Finite Group Databasebeta

← #878110

Subgroups — #878110

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2807C212yesmaximal
2#16987C212yesmaximal
3#16824C212yesmaximal
4#15817C214yesmaximal, complemented
5#15533C214yesmaximal, complemented
6#15146C212yesmaximal, complemented
7#15135C211yesmaximal, characteristic
8#15135C211yesmaximal, characteristic
9#15133C212yesmaximal, complemented
10#14697C212yesmaximal
11#14514C212yesmaximal
12#14151C212yesmaximal, complemented
13#14005C212yesmaximal, complemented
14#13876C212yesmaximal
15C4×#385C211yesmaximal, characteristic
16C23×C42C2211yescharacteristic
17#2807—216no—
18C2×#400C2214yes—
19C2×#399C2212yes—
20C2×#399C2212yes—
21C2×#398C2214yes—
22C2×#389C2212yes—
23C2×#389C2214yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#385C2214yes—
30C2×#385C2211yescharacteristic
31C2×#385C2211yescharacteristic
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2212yes—
42#1445C2212yes—
43#1444C2212yes—
44#1444C2212yes—
45#1444C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2214yes—
74C22⋊#95C2211yescharacteristic
75C4×#95C2212yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C42—44no—
102C22×C42—42no—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2312yes—
109#400—216no—
110#399—28no—
111#399—28no—
112#398—216no—
113#389—216no—
114#389—28no—
115#389—216no—
116#389—44no—
117#389—44no—
118#389—44no—
119#388—216no—
120#385—216no—
121#385D418yes—
122#385Q818yes—
123#384—28no—
124#384—42no—
125#384—42no—
126#380—28no—
127#380—44no—
128C4×C4⋊C4—44no—
129C4×C4⋊C4—44no—
130C4×C22⋊C4—42no—
131C4×C22⋊C4Q818yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C25C2411yescharacteristic
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—41no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—41no—
173C23×C4C2×D414yes—
174C23×C4C2×Q814yes—
175C23×C4C2×Q814yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×Q818yes—
235C2×C4⋊C4#4118yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×D414yes—
249C2×C22⋊C4C2×D414yes—
250C2×C22⋊C4C2×Q814yes—
251C2×C22⋊C4C2×Q814yes—
252C2×C22⋊C4C2×Q814yes—
253C2×C22⋊C4C2×Q814yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—416no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42C2×Q818yes—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—42no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—41no—
288#95—41no—
289#95—42no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95C2×D418yes—
297#95C2×Q818yes—
298#95C2×Q818yes—
299C24—22no—
300C24—21no—
301C24—22no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24C22×D412yes—
308C24C22×Q812yes—
309C24C22×Q812yes—
310C24C2511yescharacteristic
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—216no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
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—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4C22×D414yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D414yes—
399C22×C4C22×Q812yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q814yes—
402C22×C4C22×Q814yes—
403C22×C4C22×Q812yes—
404C22×C4C22×Q814yes—
405C22×C4C22×Q814yes—
406C22×C4C22×Q812yes—
407C22×C4C2×#4118yes—
408C22×C4C2×#4114yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412C4⋊C4—216no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—416no—
432C22⋊C4—88no—
433C22⋊C4C4⋊Q8116yes—
434C42—216no—
435C42—48no—
436C42—44no—
437C42—44no—
438C42—48no—
439C42—88no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—24no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23C2×C4⋊Q814yes—
463C23C23×D411yescharacteristic
464C23C23×Q811yescharacteristic
465C23C23×Q811yescharacteristic
466C23C22×#4112yes—
467C23C2×H(5,𝔽2)12yes—
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×2−1+411yescharacteristic
470C23#58511yescharacteristic
471C23#58511yescharacteristic
472C23#58512yes—
473C23#58512yes—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—416no—
503C2×C4—416no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—416no—
536C2×C4—44no—
537C2×C4—44no—
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—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4C2×C4⋊Q818yes—
558C2×C4#53918yes—
559C2×C4#54414yes—
560C2×C4#55418yes—
561C2×C4#55914yes—
562C2×C4#56014yes—
563C22—24no—
564C22—28no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22C22×C4⋊Q811yescharacteristic
575C22C2×#53912yes—
576C22C2×#54411yescharacteristic
577C22#345811yescharacteristic
578C22D4×#4112yes—
579C22C2×#55412yes—
580C22Q8×#4112yes—
581C22C2×#55911yescharacteristic
582C22#349611yescharacteristic
583C22#349612yes—
584C22#349711yescharacteristic
585C22#350111yescharacteristic
586C22#350112yes—
587C22#350211yescharacteristic
588C22#350412yes—
589C22#352112yes—
590C22#352412yes—
591C22C2×#56011yescharacteristic
592C22#353912yes—
593C22#353912yes—
594C22#354011yescharacteristic
595C22#354011yescharacteristic
596C22#355712yes—
597C4—28no—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—42no—
602C4—44no—
603C4—44no—
604C4—42no—
605C4—42no—
606C4—48no—
607C4—84no—
608C4—84no—
609C4—84no—
610C4#280718yescomplemented
611C2—44no—
612C2#3516711yescharacteristic
613C2#3626112yes—
614C2#3742811yescharacteristic
615C2#3810712yes—
616C2#3940912yes—
617C2#3966712yes—
618C2#4547811yescharacteristic
619C2#4549311yescharacteristic
620C2C2×#280711yescharacteristic
621C2#5962811yescharacteristic
622C2#5962911yescharacteristic
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
4. #15817C2semidirect
5. #15533C2semidirect
6. #15146C2semidirect
9. #15133C2semidirect
12. #14151C2semidirect
13. #14005C2semidirect
610. C4#2807semidirect

Maximal subgroups

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

subgroupindex
C2×#28072
#169872
#168242
#158172
#155332
#151462
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C4×#3852