Finite Group Databasebeta

← #535385

Subgroups — #535385

Order 512.

2730 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2346C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4#14385C214yesmaximal, complemented
5#14383C214yesmaximal
6#14151C212yesmaximal, complemented
7#14089C214yesmaximal, complemented
8#14072C214yesmaximal
9#14030C212yesmaximal, complemented
10#14029C212yesmaximal, complemented
11#13939C211yesmaximal, characteristic
12#13924C211yesmaximal, characteristic
13#13852C211yesmaximal, characteristic
14#13804C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2346—28no—
21#2344—28no—
22#2317C4116yescomplemented
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#381C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2212yes—
46#1445C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1443C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2212yes—
57#1440C2212yes—
58#1439C2212yes—
59#1439C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2214yes—
77C4×#95C2212yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—28no—
83C2×#117—28no—
84C2×#117—48no—
85C2×#117C2×C418yes—
86C2×#117C2×C418yes—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4C2×C418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119#389—28no—
120#389—28no—
121#389—48no—
122#389C2×C4132yes—
123#388—216no—
124#387—216no—
125#387—48no—
126#385C2×C4132yes—
127#384C2×C418yes—
128#384C2×C418yes—
129#382—28no—
130#382C2×C418yes—
131#382C2×C418yes—
132#381—28no—
133#380—24no—
134#380D418yes—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—28no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141#376—28no—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C25C2411yescharacteristic
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C22×C414yes—
178C23×C4C22×C418yes—
179C23×C4C2×D414yes—
180C23×C4#4114yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195#117—232no—
196#117—232no—
197#117—432no—
198C2×C4⋊C4—232no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
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—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4C22×C4116yes—
239C2×C4⋊C4C22×C4116yes—
240C2×C4⋊C4C2×D418yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C22×C418yes—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C2×D418yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C22×C414yes—
277C2×C42C22×C418yes—
278C2×C42C22×C414yes—
279C2×C42#4118yes—
280#95—22no—
281#95—28no—
282#95—22no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95C22×C4116yes—
306#95C22×C4116yes—
307#95C2×D418yes—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24C23×C412yes—
315C24C22×D412yes—
316C24C2×#4112yes—
317C24C2511yescharacteristic
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—416no—
394C22×C4C4×D4116yes—
395C22×C4C23×C414yes—
396C22×C4C23×C414yes—
397C22×C4C23×C414yes—
398C22×C4C23×C414yes—
399C22×C4C23×C414yes—
400C22×C4C23×C414yes—
401C22×C4C23×C412yes—
402C22×C4C23×C412yes—
403C22×C4C22×D418yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C2×#4114yes—
407C22×C4C2×#4114yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4112yes—
411C22×C4H(5,𝔽2)14yes—
412C22×C42−1+414yes—
413C22×C42−1+414yes—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—216no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—48no—
427C4⋊C4—88no—
428C22⋊C4—28no—
429C22⋊C4—216no—
430C22⋊C4—216no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—84no—
435C42—28no—
436C42—216no—
437C42—416no—
438C42—416no—
439C42—48no—
440C42—48no—
441C42—88no—
442C42#117116yescomplemented
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C2×C4×D414yes—
464C23C24×C411yescharacteristic
465C23C23×D411yescharacteristic
466C23C22×#4112yes—
467C23C22×#4111yescharacteristic
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+412yes—
470C23C2×2−1+412yes—
471C23#58512yes—
472C23#58512yes—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—232no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—432no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4C2×#11718yes—
555C2×C4C2×#11714yes—
556C2×C4C2×C4×D414yes—
557C2×C4C2×C4×D414yes—
558C2×C4#51914yes—
559C2×C4#54718yes—
560C2×C4#55218yes—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22C22×#11711yescharacteristic
571C22C22×C4×D411yescharacteristic
572C22C2×#51911yescharacteristic
573C22C4×H(5,𝔽2)12yes—
574C22C4×2−1+412yes—
575C22C2×#54712yes—
576C22C2×#55212yes—
577C22#348411yescharacteristic
578C22#348711yescharacteristic
579C22#348912yes—
580C22#349412yes—
581C22#349512yes—
582C22#350612yes—
583C22#350911yescharacteristic
584C22#351011yescharacteristic
585C22#351112yes—
586C22#351212yes—
587C22#354412yes—
588C22#354712yes—
589C22#356312yes—
590C22#356512yes—
591C4—24no—
592C4—28no—
593C4—48no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—44no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4#234418yes—
605C2—44no—
606C2C2×#234411yescharacteristic
607C2#3425312yes—
608C2#3426511yescharacteristic
609C2#3426811yescharacteristic
610C2#3443512yes—
611C2#4492812yes—
612C2#4495612yes—
613C2#4495812yes—
614C2#4526412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14385C2semidirect
6. #14151C2semidirect
7. #14089C2semidirect
9. #14030C2semidirect
10. #14029C2semidirect
14. #13804C2semidirect
22. #2317C4semidirect
442. C42#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23462
C2×#23442
C2×#23172
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