Finite Group Databasebeta

← #415048

Subgroups — #415048

Order 512.

1366 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×#2628C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1847C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#14340C211yesmaximal, characteristic
7#11220C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10842C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2628—24no—
20#2628—24no—
21C2×#400—21no—
22C2×#399—21no—
23C2×#395—21no—
24C2×#393—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33#1993—24no—
34#1990—24no—
35#1847—24no—
36#1750—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1441—21no—
46#1439—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8Q818yescomplemented
94C2×#97—22no—
95C2×#97—22no—
96C2×#97D414yes—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100#400—28no—
101#399—28no—
102#395—28no—
103#393—28no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#384—22no—
109#384—22no—
110#382—22no—
111#382—22no—
112#382—22no—
113#382D414yes—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—42no—
119#380—42no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#340—42no—
137#336—24no—
138C25C2×D411yescharacteristic
139C22×Q8—24no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—82no—
143C22×Q8—82no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×Q814yes—
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D414yes—
177C22×C8C2×Q814yes—
178C22×C8C2×Q814yes—
179C22×C8#4114yes—
180C4⋊Q8—28no—
181C4⋊Q8—28no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184#117—44no—
185#117—44no—
186#117—44no—
187#117—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—48no—
263#106—48no—
264C22⋊C8—216no—
265#97—28no—
266C4×C8—44no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—44no—
274#95—42no—
275#95—42no—
276#95—82no—
277#95—82no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C4⋊Q812yes—
289C24C22×D411yescharacteristic
290C2×Q8—48no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—88no—
294C2×Q8—88no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4C4⋊Q812yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q812yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8C22⋊Q818yes—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—88no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42#12114yes—
433C42#12114yes—
434C42#12314yes—
435C42#12414yes—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23C2×#12011yescharacteristic
460C23C2×#12111yescharacteristic
461C23C2×#12311yescharacteristic
462C23C2×C4⋊Q811yescharacteristic
463C23#53211yescharacteristic
464C23#53511yescharacteristic
465C23#53611yescharacteristic
466C23#57612yes—
467C23#57712yes—
468C23#57712yes—
469C23#57812yes—
470Q8—816no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
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—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4#39414yes—
535C2×C4#39414yes—
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×C22⋊Q812yes—
539C2×C4C2×#12311yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4#53312yes—
542C2×C4#53712yes—
543C2×C4#54212yes—
544C2×C4#54612yes—
545C2×C4D4×Q812yes—
546C2×C4D4×Q812yes—
547C2×C4#55012yes—
548C2×C4#55212yes—
549C2×C4#55412yes—
550C2×C4#55412yes—
551C2×C4#55612yes—
552C8—28no—
553C8—44no—
554C22—44no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—82no—
562C22—81no—
563C22—81no—
564C22—82no—
565C22C2×#39411yescharacteristic
566C22#243111yescharacteristic
567C22#243411yescharacteristic
568C22#249911yescharacteristic
569C22#250211yescharacteristic
570C22#262011yescharacteristic
571C22#285711yescharacteristic
572C22#300811yescharacteristic
573C22#300911yescharacteristic
574C22#301011yescharacteristic
575C22#301111yescharacteristic
576C22#306212yes—
577C22#306312yes—
578C22#309612yes—
579C22#309712yes—
580C22#316312yes—
581C22#316412yes—
582C22#317812yes—
583C22#317912yes—
584C22#322412yes—
585C22#322512yes—
586C22#324312yes—
587C22#324412yes—
588C4—24no—
589C4—24no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4—82no—
596C4—81no—
597C4—162no—
598C4—162no—
599C4#241712yes—
600C4#242312yes—
601C4#249212yes—
602C4#253812yes—
603C2—82no—
604C2#1518211yescharacteristic
605C2#2348311yescharacteristic
606C2#2348411yescharacteristic
607C2#2561311yescharacteristic
608C2#2561411yescharacteristic
609C2#2582011yescharacteristic
610C2#2582111yescharacteristic
611C2#2788712yes—
612C2#2795912yes—
613C2#2985712yes—
614C2#3321912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1993C2semidirect
3. C2×#1990C2semidirect
4. C2×#1847C2semidirect
7. #11220C2semidirect
8. #11219C2semidirect
11. #10842C2semidirect
93. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#19932
C2×#19902
C2×#18472
C2×#17502
#143402
#112202
#112192
#110962
#110952
#108422