Finite Group Databasebeta

← #364820

Subgroups — #364820

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2960C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#11216C212yesmaximal
7#11062C212yesmaximal
8#10942C212yesmaximal
9#10896C212yesmaximal
10#10847C211yesmaximal, characteristic
11#10844C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078C418yescomplemented
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19#2960—24no—
20#2960—24no—
21C2×#446—21no—
22C2×#446—21no—
23C2×C8×Q8—21no—
24C2×C8×Q8—21no—
25C2×#428C2212yes—
26C2×#424C2212yes—
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#419C2212yes—
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#101C2212yes—
34C2×#405C2212yes—
35C22×C4×C8C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1989—22no—
38#1988—21no—
39#1987—21no—
40#1801—24no—
41C4×#103—21no—
42C4×#103—21no—
43C4×#103—21no—
44C4×#103—21no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2212yes—
48C2×#341C2212yes—
49C2×#341C2212yes—
50#1736C2212yes—
51#1736C2212yes—
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C23×Q8C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2312yes—
57C22×M16C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60#533—22no—
61#533—22no—
62#533C2×C414yes—
63#533D414yes—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D411yescharacteristic
66C2×#125D412yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—22no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—22no—
78C2×C4×Q8C2×C414yes—
79C2×#117C2312yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#479—216no—
88#477—28no—
89#475—28no—
90#446—28no—
91#446—28no—
92C8×Q8—28no—
93C8×Q8—28no—
94#428—24no—
95#428C2×C418yes—
96C2×#107—22no—
97C2×#106—22no—
98#424—24no—
99#424—24no—
100#423—22no—
101#423—22no—
102#423—22no—
103#423—22no—
104#423C2×C414yes—
105#423D414yes—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114#419—24no—
115#419C2×C418yes—
116C2×#103—22no—
117C2×#103—21no—
118C2×#103—21no—
119C2×#103—21no—
120C2×#103—21no—
121C2×#103—22no—
122C2×#103—22no—
123C2×#103—22no—
124C2×#103—22no—
125C2×#103Q814yescomplemented
126C2×#101—24no—
127C2×#101—24no—
128C2×#101—42no—
129#405—24no—
130#405—24no—
131C2×#97—22no—
132C2×C4×C8—22no—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8D411yescharacteristic
136C2×C4×C8D411yescharacteristic
137C2×C4×C8D411yescharacteristic
138C2×C4×C8D411yescharacteristic
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C2×#95—21no—
144#343—24no—
145#343—24no—
146#343—42no—
147#341—24no—
148#341—42no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
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×Q8—21no—
169C22×Q8—21no—
170C22×Q8C22×C412yes—
171C22×Q8C2×Q812yes—
172C22×Q8#4112yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4#4111yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4C2411yescharacteristic
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16C22×C414yes—
197C2×M16C2×D414yes—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×Q812yes—
216C22×C8#4112yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8C22⋊C414yes—
220C4⋊Q8C22⋊C414yes—
221#125—24no—
222#125C22⋊C418yes—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×Q8—42no—
239#117—22no—
240#117—22no—
241#117C22×C414yes—
242#117C2×D414yes—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×Q812yes—
270C2×C4⋊C4#4112yes—
271C2×C22⋊C4—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—42no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—82no—
280C2×C42C22×C412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D412yes—
287C2×C42#4112yes—
288#107—28no—
289#106—28no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—42no—
297C4⋊C8—42no—
298#103—24no—
299#103—24no—
300#103—24no—
301#103—24no—
302#103—28no—
303#103—28no—
304#101—48no—
305#97—28no—
306C4×C8—28no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311#95—41no—
312#95—41no—
313C24C2×#4111yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
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—22no—
331C2×Q8—42no—
332C2×Q8—42no—
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—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8C4×Q814yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12312yes—
383C22×C4#12312yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C42−1+411yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398M16—24no—
399M16—24no—
400M16—24no—
401M16—24no—
402M16—42no—
403M16—42no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409M16—42no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—24no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8C22⋊Q812yes—
438C2×C8C22⋊Q812yes—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4C2×C22⋊C414yes—
460C4⋊C4C2×C22⋊C414yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C42—24no—
464C42—24no—
465C42—44no—
466C42—44no—
467C42—82no—
468C42—82no—
469C42C2×C22⋊C412yes—
470C42C2×C22⋊C412yes—
471C42C4×D412yes—
472C42C4×D412yes—
473C42#12012yes—
474C42#12012yes—
475C42#12112yes—
476C42#12112yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23C2×C4×D411yescharacteristic
482C23C2×#12111yescharacteristic
483C23C2×#12311yescharacteristic
484C23#55211yescharacteristic
485C23#55211yescharacteristic
486C23#55411yescharacteristic
487C23#55611yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—48no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—48no—
496Q8—42no—
497Q8—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4#38612yes—
537C2×C4#38612yes—
538C2×C4#38612yes—
539C2×C4#38612yes—
540C2×C4#47112yes—
541C2×C4#47312yes—
542C2×C4C22×C22⋊C411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×Q811yescharacteristic
545C2×C4C4×#4111yescharacteristic
546C2×C4#51911yescharacteristic
547C2×C4#52011yescharacteristic
548C2×C4C2×#12011yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4#52511yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4#53311yescharacteristic
554C2×C4#53611yescharacteristic
555C2×C4#53711yescharacteristic
556C2×C4#54111yescharacteristic
557C2×C4#54211yescharacteristic
558C2×C4#54611yescharacteristic
559C2×C4#54711yescharacteristic
560C2×C4#54811yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4#54811yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4D4×Q811yescharacteristic
566C2×C4D4×Q811yescharacteristic
567C2×C4D4×Q811yescharacteristic
568C2×C4#55011yescharacteristic
569C2×C4#55011yescharacteristic
570C8—22no—
571C8—22no—
572C8—44no—
573C8—41no—
574C8—41no—
575C8—44no—
576C8—82no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22C2×#38611yescharacteristic
590C22#236011yescharacteristic
591C22#236211yescharacteristic
592C22#242111yescharacteristic
593C22#242311yescharacteristic
594C22#253611yescharacteristic
595C22#253811yescharacteristic
596C22C2×#47111yescharacteristic
597C22C2×#47311yescharacteristic
598C22#306711yescharacteristic
599C22#306711yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#190814yes—
610C4C4×C22⋊Q811yescharacteristic
611C4#232311yescharacteristic
612C4Q8×C22⋊C411yescharacteristic
613C4#234111yescharacteristic
614C4#240711yescharacteristic
615C4#240911yescharacteristic
616C4#242611yescharacteristic
617C4#242911yescharacteristic
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2#1396911yescharacteristic
623C2C2×#190811yescharacteristic
624C2#2100311yescharacteristic
625C2#2821811yescharacteristic
626C2#2821911yescharacteristic
627C2#2829611yescharacteristic
628C2#2829711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
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
125. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29602
#213122
C2×#18012
C2×C4×#1032
#112162
#110622
#109422
#108962
#108472
#108442