Finite Group Databasebeta

← #1494579

Subgroups — #1494579

Order 512.

3462 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic
3C2×#2440C214yesmaximal, complemented
4#15762C212yesmaximal, complemented
5#15625C212yesmaximal, complemented
6#15614C214yesmaximal, complemented
7#15430C214yesmaximal, complemented
8#15145C212yesmaximal, complemented
9#14771C214yesmaximal, complemented
10#14401C212yesmaximal
11#14226C212yesmaximal, complemented
12#14015C212yesmaximal
13#13824C211yesmaximal, characteristic
14C22×C4⋊Q8C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20#2683—28no—
21#2542—28no—
22#2440—232no—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#398C2212yes—
27C2×#396C2214yes—
28C2×#395C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#385C2212yes—
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#384C2214yes—
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2214yes—
48#1446C2212yes—
49#1445C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1441C2214yes—
56#1441C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66C4⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C4×#95C2212yes—
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×C4⋊Q8—216no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—44no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—48no—
87C2×C4×D4—24no—
88C2×C4×D4—42no—
89C2×C4×D4—42no—
90C22×C4⋊C4—216no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#399—216no—
122#398—216no—
123#398—48no—
124#398D4116yescomplemented
125#396—216no—
126#395—28no—
127#395—216no—
128#395—216no—
129#395—44no—
130#390—24no—
131#390D418yes—
132#389—216no—
133#389—28no—
134#389—216no—
135#386—28no—
136#386Q8116yescomplemented
137#385—216no—
138#385—216no—
139#385—44no—
140#384—216no—
141#384—24no—
142#384—24no—
143#380—216no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—216no—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C25C2412yes—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—28no—
168C22×Q8—28no—
169C22×Q8—24no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—42no—
178C22×D4C2×D414yes—
179C22×D4C2×Q818yes—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4C2×Q818yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C4⋊Q8—416no—
208C4⋊Q8—432no—
209C22⋊Q8—464no—
210C22⋊Q8—464no—
211C4×D4—416no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—432no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D4116yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D414yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4C2×D4116yes—
282C2×C22⋊C4C2×Q8116yes—
283C2×C22⋊C4C2×Q818yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—22no—
290C2×C42—28no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42C2×D414yes—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95C2×D418yes—
312#95C2×D4116yes—
313#95C2×Q8116yes—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24C22×D414yes—
323C24C22×Q814yes—
324C24C22×Q814yes—
325C24C2511yescharacteristic
326C2×Q8—416no—
327C2×Q8—416no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—88no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8—88no—
335C2×Q8—84no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4C4⋊Q8116yescomplemented
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—232no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—416no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—48no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×D418yes—
411C22×C4C22×D412yes—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D418yes—
416C22×C4C22×Q818yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q818yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+418yes—
423C4⋊C4—232no—
424C4⋊C4—216no—
425C4⋊C4—216no—
426C4⋊C4—44no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—416no—
436C4⋊C4—416no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C22⋊C4—232no—
441C22⋊C4—232no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—84no—
447C42—28no—
448C42—44no—
449C42—416no—
450C42—48no—
451C42—44no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23C2×C4⋊Q818yes—
475C23C23×D411yescharacteristic
476C23C23×D412yes—
477C23C23×Q812yes—
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×H(5,𝔽2)12yes—
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×2−1+414yes—
482C23#58512yes—
483Q8—816no—
484Q8—816no—
485D4—416no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—432no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—816no—
551C2×C4—816no—
552C2×C4C2×C4⋊Q814yes—
553C2×C4D4218yes—
554C2×C4D4214yes—
555C2×C4D4×Q8116yes—
556C22—216no—
557C22—28no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—48no—
564C22—48no—
565C22—48no—
566C22—44no—
567C22C22×C4⋊Q811yescharacteristic
568C22C2×D4212yes—
569C22C2×D4211yescharacteristic
570C22C2×D4×Q814yes—
571C22#348411yescharacteristic
572C22#348511yescharacteristic
573C22#348914yes—
574C22#349812yes—
575C22#349912yes—
576C22#350112yes—
577C22#350112yes—
578C22#350314yes—
579C22#353812yes—
580C22#354311yescharacteristic
581C22#354412yes—
582C22#355914yes—
583C4—24no—
584C4—216no—
585C4—28no—
586C4—24no—
587C4—42no—
588C4—48no—
589C4—44no—
590C4—48no—
591C4—44no—
592C4—84no—
593C4—82no—
594C4—82no—
595C2—48no—
596C2#3539111yescharacteristic
597C2D4×C4⋊Q812yes—
598C2#4444712yes—
599C2#4494114yes—
600C2#5070012yes—
601C2#5154211yescharacteristic
602C2#5899812yes—
603C2#5912611yescharacteristic
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
3. C2×#2440C2semidirect
4. #15762C2semidirect
5. #15625C2semidirect
6. #15614C2semidirect
7. #15430C2semidirect
8. #15145C2semidirect
9. #14771C2semidirect
11. #14226C2semidirect
124. #398D4semidirect
136. #386Q8semidirect
340. C2×D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#25422
C2×#24402
#157622
#156252
#156142
#154302
#151452
#147712
#144012
#142262
#140152
#138242