Finite Group Databasebeta

← #368954

Subgroups — #368954

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4C2×#1992C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10843C211yesmaximal, characteristic
11#10693C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×Q8—21no—
15C2×#501—24no—
16C2×#501—24no—
17C2×#501C418yescomplemented
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×#97C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C4×C4⋊Q8—24no—
23C4×C4⋊Q8—24no—
24C2×#391—21no—
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C43C2211yescharacteristic
32#1993—24no—
33#1992—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1436—21no—
43#1436—21no—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2211yescharacteristic
49C4⋊#95C2211yescharacteristic
50C23×Q8—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8D414yes—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125C2×C414yes—
65C2×C4×Q8—28no—
66C2×C4×Q8—44no—
67C2×C4×Q8—44no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C42—22no—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#501—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107C2×C418yes—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106C2×C418yes—
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97C2×C414yes—
101#391—28no—
102#389—28no—
103#389—24no—
104#389—24no—
105#384—22no—
106#384D414yes—
107#382—22no—
108#382—22no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C43—22no—
125C43—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#336—24no—
133#336Q818yescomplemented
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C22×C414yes—
165C22×C8C2×Q814yes—
166C22×C8C2×Q814yes—
167C22×C8#4114yes—
168C4⋊Q8—28no—
169C4⋊Q8—28no—
170C4⋊Q8—28no—
171C4⋊Q8C22⋊C4116yes—
172#125—28no—
173#125—28no—
174#125—44no—
175#125—44no—
176C4×Q8—432no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42C22×C412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×Q814yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256#107—216no—
257#106—216no—
258#97—28no—
259#97—44no—
260#95—28no—
261#95—28no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272C24C22×D411yescharacteristic
273C2×Q8—216no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—48no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C4×D412yes—
353C22×C4C4×D412yes—
354C22×C4#12012yes—
355C22×C4#12012yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4C4⋊Q812yes—
359C22×C4C4⋊Q812yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4#13712yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q812yes—
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4111yescharacteristic
372C22×C42−1+412yes—
373C22×C42−1+412yes—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8C4×Q818yescomplemented
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—216no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—416no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4C2×C22⋊C418yes—
414C42—28no—
415C42—28no—
416C42—24no—
417C42—24no—
418C42—44no—
419C42—44no—
420C42—416no—
421C42—42no—
422C42—42no—
423C42C2×C22⋊C414yes—
424C42C4×D414yes—
425C42#12114yes—
426C42#12314yes—
427C23C2×C4×D411yescharacteristic
428C23C2×#12011yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×C4⋊Q811yescharacteristic
431C23#53211yescharacteristic
432C23#53611yescharacteristic
433C23#53611yescharacteristic
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23C2×AGL(1,ℤ/8)11yescharacteristic
436C23C2×#13711yescharacteristic
437C23C2×#13711yescharacteristic
438C23#57511yescharacteristic
439C23#57511yescharacteristic
440C23#57511yescharacteristic
441C23#57511yescharacteristic
442Q8—432no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4#38614yes—
525C2×C4#38614yes—
526C2×C4#50114yescomplemented
527C2×C4C22×C22⋊C411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×Q812yes—
530C2×C4#51912yes—
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4#53312yes—
534C2×C4#54112yes—
535C2×C4#54712yes—
536C2×C4#54812yes—
537C2×C4#54812yes—
538C2×C4D4×Q812yes—
539C2×C4D4×Q812yes—
540C2×C4D4×Q812yes—
541C2×C4#55612yes—
542C2×C4#55712yes—
543C8—28no—
544C8—44no—
545C22C2×#38611yescharacteristic
546C22C4×C4⋊Q811yescharacteristic
547C22#232711yescharacteristic
548C22#243211yescharacteristic
549C22#243411yescharacteristic
550C22#254511yescharacteristic
551C22#254611yescharacteristic
552C22C4×AGL(1,ℤ/8)11yescharacteristic
553C22C4×AGL(1,ℤ/8)11yescharacteristic
554C22C4×#13711yescharacteristic
555C22C4×#13711yescharacteristic
556C22#300311yescharacteristic
557C22#300411yescharacteristic
558C22#300511yescharacteristic
559C22#300511yescharacteristic
560C22#303611yescharacteristic
561C22#303611yescharacteristic
562C22#303711yescharacteristic
563C22#303711yescharacteristic
564C22C2×#50111yescharacteristic
565C22#316211yescharacteristic
566C22#316511yescharacteristic
567C22#316611yescharacteristic
568C22#317111yescharacteristic
569C22#317211yescharacteristic
570C22#318011yescharacteristic
571C22#318111yescharacteristic
572C22#323411yescharacteristic
573C22#323511yescharacteristic
574C22#323611yescharacteristic
575C22#323711yescharacteristic
576C22#323811yescharacteristic
577C22#323911yescharacteristic
578C22#324011yescharacteristic
579C22#324011yescharacteristic
580C4—24no—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—48no—
586C4—42no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4—161no—
591C4—161no—
592C4Q8×C22⋊C412yes—
593C4#236512yes—
594C4#242912yes—
595C4#244612yes—
596C2#1402411yescharacteristic
597C2#2092511yescharacteristic
598C2#2092611yescharacteristic
599C2C4×#50111yescharacteristic
600C2#2173111yescharacteristic
601C2#2225011yescharacteristic
602C2#2225111yescharacteristic
603C2#2567611yescharacteristic
604C2#2567711yescharacteristic
605C2#2581611yescharacteristic
606C2#2581711yescharacteristic
607C2#2865811yescharacteristic
608C2#2865911yescharacteristic
609C2#2869211yescharacteristic
610C2#2869311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#501C4semidirect
133. #336Q8semidirect
387. C2×C8C4×Q8semidirect
526. C2×C4#501semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×C4×C4⋊Q82
C2×#19932
C2×#19922
#147092
#112202
#112192
#109082
#109072
#108432
#106932