Finite Group Databasebeta

← #403383

Subgroups — #403383

Order 512.

2488 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2059C212yesmaximal
3C2×#2057C211yesmaximal, characteristic
4C2×#1865C211yesmaximal, characteristic
5C2×#1863C212yesmaximal
6#15158C211yesmaximal, characteristic
7#11385C212yesmaximal
8#11384C211yesmaximal, characteristic
9#11380C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic
11#10789C212yesmaximal
12C2×#557—24no—
13C2×#557—28no—
14C2×#557—28no—
15C2×#557—24no—
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—22no—
19C22×#125C2212yes—
20C22×C4×Q8—21no—
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×#107C2212yes—
25C22×#106C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2212yes—
28C22×#103C2211yescharacteristic
29C2×#395—21no—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32#2059—28no—
33#2057—24no—
34#1865—24no—
35#1863—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45#1441—21no—
46#1441—22no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55#557—264no—
56#557—232no—
57#557—232no—
58C2×C4⋊Q8—216no—
59C2×C4⋊Q8—28no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8D414yes—
64C2×#125—216no—
65C2×#125—216no—
66C2×#125—216no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—24no—
72C2×#125D418yes—
73C2×C4×Q8—28no—
74C2×C4×Q8—216no—
75C2×C4×Q8—24no—
76C2×C4×Q8—24no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8—24no—
80C2×C4×Q8—24no—
81C2×C4×Q8—24no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8D418yes—
88C2×C4×Q8D414yes—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D412yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C42—22no—
104C22×C42—21no—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C2×#107—24no—
108C2×#107—24no—
109C2×#107—24no—
110C2×#106—24no—
111C2×#106—22no—
112C2×#103—24no—
113C2×#103—22no—
114C2×#103—24no—
115C2×#103—22no—
116C2×#103—24no—
117C2×#103—22no—
118C2×#103Q818yescomplemented
119C2×#103Q814yescomplemented
120#395—44no—
121#389—22no—
122#389—24no—
123#389—24no—
124#389—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—22no—
136#336—22no—
137#336—24no—
138#336—24no—
139#336—22no—
140C22×Q8—28no—
141C22×Q8—24no—
142C22×Q8—22no—
143C22×Q8—24no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—44no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8C2×D414yes—
156C22×Q8C2×D412yes—
157C22×Q8C2×Q812yes—
158C22×Q8C2×Q814yes—
159C22×Q8#4112yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8C2×Q812yes—
187C22×C8C2×Q814yes—
188C4⋊Q8—264no—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8—216no—
192#125—2128no—
193#125—216no—
194#125—216no—
195#125—216no—
196#125—216no—
197C4×Q8—264no—
198C4×Q8—216no—
199C4×Q8—216no—
200C4×Q8—216no—
201C4×Q8—28no—
202C4×Q8—28no—
203C4×Q8—216no—
204C4×Q8—216no—
205C4×Q8—28no—
206C4×Q8—216no—
207C4×Q8—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×Q812yes—
257C2×C4⋊C4C2×Q814yes—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42C2×D414yes—
270C2×C42C2×D412yes—
271C2×C42C2×D414yes—
272C2×C42C2×D412yes—
273#107—48no—
274#106—44no—
275#103—216no—
276#103—28no—
277#103—44no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—42no—
282#95—81no—
283#95—82no—
284C24C22×D411yescharacteristic
285C2×Q8—28no—
286C2×Q8—216no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—416no—
303C2×Q8—44no—
304C2×Q8—48no—
305C2×Q8—416no—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×Q8#12014yes—
309C2×Q8C22⋊Q814yes—
310C2×Q8C22⋊Q818yes—
311C2×Q8C4⋊Q814yes—
312C2×Q8C4⋊Q818yes—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4#12014yes—
368C22×C4#12012yes—
369C22×C4#12014yes—
370C22×C4#12012yes—
371C22×C4#13512yes—
372C22×C4AGL(1,ℤ/8)12yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×Q811yescharacteristic
378C22×C4C22×Q812yes—
379C22×C4C2×#4111yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+412yes—
382C22×C42−1+411yescharacteristic
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—42no—
400C4⋊C4—28no—
401C4⋊C4—216no—
402C4⋊C4—216no—
403C4⋊C4—232no—
404C4⋊C4—232no—
405C4⋊C4—216no—
406C4⋊C4—232no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—88no—
424C4⋊C4—84no—
425C4⋊C4#12014yes—
426C4⋊C4#12018yes—
427C42—232no—
428C42—216no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—22no—
434C42—24no—
435C42—22no—
436C23C2×#12011yescharacteristic
437C23C2×#12012yes—
438C23#54811yescharacteristic
439C23D4×Q811yescharacteristic
440C23D4×Q812yes—
441C23C2×#13511yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23#57512yes—
444C23#57711yescharacteristic
445C23#57811yescharacteristic
446C23#57812yes—
447Q8—24no—
448Q8—24no—
449Q8—28no—
450Q8—28no—
451Q8—24no—
452Q8—416no—
453Q8—432no—
454Q8#39518yescomplemented
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4#44914yes—
515C2×C4C2×#12011yescharacteristic
516C2×C4C2×#12012yes—
517C2×C4C2×#12011yescharacteristic
518C2×C4C2×C22⋊Q811yescharacteristic
519C2×C4C2×C22⋊Q812yes—
520C2×C4C2×C4⋊Q811yescharacteristic
521C2×C4C2×C4⋊Q812yes—
522C2×C4#53611yescharacteristic
523C2×C4#53612yes—
524C2×C4#53611yescharacteristic
525C2×C4#53612yes—
526C2×C4#54811yescharacteristic
527C2×C4#54812yes—
528C2×C4D4×Q812yes—
529C2×C4D4×Q812yes—
530C2×C4D4×Q812yes—
531C2×C4D4×Q811yescharacteristic
532C2×C4D4×Q812yes—
533C8—42no—
534C8—44no—
535C8—81no—
536C22#240911yescharacteristic
537C22#240912yes—
538C22#243411yescharacteristic
539C22#243412yes—
540C22#284611yescharacteristic
541C22C2×#44911yescharacteristic
542C22#300212yes—
543C22#300511yescharacteristic
544C22#300612yes—
545C22#301112yes—
546C22#301111yescharacteristic
547C22#301311yescharacteristic
548C22#301312yes—
549C22#336411yescharacteristic
550C22#336811yescharacteristic
551C22#337712yes—
552C22#338111yescharacteristic
553C22#338312yes—
554C22#338611yescharacteristic
555C22#338612yes—
556C22#338812yes—
557C22#339012yes—
558C22#339012yes—
559C4—28no—
560C4—24no—
561C4—44no—
562C4—48no—
563C4—44no—
564C4—42no—
565C4—44no—
566C4—82no—
567C4—84no—
568C4—161no—
569C4C2×#39511yescharacteristic
570C4#240911yescharacteristic
571C4#240912yes—
572C4#243412yes—
573C4#243411yescharacteristic
574C4#261411yescharacteristic
575C2#1487411yescharacteristic
576C2#2432811yescharacteristic
577C2#2433612yes—
578C2#2434411yescharacteristic
579C2#2434612yes—
580C2#2584011yescharacteristic
581C2#2584212yes—
582C2#2584611yescharacteristic
583C2#2584912yes—
584C2#3288011yescharacteristic
585C2#3288711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
118. C2×#103Q8semidirect
119. C2×#103Q8semidirect
454. Q8#395semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#20592
C2×#20572
C2×#18652
C2×#18632
#151582
#113852
#113842
#113802
#107892