Finite Group Databasebeta

← #945707

Subgroups — #945707

Order 512.

3628 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2434C212yesmaximal, complemented
3C2×#2409C211yesmaximal, characteristic
4C2×#2621C212yesmaximal, complemented
5#17106C214yesmaximal, complemented
6#16522C214yesmaximal, complemented
7#16321C214yesmaximal, complemented
8#15342C214yesmaximal
9#15236C212yesmaximal, complemented
10#15190C212yesmaximal, complemented
11#15178C212yesmaximal, complemented
12#14003C212yesmaximal, complemented
13#13839C211yesmaximal, characteristic
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2621—216no—
24#2542—28no—
25#2434—216no—
26#2409—28no—
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#398C2218yes—
30C2×#397C2218yes—
31C2×#395C2212yes—
32C2×#395C2218yes—
33C2×#393C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2218yes—
43C2×#384C2212yes—
44C2×#382C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49#1444C2218yes—
50#1444C2218yes—
51#1444C2218yes—
52#1443C2218yes—
53#1442C2218yes—
54#1441C2212yes—
55#1440C2212yes—
56#1440C2218yes—
57#1439C2212yes—
58#1439C2218yes—
59#1438C2212yes—
60C4⋊#95C2212yes—
61#1434C2218yes—
62C4×#95C2212yes—
63C23×Q8—28no—
64C23×Q8—24no—
65C23×Q8—42no—
66C23×Q8—42no—
67C23×Q8—44no—
68C23×Q8C2312yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—28no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—44no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—44no—
83C2×C4⋊Q8—44no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4D418yes—
96C2×#117—28no—
97C2×#117—42no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2318yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#399—28no—
123#399—28no—
124#398—232no—
125#397—232no—
126#395—28no—
127#395—232no—
128#393—28no—
129#391—28no—
130#391—28no—
131#391—28no—
132#391—44no—
133#391D4116yescomplemented
134#390—28no—
135#386—28no—
136#385—28no—
137#385—232no—
138#384—28no—
139#382—28no—
140#380—24no—
141C4×C4⋊C4—24no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—42no—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2312yes—
157C2×#95C2318yes—
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—28no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—44no—
178C22×Q8—432no—
179C22×Q8—416no—
180C22×Q8—416no—
181C22×Q8—432no—
182C22×Q8—416no—
183C22×Q8C2×D418yes—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4C2×D414yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C4⋊Q8—432no—
216C4⋊Q8—416no—
217C4⋊Q8—432no—
218C22⋊Q8—432no—
219C22⋊Q8—432no—
220C22⋊Q8—432no—
221C4×D4—416no—
222#117—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42C2×D418yes—
289C2×C42C2×D414yes—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—216no—
294#95—28no—
295#95—216no—
296#95—28no—
297#95—216no—
298#95—28no—
299#95—28no—
300#95—216no—
301#95—42no—
302#95—48no—
303#95C2×D4132yes—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24C22×D412yes—
318C24C22×D412yes—
319C242−1+414yes—
320C24C2511yescharacteristic
321C2×Q8—216no—
322C2×Q8—216no—
323C2×Q8—216no—
324C2×Q8—416no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—416no—
328C2×Q8—4128no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4C22×D412yes—
410C22×C4C22×D418yes—
411C22×C4C22×D418yes—
412C22×C4C22×D412yes—
413C22×C4C22×D418yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D414yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C42−1+418yes—
422C22×C42−1+414yes—
423C4⋊C4—216no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—416no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—432no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—432no—
433C4⋊C4—48no—
434C22⋊C4—232no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C42—28no—
444C42—216no—
445C42—48no—
446C42—48no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—28no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—28no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—41no—
462C23—44no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—42no—
468C23—41no—
469C23—44no—
470C23—44no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23#53614yes—
479C23C23×D411yescharacteristic
480C23C23×D412yes—
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+412yes—
485C23C2×2−1+412yes—
486C23C2×2−1+414yes—
487Q8—432no—
488Q8—432no—
489Q8—816no—
490D4—416no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—216no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—432no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—416no—
537C2×C4—416no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4#53414yes—
551C2×C4#53518yes—
552C2×C4#53614yes—
553C2×C4D4214yes—
554C2×C4D4218yes—
555C2×C4#56014yes—
556C22—28no—
557C22—24no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—48no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—82no—
570C22—82no—
571C22—84no—
572C22C2×#53411yescharacteristic
573C22C2×#53512yes—
574C22C2×#53611yescharacteristic
575C22C2×#53611yescharacteristic
576C22C2×D4211yescharacteristic
577C22C2×D4212yes—
578C22#348614yes—
579C22#349914yes—
580C22#349912yes—
581C22C2×#56011yescharacteristic
582C22#353812yes—
583C22#353812yes—
584C22#353812yes—
585C22#355012yes—
586C22#355214yes—
587C22#355214yes—
588C4—24no—
589C4—48no—
590C4—42no—
591C4—48no—
592C4—42no—
593C4—42no—
594C4—48no—
595C4—48no—
596C4—84no—
597C4—84no—
598C4#266918yes—
599C2—44no—
600C2—82no—
601C2#3738611yescharacteristic
602C2#3745311yescharacteristic
603C2#3767512yes—
604C2#3936012yes—
605C2#4279214yes—
606C2#4532112yes—
607C2C2×#266911yescharacteristic
608C2#5300812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2434C2semidirect
4. C2×#2621C2semidirect
5. #17106C2semidirect
6. #16522C2semidirect
7. #16321C2semidirect
9. #15236C2semidirect
10. #15190C2semidirect
11. #15178C2semidirect
12. #14003C2semidirect
133. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24342
C2×#24092
C2×#26212
#171062
#165222
#163212
#153422
#152362
#151902
#151782
#140032
#138392