Finite Group Databasebeta

← #626970

Subgroups — #626970

Order 512.

2464 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic
3C2×#2629C211yesmaximal, characteristic
4C2×#2330C211yesmaximal, characteristic, complemented
5#15620C214yesmaximal
6#15618C214yesmaximal, complemented
7#15518C214yesmaximal, complemented
8#15361C214yesmaximal
9#15220C212yesmaximal
10#15137C212yesmaximal, complemented
11#14712C211yesmaximal, characteristic, complemented
12#14283C211yesmaximal, characteristic, complemented
13#14006C212yesmaximal, complemented
14#13876C212yesmaximal
15#13761C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2859—28no—
20#2629—28no—
21#2629—28no—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27#2330—28no—
28C2×#389C2214yes—
29C2×#389C2214yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#384C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×#376C2211yescharacteristic
48C2×C43C2211yescharacteristic
49#1444C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#125—24no—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125—24no—
81C2×#125D418yes—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115#398—216no—
116#398—216no—
117#398—48no—
118#396—216no—
119#396—216no—
120#389—216no—
121#389—28no—
122#389—216no—
123#389—216no—
124#389—28no—
125#389—28no—
126#389—28no—
127#389—48no—
128#388—48no—
129#387—216no—
130#384—28no—
131#384—24no—
132#384D418yes—
133#382—28no—
134#380—28no—
135#380—28no—
136C4×C4⋊C4—28no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4Q818yes—
139#376—24no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C43—24no—
165C25C2411yescharacteristic
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4C2×Q814yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187#125—232no—
188#125—416no—
189#117—416no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
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—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C2×D418yes—
232C2×C4⋊C4C2×D4116yes—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×D4116yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C2×Q818yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—28no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42C2×D414yes—
273C2×C42C2×D414yes—
274C2×C42C2×Q814yes—
275C2×C42C2×Q814yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
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—22no—
289#95—22no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—42no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95C2×Q818yes—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24C22×Q812yes—
313C24C2511yescharacteristic
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4C22×D418yes—
393C22×C4C22×D418yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q814yes—
400C22×C4C22×Q814yes—
401C22×C4C22×Q812yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C42−1+414yes—
409C22×C42−1+412yes—
410C22×C42−1+412yes—
411C4⋊C4—232no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—416no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C22⋊C4—216no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C42—28no—
433C42—28no—
434C42—28no—
435C42—48no—
436C42—48no—
437C42—44no—
438C42—44no—
439C42—88no—
440C42#121116yes—
441C23—22no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C23×D411yescharacteristic
465C23C23×D411yescharacteristic
466C23C23×Q811yescharacteristic
467C23C22×#4111yescharacteristic
468C23C22×#4111yescharacteristic
469C23C2×H(5,𝔽2)12yes—
470C23C2×2−1+411yescharacteristic
471C23C2×2−1+411yescharacteristic
472C23C2×2−1+412yes—
473C23#58512yes—
474C23#58512yes—
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—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4C2×#12118yes—
558C2×C4C2×#12114yes—
559C2×C4#53314yes—
560C2×C4#54114yes—
561C2×C4#54418yes—
562C2×C4#54714yes—
563C2×C4#54814yes—
564C2×C4D4×Q814yes—
565C2×C4D4×Q814yes—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—82no—
577C22C22×#12111yescharacteristic
578C22C2×#53311yescharacteristic
579C22C2×#54111yescharacteristic
580C22#345511yescharacteristic
581C22#345511yescharacteristic
582C22C2×#54412yes—
583C22C2×#54711yescharacteristic
584C22C2×#54811yescharacteristic
585C22C2×D4×Q811yescharacteristic
586C22C2×D4×Q811yescharacteristic
587C22#348812yes—
588C22#348912yes—
589C22#349612yes—
590C22#349712yes—
591C22#350412yes—
592C22#350811yescharacteristic
593C22#350911yescharacteristic
594C22#351112yes—
595C22#351212yes—
596C22#354712yes—
597C22#355512yes—
598C22#355912yes—
599C22#356312yes—
600C4—28no—
601C4—28no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—44no—
606C4—48no—
607C4—48no—
608C4—84no—
609C4—81no—
610C4—81no—
611C4—82no—
612C4—84no—
613C4#242918yes—
614C2—82no—
615C2C2×#242911yescharacteristic
616C2#3529411yescharacteristic
617C2#3533611yescharacteristic
618C2#3556412yes—
619C2#3732812yes—
620C2#3768712yes—
621C2#3770911yescharacteristic
622C2#3771011yescharacteristic
623C2#3937012yes—
624C2#3937211yescharacteristic
625C2#3937311yescharacteristic
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
1. C2×#2859C2semidirect
4. C2×#2330C2semidirect
6. #15618C2semidirect
7. #15518C2semidirect
10. #15137C2semidirect
11. #14712C2semidirect
12. #14283C2semidirect
13. #14006C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#26292
C2×#23302
#156202
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