Finite Group Databasebeta

← #364807

Subgroups — #364807

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4C4×#419C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#11217C212yesmaximal
7#11064C212yesmaximal
8#10957C212yesmaximal
9#10899C212yesmaximal
10#10858C211yesmaximal, characteristic
11#10852C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078C418yescomplemented
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19C2×#428C2212yes—
20C2×#423C2211yescharacteristic
21C2×#419C2211yescharacteristic
22C2×#419C2212yes—
23C2×#419C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2368—22no—
26#2368—22no—
27#2355—22no—
28#2355—22no—
29#2331—21no—
30#2331—21no—
31#2317C2212yes—
32C4×#125—22no—
33C2×#391C2212yes—
34C2×#382C2211yescharacteristic
35#2274C2212yes—
36C4×#117C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1986—24no—
39#1763—22no—
40C4×#103—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2211yescharacteristic
44#1736C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49#1441—22no—
50C4⋊#95C2212yes—
51#1434—22no—
52C23×Q8C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533D414yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125D412yes—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8C2×C414yes—
75C2×#117C2311yescharacteristic
76C2×#117C2312yes—
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#479—216no—
86#477—28no—
87#475—28no—
88#428—24no—
89#428C2×C418yes—
90C2×#107—22no—
91C2×#106—22no—
92#423—22no—
93#423C2×C414yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96#419—22no—
97#419—22no—
98#419—24no—
99#419—22no—
100#419C2×C418yes—
101#419Q814yescomplemented
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C4×M16—22no—
107C4×M16—22no—
108C2×#97—21no—
109C2×C4×C8—21no—
110#391—24no—
111#391—24no—
112#391—24no—
113#391—42no—
114#389—22no—
115#389—22no—
116#387—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#385—24no—
121#384—22no—
122#382—22no—
123#382—21no—
124#382—21no—
125#382—21no—
126#382—21no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4D412yes—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—24no—
140#376—21no—
141C2×#95—21no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95D412yes—
145C2×#95D412yes—
146C43—21no—
147#343—24no—
148#343—42no—
149#340—22no—
150#340—22no—
151#339—24no—
152#339—24no—
153#339—24no—
154#339—42no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8C22×C412yes—
169C22×Q8C2×Q812yes—
170C22×Q8#4112yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4C2×D414yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16C22×C414yes—
194C2×M16C2×Q812yes—
195C2×M16#4112yes—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8C22⋊C414yes—
202C4⋊Q8C22⋊C414yes—
203#125—24no—
204#125C22⋊C418yes—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×Q8—42no—
220C4×Q8—42no—
221#117—22no—
222#117—22no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117C22×C414yes—
230#117C2×D414yes—
231#117C2×Q812yes—
232#117#4112yes—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—24no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—42no—
281C2×C42C22×C412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42#4112yes—
288C2×C42#4112yes—
289#107—28no—
290#106—28no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293#103—28no—
294#103—28no—
295#103—28no—
296#103—28no—
297#97—42no—
298C4×C8—42no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—48no—
310#95—48no—
311#95—42no—
312C24C22×D411yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8C4×Q814yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12011yescharacteristic
377C22×C4#12011yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12112yes—
381C22×C4#12112yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396M16—24no—
397M16—24no—
398M16—22no—
399M16—22no—
400M16—22no—
401M16—22no—
402M16—42no—
403M16—41no—
404M16—42no—
405M16—41no—
406M16—41no—
407M16—41no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—42no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—48no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4C2×C22⋊C414yes—
440C4⋊C4C2×C22⋊C414yes—
441C4⋊C4C22⋊Q814yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—44no—
455C42—44no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—82no—
461C42—82no—
462C42—84no—
463C42—84no—
464C42C2×C22⋊C412yes—
465C42C2×C22⋊C412yes—
466C42C4×D412yes—
467C42C4×D412yes—
468C42#12112yes—
469C42#12112yes—
470C42#12312yes—
471C42#12312yes—
472C23—22no—
473C23—22no—
474C23C2×C4×D411yescharacteristic
475C23C2×#12011yescharacteristic
476C23C2×#12111yescharacteristic
477C23#54611yescharacteristic
478C23#54711yescharacteristic
479C23#54811yescharacteristic
480C23D4×Q811yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—84no—
490Q8—84no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4#38612yes—
535C2×C4#38612yes—
536C2×C4#38612yes—
537C2×C4#38612yes—
538C2×C4C22×C22⋊C411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×Q811yescharacteristic
541C2×C4C4×#4111yescharacteristic
542C2×C4#51911yescharacteristic
543C2×C4#52011yescharacteristic
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×C22⋊Q811yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4#52511yescharacteristic
548C2×C4#52911yescharacteristic
549C2×C4#53311yescharacteristic
550C2×C4#53611yescharacteristic
551C2×C4#53711yescharacteristic
552C2×C4#54111yescharacteristic
553C2×C4#54211yescharacteristic
554C2×C4#54811yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4#54811yescharacteristic
557C2×C4#54811yescharacteristic
558C2×C4D4×Q811yescharacteristic
559C2×C4D4×Q811yescharacteristic
560C2×C4#55011yescharacteristic
561C2×C4#55011yescharacteristic
562C2×C4#55211yescharacteristic
563C2×C4#55211yescharacteristic
564C2×C4#55411yescharacteristic
565C2×C4#55611yescharacteristic
566C8—44no—
567C8—42no—
568C8—42no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22C2×#38611yescharacteristic
578C22#236011yescharacteristic
579C22#236211yescharacteristic
580C22#240711yescharacteristic
581C22#240911yescharacteristic
582C22#242611yescharacteristic
583C22#242911yescharacteristic
584C22#301812yes—
585C22#302212yes—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—44no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—84no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4C4×C22⋊Q811yescharacteristic
599C4#232311yescharacteristic
600C4Q8×C22⋊C411yescharacteristic
601C4#234111yescharacteristic
602C4#242111yescharacteristic
603C4#242311yescharacteristic
604C4#253611yescharacteristic
605C4#253811yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2#1396911yescharacteristic
611C2#2094211yescharacteristic
612C2#2094411yescharacteristic
613C2#2414912yes—
614C2#2432512yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #3078C4semidirect
101. #419Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#342152
C2×#19862
C4×#4192
#139422
#112172
#110642
#109572
#108992
#108582
#108522