Finite Group Databasebeta

← #403384

Subgroups — #403384

Order 512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2059C212yesmaximal
3C2×#2058C211yesmaximal, characteristic
4C2×#1865C212yesmaximal
5C2×#1863C211yesmaximal, characteristic
6#15202C211yesmaximal, characteristic
7#11385C212yesmaximal
8#11384C211yesmaximal, characteristic
9#11380C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C211yesmaximal, characteristic
12C2×#557—24no—
13C2×#557—28no—
14C2×#557—24no—
15C2×#557—28no—
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—22no—
19C22×#125C2212yes—
20C22×C4×Q8—21no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2212yes—
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31#2059—28no—
32#2058—24no—
33#1865—28no—
34#1863—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1444—22no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
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×Q8D414yes—
88C2×C4×Q8D418yes—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D412yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C42—22no—
104C22×C42—21no—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C2×#107—24no—
108C2×#107—22no—
109C2×#106—24no—
110C2×#106—24no—
111C2×#106—24no—
112C2×#103—22no—
113C2×#103—24no—
114C2×#103—22no—
115C2×#103—24no—
116C2×#103—22no—
117C2×#103—24no—
118C2×#103Q814yescomplemented
119C2×#103Q818yescomplemented
120#389—22no—
121#389—24no—
122#389—22no—
123#389—24no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—22no—
134#340—24no—
135#336—22no—
136#336—24no—
137#336—22no—
138#336—24no—
139C22×Q8—28no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8C2×D412yes—
155C22×Q8C2×D414yes—
156C22×Q8C2×Q814yes—
157C22×Q8C2×Q812yes—
158C22×Q8#4112yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
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—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×Q814yes—
186C22×C8C2×Q812yes—
187C4⋊Q8—264no—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—216no—
191#125—2128no—
192#125—216no—
193#125—216no—
194#125—216no—
195#125—216no—
196C4×Q8—264no—
197C4×Q8—216no—
198C4×Q8—216no—
199C4×Q8—216no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—216no—
203C4×Q8—216no—
204C4×Q8—28no—
205C4×Q8—216no—
206C4×Q8—28no—
207C2×C4⋊C4—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—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
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—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×Q814yes—
255C2×C4⋊C4C2×Q812yes—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42C2×D414yes—
268C2×C42C2×D412yes—
269C2×C42C2×D414yes—
270C2×C42C2×D412yes—
271#107—44no—
272#106—48no—
273#103—28no—
274#103—216no—
275#103—44no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—81no—
281#95—82no—
282C24C22×D411yescharacteristic
283C2×Q8—28no—
284C2×Q8—216no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—416no—
301C2×Q8—44no—
302C2×Q8—48no—
303C2×Q8—416no—
304C2×Q8—44no—
305C2×Q8—48no—
306C2×Q8#12014yes—
307C2×Q8C22⋊Q814yes—
308C2×Q8C22⋊Q818yes—
309C2×Q8C4⋊Q814yes—
310C2×Q8C4⋊Q818yes—
311C22×C4—22no—
312C22×C4—24no—
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—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4#12014yes—
364C22×C4#12014yes—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#13512yes—
368C22×C4#13712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×Q812yes—
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C42−1+412yes—
377C22×C42−1+411yescharacteristic
378C22×C42−1+411yescharacteristic
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—42no—
396C4⋊C4—28no—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—232no—
400C4⋊C4—232no—
401C4⋊C4—216no—
402C4⋊C4—232no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—88no—
420C4⋊C4—84no—
421C4⋊C4#12018yes—
422C4⋊C4#12014yes—
423C42—232no—
424C42—216no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—22no—
430C42—24no—
431C42—22no—
432C23C2×#12011yescharacteristic
433C23C2×#12012yes—
434C23#54811yescharacteristic
435C23D4×Q812yes—
436C23D4×Q811yescharacteristic
437C23C2×#13511yescharacteristic
438C23C2×#13711yescharacteristic
439C23#57512yes—
440C23#57711yescharacteristic
441C23#57712yes—
442C23#57811yescharacteristic
443Q8—24no—
444Q8—28no—
445Q8—24no—
446Q8—28no—
447Q8—24no—
448Q8—416no—
449Q8—432no—
450Q8#39518yes—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4#45214yes—
509C2×C4C2×#12011yescharacteristic
510C2×C4C2×#12012yes—
511C2×C4C2×#12011yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×C22⋊Q812yes—
514C2×C4C2×C4⋊Q811yescharacteristic
515C2×C4C2×C4⋊Q812yes—
516C2×C4#53611yescharacteristic
517C2×C4#53612yes—
518C2×C4#53611yescharacteristic
519C2×C4#53612yes—
520C2×C4#54811yescharacteristic
521C2×C4#54812yes—
522C2×C4D4×Q812yes—
523C2×C4D4×Q812yes—
524C2×C4D4×Q812yes—
525C2×C4D4×Q811yescharacteristic
526C2×C4D4×Q812yes—
527C8—44no—
528C8—42no—
529C8—81no—
530C22#240911yescharacteristic
531C22#240912yes—
532C22#243411yescharacteristic
533C22#243412yes—
534C22#284611yescharacteristic
535C22C2×#45211yescharacteristic
536C22#300212yes—
537C22#300511yescharacteristic
538C22#300712yes—
539C22#301012yes—
540C22#301011yescharacteristic
541C22#301311yescharacteristic
542C22#301312yes—
543C22#336511yescharacteristic
544C22#337112yes—
545C22#337411yescharacteristic
546C22#338112yes—
547C22#338311yescharacteristic
548C22#338711yescharacteristic
549C22#338712yes—
550C22#338812yes—
551C22#338812yes—
552C22#339012yes—
553C4—28no—
554C4—24no—
555C4—44no—
556C4—48no—
557C4—44no—
558C4—42no—
559C4—44no—
560C4—84no—
561C4—82no—
562C4—161no—
563C4C2×#39511yescharacteristic
564C4#240912yes—
565C4#240911yescharacteristic
566C4#243412yes—
567C4#243411yescharacteristic
568C4#261411yescharacteristic
569C2#1487411yescharacteristic
570C2#2433211yescharacteristic
571C2#2433412yes—
572C2#2434011yescharacteristic
573C2#2434812yes—
574C2#2583611yescharacteristic
575C2#2584212yes—
576C2#2584611yescharacteristic
577C2#2584812yes—
578C2#3288311yescharacteristic
579C2#3288511yescharacteristic
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

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×#20582
C2×#18652
C2×#18632
#152022
#113852
#113842
#113802
#107892