Finite Group Databasebeta

← #822373

Subgroups — #822373

Order 512.

2846 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C212yesmaximal
2C2×#2551C212yesmaximal
3#17072C212yesmaximal
4#15762C212yesmaximal
5#15625C214yesmaximal
6#15510C212yesmaximal
7#15143C214yesmaximal
8#15143C212yesmaximal
9#15128C211yesmaximal, characteristic
10#14236C212yesmaximal
11#14045C212yesmaximal
12#14005C212yesmaximal
13#13910C211yesmaximal, characteristic
14C4×#395C211yesmaximal, characteristic
15#13818C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C4⋊Q8C2214yes—
19C22×C4×Q8C2211yescharacteristic
20#2567—216no—
21#2551—216no—
22C2×#395C2214yes—
23C2×#395C2214yes—
24C2×#395C2214yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#395C2212yes—
28C2×#395C2214yes—
29C2×#395C2211yescharacteristic
30C2×#391C2214yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#384C2214yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#1446C2212yes—
49#1446C2211yescharacteristic
50#1444C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2211yescharacteristic
56#1444C2214yes—
57#1444C2214yes—
58#1444C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2211yescharacteristic
63#1441C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C23×Q8C2314yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—216no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—44no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—48no—
87C2×C4⋊Q8—48no—
88C2×C4×Q8—28no—
89C2×C4×Q8—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120#395—216no—
121#395—216no—
122#395—216no—
123#395—216no—
124#395—48no—
125#395—44no—
126#395—48no—
127#395Q818yescomplemented
128#391—216no—
129#389—28no—
130#389—28no—
131#389—28no—
132#389—28no—
133#389—216no—
134#389—48no—
135#389D4116yes—
136#389Q8116yescomplemented
137#384—216no—
138#384—28no—
139#384—42no—
140#384—44no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—42no—
145C4×C4⋊C4Q818yescomplemented
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2312yes—
172C22×Q8—24no—
173C22×Q8—28no—
174C22×Q8—28no—
175C22×Q8—28no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8—42no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—42no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8C2×Q814yes—
189C23×C4C2411yescharacteristic
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C4⋊Q8—432no—
205C4⋊Q8—416no—
206C4⋊Q8—464no—
207C4×Q8—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D4116yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×Q818yes—
266C2×C4⋊C4C2×Q818yes—
267C2×C4⋊C4C2×Q8116yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C4⋊C4C2×Q818yes—
270C2×C4⋊C4C2×Q818yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D418yes—
290C2×C42C2×Q814yes—
291C2×C42C2×Q818yes—
292C2×C42C2×Q818yes—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—42no—
308#95—41no—
309#95—41no—
310#95—41no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95C2×D4116yes—
315#95C2×Q818yes—
316#95C2×Q8116yes—
317#95C2×Q814yes—
318#95C2×Q818yes—
319C24C2511yescharacteristic
320C2×Q8—216no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—44no—
325C2×Q8—416no—
326C2×Q8—48no—
327C2×Q8—416no—
328C2×Q8—416no—
329C2×Q8—88no—
330C2×Q8—88no—
331C2×Q8—88no—
332C2×Q8—88no—
333C2×Q8—84no—
334C2×Q8—88no—
335C2×Q8—84no—
336C2×Q8—88no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4C22×D418yes—
406C22×C4C22×D414yes—
407C22×C4C22×D418yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×Q812yes—
411C22×C4C22×Q818yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q814yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q818yes—
416C22×C4C22×Q818yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q814yes—
419C22×C4C22×Q814yes—
420C22×C4C22×Q812yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q812yes—
423C22×C4C22×Q814yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+414yes—
429C22×C42−1+412yes—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—216no—
433C4⋊C4—216no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—88no—
446C4⋊C4—84no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—84no—
451C4⋊C4C4⋊Q8132yescomplemented
452C42—28no—
453C42—216no—
454C42—216no—
455C42—48no—
456C42—416no—
457C42—416no—
458C42—44no—
459C42—44no—
460C23C23×D412yes—
461C23C23×Q811yescharacteristic
462C23C23×Q812yes—
463C23C23×Q811yescharacteristic
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×H(5,𝔽2)12yes—
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×2−1+411yescharacteristic
469C23C2×2−1+412yes—
470C23#58511yescharacteristic
471Q8—416no—
472Q8—816no—
473Q8—832no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4C2×C4⋊Q8116yes—
559C2×C4C2×C4⋊Q818yes—
560C2×C4#54414yes—
561C2×C4D4214yes—
562C2×C4D4×Q818yes—
563C2×C4D4×Q818yes—
564C2×C4Q8218yes—
565C2×C4Q8214yes—
566C2×C4Q8214yes—
567C22C22×C4⋊Q812yes—
568C22C2×#54411yescharacteristic
569C22C2×D4211yescharacteristic
570C22C2×D4×Q812yes—
571C22C2×D4×Q812yes—
572C22C2×Q8212yes—
573C22C2×Q8211yescharacteristic
574C22C2×Q8211yescharacteristic
575C22#348412yes—
576C22#348912yes—
577C22#349812yes—
578C22#350112yes—
579C22#350111yescharacteristic
580C22#350111yescharacteristic
581C22#350311yescharacteristic
582C22#350312yes—
583C22#350312yes—
584C22#350312yes—
585C22#353711yescharacteristic
586C22#353811yescharacteristic
587C22#354412yes—
588C22#354911yescharacteristic
589C22#355911yescharacteristic
590C4—28no—
591C4—216no—
592C4—28no—
593C4—24no—
594C4—44no—
595C4—48no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—48no—
601C4—42no—
602C4—81no—
603C4—84no—
604C4#255118yes—
605C2C2×#255111yescharacteristic
606C2#3592812yes—
607C2Q8×C4⋊Q812yes—
608C2#3962311yescharacteristic
609C2#4297311yescharacteristic
610C2#4487011yescharacteristic
611C2#4494112yes—
612C2#4543511yescharacteristic
613C2#5899811yescharacteristic
614C2#5915112yes—
615C2#5992411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
127. #395Q8semidirect
136. #389Q8semidirect
145. C4×C4⋊C4Q8semidirect
451. C4⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25512
#170722
#157622
#156252
#155102
#151432
#151282
#142362
#140452
#140052
#139102
C4×#3952
#138182