Finite Group Databasebeta

← #1409130

Subgroups — #1409130

Order 512.

4091 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2416C213yesmaximal, complemented
2C2×#2425C213yesmaximal, complemented
3#17071C213yesmaximal, complemented
4#16984C213yesmaximal, complemented
5#16807C213yesmaximal, complemented
6#16363C213yesmaximal, complemented
7#15211C211yesmaximal, characteristic, complemented
8#14926C213yesmaximal, complemented
9#14777C213yesmaximal, complemented
10#14757C211yesmaximal, characteristic, complemented
11#14757C211yesmaximal, characteristic, complemented
12#14353C211yesmaximal, characteristic, complemented
13#14321C211yesmaximal, characteristic, complemented
14C2×#1444C211yesmaximal, characteristic, complemented
15D4×#95C211yesmaximal, characteristic, complemented
16C22×#123C2211yescharacteristic
17C22×#123C2213yes—
18C22×#121C2213yes—
19C22×#121C2213yes—
20C22×#121C2213yes—
21C22×C4×D4C2213yes—
22C23×C4⋊C4C2213yes—
23#2425—224no—
24#2416—224no—
25C2×#399C2216yes—
26C2×#398C2213yes—
27C2×#397C2213yes—
28C2×#396C2216yes—
29C2×#396C2213yes—
30C2×#394C2213yes—
31C2×#394C2216yes—
32C2×#392C2211yescharacteristic
33C2×#390C2213yes—
34C2×#389C2213yes—
35C2×#389C2213yes—
36C2×#388C2213yes—
37C2×#386C2216yes—
38C2×#386C2213yes—
39C2×#386C2213yes—
40C2×#385C2213yes—
41C2×#380C2213yes—
42C2×#380C2213yes—
43C22×#95C2211yescharacteristic
44C22×#95C2213yes—
45C22×#95C2211yescharacteristic
46#1445C2216yes—
47#1444C2211yescharacteristic
48#1444C2213yes—
49#1444C2211yescharacteristic
50#1444C2213yes—
51#1444C2213yes—
52#1444C2211yescharacteristic
53#1444C2213yes—
54#1444C2213yes—
55#1444C2213yes—
56#1439C2216yes—
57#1439C2216yes—
58#1439C2213yes—
59#1439C2213yes—
60#1439C2213yes—
61#1438C2213yes—
62#1438C2213yes—
63#1438C2213yes—
64C4⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C4×#95C2211yescharacteristic
68C23×D4C2313yes—
69C23×D4C2313yes—
70C23×D4C2311yescharacteristic
71C23×D4C2313yes—
72C24×C4C2311yescharacteristic
73C24×C4C2313yes—
74C24×C4C2313yes—
75C24×C4C2313yes—
76C2×#123—212no—
77C2×#123—212no—
78C2×#123—412no—
79C2×#121—224no—
80C2×#121—212no—
81C2×#121—212no—
82C2×#121—212no—
83C2×#121—412no—
84C2×#121—412no—
85C2×C4×D4—412no—
86C2×C4×D4D4124yescomplemented
87C22×C4⋊C4—212no—
88C22×C4⋊C4—412no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2313yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2313yes—
93C22×C4⋊C4C2316yes—
94C22×C4⋊C4C2313yes—
95C22×C4⋊C4C2313yes—
96C22×C4⋊C4C2313yes—
97C22×C4⋊C4C2313yes—
98C22×C4⋊C4C2313yes—
99C22×C4⋊C4C2313yes—
100C22×C4⋊C4C2313yes—
101C22×C22⋊C4C2313yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2313yes—
104C22×C22⋊C4C2313yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2316yes—
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2316yes—
111C22×C22⋊C4C2316yes—
112C22×C22⋊C4C2313yes—
113C22×C22⋊C4C2313yes—
114C22×C22⋊C4C2313yes—
115C22×C42C2313yes—
116#399—412no—
117#398—212no—
118#397—212no—
119#396—224no—
120#396—212no—
121#394—224no—
122#394—212no—
123#392—24no—
124#390—212no—
125#389—212no—
126#389—212no—
127#388—212no—
128#386—212no—
129#386—212no—
130#386—224no—
131#385—212no—
132#380—212no—
133#380—212no—
134C2×#95—212no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95C2313yes—
138C2×#95C2316yes—
139C2×#95C2313yes—
140C2×#95C2316yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2316yes—
151C2×#95C2311yescharacteristic
152C25C2413yes—
153C25C2411yescharacteristic
154C25C2411yescharacteristic
155C22×D4—26no—
156C22×D4—22no—
157C22×D4—212no—
158C22×D4—26no—
159C22×D4—26no—
160C22×D4—212no—
161C22×D4—26no—
162C22×D4—212no—
163C22×D4—26no—
164C22×D4—26no—
165C22×D4—26no—
166C22×D4—26no—
167C22×D4—26no—
168C22×D4—46no—
169C22×D4—46no—
170C22×D4—46no—
171C22×D4—43no—
172C22×D4C2×D4112yes—
173C23×C4—212no—
174C23×C4—26no—
175C23×C4—26no—
176C23×C4—26no—
177C23×C4—26no—
178C23×C4—26no—
179C23×C4—26no—
180C23×C4—22no—
181C23×C4—212no—
182C23×C4—26no—
183C23×C4—26no—
184C23×C4—46no—
185C23×C4—46no—
186C23×C4—46no—
187C23×C4—46no—
188C23×C4—43no—
189C23×C4C2×D4112yes—
190C23×C4C2×D4112yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2413yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2413yes—
197C23×C4C2413yes—
198C23×C4C2413yes—
199C23×C4C2413yes—
200C23×C4C2413yes—
201C23×C4C2411yescharacteristic
202C23×C4C2413yes—
203#123—448no—
204#123—416no—
205#121—448no—
206#121—448no—
207#121—448no—
208C4×D4—448no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—212no—
211C2×C4⋊C4—26no—
212C2×C4⋊C4—212no—
213C2×C4⋊C4—26no—
214C2×C4⋊C4—212no—
215C2×C4⋊C4—26no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—26no—
220C2×C4⋊C4—26no—
221C2×C4⋊C4—212no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—448no—
225C2×C4⋊C4—46no—
226C2×C4⋊C4—46no—
227C2×C4⋊C4—43no—
228C2×C4⋊C4—46no—
229C2×C4⋊C4—46no—
230C2×C4⋊C4—46no—
231C2×C4⋊C4—46no—
232C2×C4⋊C4—46no—
233C2×C4⋊C4C2×D4112yes—
234C2×C22⋊C4—212no—
235C2×C22⋊C4—212no—
236C2×C22⋊C4—26no—
237C2×C22⋊C4—26no—
238C2×C22⋊C4—26no—
239C2×C22⋊C4—26no—
240C2×C22⋊C4—212no—
241C2×C22⋊C4—26no—
242C2×C22⋊C4—212no—
243C2×C22⋊C4—212no—
244C2×C22⋊C4—26no—
245C2×C22⋊C4—26no—
246C2×C22⋊C4—26no—
247C2×C22⋊C4—26no—
248C2×C22⋊C4—212no—
249C2×C22⋊C4—26no—
250C2×C22⋊C4—212no—
251C2×C22⋊C4—26no—
252C2×C22⋊C4—46no—
253C2×C22⋊C4—46no—
254C2×C22⋊C4—46no—
255C2×C22⋊C4—46no—
256C2×C22⋊C4—46no—
257C2×C22⋊C4—46no—
258C2×C22⋊C4—46no—
259C2×C22⋊C4—46no—
260C2×C22⋊C4—46no—
261C2×C22⋊C4C2×D4112yes—
262C2×C22⋊C4C2×D4112yes—
263C2×C42—46no—
264C2×C42C2×D4112yes—
265#95—26no—
266#95—26no—
267#95—26no—
268#95—26no—
269#95—26no—
270#95—212no—
271#95—26no—
272#95—26no—
273#95—212no—
274#95—26no—
275#95—26no—
276#95—26no—
277#95—26no—
278#95—46no—
279#95#4114yes—
280C24—21no—
281C24—23no—
282C24—23no—
283C24—21no—
284C24—21no—
285C24—23no—
286C24—23no—
287C24—23no—
288C24—21no—
289C24—21no—
290C24—23no—
291C24—26no—
292C24—23no—
293C24—26no—
294C24—26no—
295C24—23no—
296C24—23no—
297C24—23no—
298C24—23no—
299C24—26no—
300C24—23no—
301C24C22×D416yes—
302C24C22×D416yes—
303C24H(5,𝔽2)16yes—
304C242−1+412yes—
305C24C2511yescharacteristic
306C2×D4—224no—
307C2×D4—224no—
308C2×D4—424no—
309C2×D4—412no—
310C2×D4—412no—
311C2×D4—412no—
312C2×D4—412no—
313C2×D4—412no—
314C2×D4—424no—
315C2×D4—412no—
316C2×D4—424no—
317C2×D4—412no—
318C2×D4—412no—
319C2×D4—812no—
320C2×D4—812no—
321C2×D4—86no—
322C2×D4—812no—
323C2×D4#127116yes—
324C22×C4—23no—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—21no—
328C22×C4—224no—
329C22×C4—23no—
330C22×C4—26no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—26no—
335C22×C4—26no—
336C22×C4—26no—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—26no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—224no—
346C22×C4—224no—
347C22×C4—23no—
348C22×C4—23no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4—26no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—26no—
355C22×C4—21no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—23no—
359C22×C4—23no—
360C22×C4—26no—
361C22×C4—26no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—23no—
367C22×C4—23no—
368C22×C4—21no—
369C22×C4—412no—
370C22×C4—412no—
371C22×C4—412no—
372C22×C4—424no—
373C22×C4—424no—
374C22×C4—424no—
375C22×C4—412no—
376C22×C4—412no—
377C22×C4—44no—
378C22×C4—412no—
379C22×C4—412no—
380C22×C4—812no—
381C22×C4—812no—
382C22×C4—86no—
383C22×C4—812no—
384C22×C4C22×D416yes—
385C22×C4C22×D416yes—
386C22×C4C22×D416yes—
387C22×C4C22×D416yes—
388C22×C4C22×D416yes—
389C22×C4C2×#4116yes—
390C22×C4H(5,𝔽2)16yes—
391C22×C42−1+416yes—
392C4⋊C4—224no—
393C4⋊C4—412no—
394C4⋊C4—412no—
395C4⋊C4—424no—
396C4⋊C4—412no—
397C4⋊C4—412no—
398C4⋊C4—412no—
399C4⋊C4—412no—
400C4⋊C4—412no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—86no—
404C22⋊C4—224no—
405C22⋊C4—412no—
406C22⋊C4—424no—
407C22⋊C4—424no—
408C22⋊C4—412no—
409C22⋊C4—412no—
410C22⋊C4—412no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—412no—
415C22⋊C4—412no—
416C22⋊C4—86no—
417C22⋊C4—812no—
418C42—412no—
419C23—26no—
420C23—26no—
421C23—26no—
422C23—26no—
423C23—22no—
424C23—26no—
425C23—212no—
426C23—26no—
427C23—26no—
428C23—26no—
429C23—26no—
430C23—26no—
431C23—26no—
432C23—26no—
433C23—26no—
434C23—41no—
435C23—43no—
436C23—46no—
437C23—43no—
438C23—43no—
439C23—43no—
440C23—43no—
441C23—46no—
442C23—43no—
443C23—46no—
444C23—43no—
445C23—46no—
446C23—46no—
447C23—43no—
448C23—46no—
449C23—43no—
450C23—43no—
451C23—46no—
452C23—43no—
453C23—43no—
454C23—46no—
455C23—46no—
456C23—46no—
457C23—46no—
458C23—46no—
459C23—43no—
460C23—46no—
461C23—43no—
462C23C2×#12714yes—
463C23C2×#12714yes—
464C23#534112yes—
465C23#537112yes—
466C23C23×D413yes—
467C23C22×#4111yescharacteristic
468C23C2×H(5,𝔽2)13yes—
469C23C2×H(5,𝔽2)13yes—
470C23C2×2−1+411yescharacteristic
471C23C2×2−1+413yes—
472C23#58511yescharacteristic
473D4—232no—
474D4—824no—
475D4—824no—
476D4—824no—
477C2×C4—26no—
478C2×C4—26no—
479C2×C4—26no—
480C2×C4—26no—
481C2×C4—212no—
482C2×C4—212no—
483C2×C4—26no—
484C2×C4—26no—
485C2×C4—26no—
486C2×C4—26no—
487C2×C4—26no—
488C2×C4—26no—
489C2×C4—212no—
490C2×C4—26no—
491C2×C4—26no—
492C2×C4—26no—
493C2×C4—26no—
494C2×C4—448no—
495C2×C4—43no—
496C2×C4—43no—
497C2×C4—46no—
498C2×C4—46no—
499C2×C4—46no—
500C2×C4—46no—
501C2×C4—46no—
502C2×C4—46no—
503C2×C4—46no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—43no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—43no—
510C2×C4—43no—
511C2×C4—43no—
512C2×C4—46no—
513C2×C4—43no—
514C2×C4—43no—
515C2×C4—43no—
516C2×C4—46no—
517C2×C4—43no—
518C2×C4—43no—
519C2×C4—43no—
520C2×C4—46no—
521C2×C4—43no—
522C2×C4—46no—
523C2×C4—43no—
524C2×C4—43no—
525C2×C4—43no—
526C2×C4—824no—
527C2×C4—824no—
528C2×C4—88no—
529C2×C4C2×#12714yes—
530C2×C4#547112yes—
531C22—212no—
532C22—24no—
533C22—24no—
534C22—212no—
535C22—212no—
536C22—212no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—46no—
541C22—46no—
542C22—46no—
543C22—46no—
544C22—83no—
545C22—86no—
546C22—83no—
547C22—83no—
548C22—83no—
549C22—86no—
550C22—83no—
551C22—83no—
552C22—86no—
553C22—83no—
554C22—86no—
555C22#265618yes—
556C22C22×#12711yescharacteristic
557C22C2×#53413yes—
558C22C2×#53713yes—
559C22C2×#54713yes—
560C22#348413yes—
561C22#348813yes—
562C22#349913yes—
563C22#351611yescharacteristic
564C22#351913yes—
565C22#353616yes—
566C22#355213yes—
567C22#355913yes—
568C4—24no—
569C4—212no—
570C4—46no—
571C4—46no—
572C4—46no—
573C4—46no—
574C4—81no—
575C4—81no—
576C4—83no—
577C4—81no—
578C4—83no—
579C4—81no—
580C2—28no—
581C2—44no—
582C2—163no—
583C2C2×#265611yescharacteristic
584C2#3592111yescharacteristic
585C2#3617911yescharacteristic
586C2#3803613yes—
587C2#3804213yes—
588C2#4049313yes—
589C2#4193713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2416C2semidirect
2. C2×#2425C2semidirect
3. #17071C2semidirect
4. #16984C2semidirect
5. #16807C2semidirect
6. #16363C2semidirect
7. #15211C2semidirect
8. #14926C2semidirect
9. #14777C2semidirect
10. #14757C2semidirect
11. #14757C2semidirect
12. #14353C2semidirect
13. #14321C2semidirect
14. C2×#1444C2semidirect
15. D4×#95C2semidirect
86. C2×C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24162
C2×#24252
#170712
#169842
#168072
#163632
#152112
#149262
#147772
#147572
#143532
#143212
C2×#14442
D4×#952