—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2316yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2313yes—
161C2×#95C2313yes—
162C2×#95C2316yes—
163C2×#95C2316yes—
164C2×#95C2316yes—
165C2×#95C2313yes—
166C25C2413yes—
167C25C2411yescharacteristic
168C2×#41—232no—
169C2×#41—448no—
170C2×#41—448no—
171C22×Q8—232no—
172C22×Q8—26no—
173C22×Q8—212no—
174C22×Q8—26no—
175C22×Q8—22no—
176C22×Q8—26no—
177C22×Q8—26no—
178C22×Q8—26no—
179C22×Q8—26no—
180C22×Q8—46no—
181C22×Q8—43no—
182C22×Q8—448no—
183C22×Q8—448no—
184C22×Q8C2×D414yes—
185C22×D4—212no—
186C22×D4—212no—
187C22×D4—26no—
188C22×D4—212no—
189C22×D4—26no—
190C22×D4—26no—
191C22×D4—26no—
192C22×D4—26no—
193C22×D4—46no—
194C22×D4C2×D4112yes—
195C23×C4—212no—
196C23×C4—26no—
197C23×C4—26no—
198C23×C4—26no—
199C23×C4—212no—
200C23×C4—26no—
201C23×C4—26no—
202C23×C4—26no—
203C23×C4—26no—
204C23×C4—26no—
205C23×C4—26no—
206C23×C4—26no—
207C23×C4—22no—
208C23×C4—43no—
209C23×C4—43no—
210C23×C4—46no—
211C23×C4—46no—
212C23×C4—43no—
213C23×C4C2×D4112yes—
214C23×C4C2×D4112yes—
215C23×C4C2411yescharacteristic
216C23×C4C2413yes—
217C23×C4C2411yescharacteristic
218C23×C4C2413yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2413yes—
222C23×C4C2413yes—
223C23×C4C2413yes—
224C23×C4C2413yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2413yes—
228#124—296no—
229C22⋊Q8—296no—
230C22⋊Q8—448no—
231C22⋊Q8—416no—
232#121—296no—
233C4×Q8—232no—
234C4×D4—296no—
235C2×C4⋊C4—26no—
236C2×C4⋊C4—212no—
237C2×C4⋊C4—212no—
238C2×C4⋊C4—26no—
239C2×C4⋊C4—26no—
240C2×C4⋊C4—26no—
241C2×C4⋊C4—26no—
242C2×C4⋊C4—26no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—26no—
246C2×C4⋊C4—26no—
247C2×C4⋊C4—26no—
248C2×C4⋊C4—26no—
249C2×C4⋊C4—26no—
250C2×C4⋊C4—26no—
251C2×C4⋊C4—26no—
252C2×C4⋊C4—26no—
253C2×C4⋊C4—26no—
254C2×C4⋊C4—26no—
255C2×C4⋊C4—26no—
256C2×C4⋊C4—26no—
257C2×C4⋊C4—26no—
258C2×C4⋊C4—448no—
259C2×C4⋊C4—43no—
260C2×C4⋊C4—43no—
261C2×C4⋊C4—43no—
262C2×C4⋊C4—43no—
263C2×C4⋊C4—46no—
264C2×C4⋊C4C2×D4112yes—
265C2×C4⋊C4C2×D4112yes—
266C2×C22⋊C4—212no—
267C2×C22⋊C4—212no—
268C2×C22⋊C4—212no—
269C2×C22⋊C4—212no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—26no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—26no—
274C2×C22⋊C4—26no—
275C2×C22⋊C4—26no—
276C2×C22⋊C4—26no—
277C2×C22⋊C4—26no—
278C2×C22⋊C4—26no—
279C2×C22⋊C4—26no—
280C2×C22⋊C4—26no—
281C2×C22⋊C4—212no—
282C2×C22⋊C4—212no—
283C2×C22⋊C4—43no—
284C2×C22⋊C4—43no—
285C2×C22⋊C4—43no—
286C2×C22⋊C4—46no—
287C2×C22⋊C4—46no—
288C2×C22⋊C4—46no—
289C2×C22⋊C4—46no—
290C2×C22⋊C4—46no—
291C2×C22⋊C4C2×D4112yes—
292C2×C22⋊C4C2×D4112yes—
293C2×C42—212no—
294C2×C42C2×D4112yes—
295#95—26no—
296#95—212no—
297#95—26no—
298#95—26no—
299#95—26no—
300#95—26no—
301#95—26no—
302#95—26no—
303#95—26no—
304#95—26no—
305#95—26no—
306#95—26no—
307#95—46no—
308#95—46no—
309#95—46no—
310#95C2×D4112yes—
311C24—21no—
312C24—23no—
313C24—23no—
314C24—21no—
315C24—23no—
316C24—26no—
317C24—23no—
318C24—23no—
319C24—26no—
320C24—23no—
321C24—26no—
322C24—26no—
323C24—23no—
324C24—23no—
325C24C22×D416yes—
326C24C22×D416yes—
327C24H(5,𝔽2)12yes—
328C242−1+416yes—
329C24C2511yescharacteristic
330#41—4128no—
331C2×Q8—28no—
332C2×Q8—224no—
333C2×Q8—224no—
334C2×Q8—224no—
335C2×Q8—424no—
336C2×Q8—424no—
337C2×Q8—412no—
338C2×Q8—412no—
339C2×Q8—4128no—
340C2×Q8—44no—
341C2×Q8—412no—
342C2×Q8—86no—
343C2×D4—224no—
344C2×D4—224no—
345C2×D4—224no—
346C2×D4—224no—
347C2×D4—424no—
348C2×D4—424no—
349C2×D4—412no—
350C2×D4—424no—
351C2×D4—424no—
352C2×D4—412no—
353C22×C4—26no—
354C22×C4—21no—
355C22×C4—23no—
356C22×C4—26no—
357C22×C4—23no—
358C22×C4—21no—
359C22×C4—224no—
360C22×C4—224no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—26no—
367C22×C4—26no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—26no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—26no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—224no—
384C22×C4—224no—
385C22×C4—224no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—26no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—26no—
404C22×C4—23no—
405C22×C4—21no—
406C22×C4—23no—
407C22×C4—412no—
408C22×C4—412no—
409C22×C4—424no—
410C22×C4—424no—
411C22×C4—412no—
412C22×C4—412no—
413C22×C4—412no—
414C22×C4—412no—
415C22×C4—424no—
416C22×C4—424no—
417C22×C4—412no—
418C22×C4—412no—
419C22×C4—412no—
420C22×C4—44no—
421C22×C4—86no—
422C22×C4C22×D416yes—
423C22×C4C22×D416yes—
424C22×C4C22×D416yes—
425C22×C4C22×D412yes—
426C22×C4C22×D416yes—
427C22×C4C22×D416yes—
428C22×C4C22×D416yes—
429C22×C4C22×D416yes—
430C22×C4H(5,𝔽2)16yes—
431C22×C42−1+416yes—
432C4⋊C4—224no—
433C4⋊C4—224no—
434C4⋊C4—224no—
435C4⋊C4—412no—
436C4⋊C4—412no—
437C4⋊C4—412no—
438C4⋊C4—412no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—412no—
443C4⋊C4—412no—
444C4⋊C4—86no—
445C22⋊C4—248no—
446C22⋊C4—224no—
447C22⋊C4—412no—
448C22⋊C4—412no—
449C22⋊C4—412no—
450C22⋊C4—44no—
451C22⋊C4—412no—
452C22⋊C4—812no—
453C22⋊C4—812no—
454C22⋊C4—86no—
455C22⋊C4#120116yes—
456C42—224no—
457C23—26no—
458C23—26no—
459C23—26no—
460C23—22no—
461C23—26no—
462C23—26no—
463C23—26no—
464C23—26no—
465C23—212no—
466C23—26no—
467C23—26no—
468C23—26no—
469C23—26no—
470C23—26no—
471C23—26no—
472C23—26no—
473C23—46no—
474C23—46no—
475C23—46no—
476C23—46no—
477C23—43no—
478C23—46no—
479C23—43no—
480C23—46no—
481C23—46no—
482C23—46no—
483C23—43no—
484C23—43no—
485C23—46no—
486C23—46no—
487C23—46no—
488C23—46no—
489C23C2×#12014yes—
490C23#536112yes—
491C23#537112yes—
492C23C23×D413yes—
493C23C23×D411yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)13yes—
496C23C2×2−1+413yes—
497C23C2×2−1+413yes—
498C23#58511yescharacteristic
499Q8—232no—
500Q8—448no—
501Q8—88no—
502D4—448no—
503D4—448no—
504C2×C4—26no—
505C2×C4—26no—
506C2×C4—212no—
507C2×C4—26no—
508C2×C4—26no—
509C2×C4—212no—
510C2×C4—26no—
511C2×C4—26no—
512C2×C4—26no—
513C2×C4—26no—
514C2×C4—26no—
515C2×C4—26no—
516C2×C4—26no—
517C2×C4—26no—
518C2×C4—26no—
519C2×C4—26no—
520C2×C4—26no—
521C2×C4—26no—
522C2×C4—26no—
523C2×C4—26no—
524C2×C4—26no—
525C2×C4—448no—
526C2×C4—448no—
527C2×C4—448no—
528C2×C4—41no—
529C2×C4—46no—
530C2×C4—43no—
531C2×C4—43no—
532C2×C4—46no—
533C2×C4—43no—
534C2×C4—46no—
535C2×C4—46no—
536C2×C4—43no—
537C2×C4—43no—
538C2×C4—43no—
539C2×C4—43no—
540C2×C4—43no—
541C2×C4—43no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—43no—
546C2×C4—43no—
547C2×C4—43no—
548C2×C4—43no—
549C2×C4—43no—
550C2×C4—43no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4—43no—
554C2×C4—43no—
555C2×C4—43no—
556C2×C4—43no—
557C2×C4—43no—
558C2×C4—43no—
559C2×C4—46no—
560C2×C4—43no—
561C2×C4—43no—
562C2×C4—46no—
563C2×C4—88no—
564C2×C4C2×#12014yes—
565C2×C4C2×#12014yes—
566C2×C4D42112yes—
567C22—212no—
568C22—212no—
569C22—212no—
570C22—212no—
571C22—24no—
572C22—24no—
573C22—46no—
574C22—46no—
575C22—46no—
576C22—46no—
577C22—46no—
578C22—46no—
579C22—46no—
580C22—86no—
581C22—86no—
582C22—86no—
583C22—86no—
584C22#261618yescomplemented
585C22C22×#12011yescharacteristic
586C22C2×#53613yes—
587C22C2×#53713yes—
588C22C2×D4213yes—
589C22#348413yes—
590C22#348613yes—
591C22#348811yescharacteristic
592C22#348913yes—
593C22#349913yes—
594C22#353613yes—
595C22#354113yes—
596C22#355216yes—
597C4—24no—
598C4—24no—
599C4—212no—
600C4—46no—
601C4—46no—
602C4—46no—
603C4—46no—
604C4—42no—
605C4—83no—
606C4—83no—
607C4—81no—
608C4—81no—
609C4—81no—
610C2—28no—
611C2—86no—
612C2C2×#261611yescharacteristic
613C2#3608511yescharacteristic
614C2D4×#12011yescharacteristic
615C2#3764913yes—
616C2#3767513yes—
617C2#4049313yes—
618C2#4058913yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#548C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2449C2semidirect
3. C2×#2616C2semidirect
4. C22×#395C2semidirect
5. #16510C2semidirect
6. #16505C2semidirect
7. #16264C2semidirect
8. #16254C2semidirect
9. #15158C2semidirect
10. #14889C2semidirect
11. #14874C2semidirect
12. #14747C2semidirect
13. #14719C2semidirect
14. #14302C2semidirect
15. #13899C2semidirect
136. #395D4semidirect
584. C22#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#5482
C2×#24492
C2×#26162
C22×#3952
#165102
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