Finite Group Databasebeta

← #362933

Subgroups — #362933

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3#14686C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5#11203C214yesmaximal
6#10854C212yesmaximal
7#10846C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C4×C22⋊Q8—21no—
12C4×C22⋊Q8—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C2×#391C411yescharacteristic, complemented
20C2×#389C411yescharacteristic, complemented
21C2×#387C412yescomplemented
22C2×#385C2212yes—
23C2×#384—21no—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C412yescomplemented
31C2×C4×C4⋊C4C411yescharacteristic, complemented
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34#1996—28no—
35C2×#342C2218yes—
36C2×#342C2218yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41#1434—21no—
42C22⋊#95C2212yes—
43C4×#95—21no—
44C23×Q8C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8D414yes—
55C2×C22⋊Q8Q814yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—44no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C412yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C42—22no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42C2×C412yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384—44no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—28no—
116#380—44no—
117#380Q818yescomplemented
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—44no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144C2×#95C2×C412yes—
145#342—216no—
146#342—48no—
147C25C2411yescharacteristic
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8C4⋊C414yescomplemented
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×Q812yes—
188C23×C4C2×Q814yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4#4114yes—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C4×Q8—44no—
201C4×Q8—44no—
202C4×Q8—44no—
203C4×Q8—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C4⋊C414yescomplemented
235C2×C4⋊C4C4⋊C418yescomplemented
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4C2×D414yes—
254C2×C22⋊C4C2×Q818yes—
255C2×C22⋊C4C2×Q814yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24C23×C411yescharacteristic
315C24C22×D411yescharacteristic
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×Q811yescharacteristic
319C24C22×Q812yes—
320C24C2×#4111yescharacteristic
321C24C2×#4111yescharacteristic
322C24C2×#4112yes—
323C24H(5,𝔽2)12yes—
324C242−1+412yes—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—48no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—48no—
331C2×Q8—42no—
332C2×Q8—42no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C4⋊C412yes—
428C22×C4C2×C4⋊C414yes—
429C22×C4C2×C4⋊C414yes—
430C22×C4C2×C4⋊C412yes—
431C22×C4#11712yes—
432C22×C4#11712yes—
433C22×C4C4×Q814yes—
434C22×C4C4×Q814yes—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4#12112yes—
439C22×C4C22⋊Q812yes—
440C22×C4C22⋊Q812yes—
441C22×C4C22⋊Q814yes—
442C22×C4C22⋊Q812yes—
443C22×C4C22⋊Q812yes—
444C22×C4C22⋊Q814yes—
445C22×C4#12412yes—
446C22×C4#12412yes—
447C22×C4C4⋊Q812yes—
448C22×C4C4⋊Q812yes—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—42no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—44no—
494C42—44no—
495C42—44no—
496C42—44no—
497C42—88no—
498C42#9914yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23C22×C22⋊C411yescharacteristic
528C23C22×C4⋊C411yescharacteristic
529C23C2×#11711yescharacteristic
530C23C2×C4×Q812yes—
531C23#52012yes—
532C23C2×#12111yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×C22⋊Q811yescharacteristic
535C23C2×C22⋊Q811yescharacteristic
536C23C2×C22⋊Q812yes—
537C23C2×#12411yescharacteristic
538C23C2×C4⋊Q811yescharacteristic
539C23#53511yescharacteristic
540C23#53611yescharacteristic
541C23#53712yes—
542C23D4×Q812yes—
543C23#55012yes—
544C23#55412yes—
545C23#55612yes—
546C23#55712yes—
547C23#55712yes—
548C23Q8212yes—
549C23#55912yes—
550Q8—48no—
551Q8—48no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—48no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4#38012yes—
635C2×C4#38012yes—
636C2×C4#38012yes—
637C2×C4#38012yes—
638C2×C4#39112yes—
639C2×C4#39112yes—
640C2×C4#39112yes—
641C2×C4#39112yes—
642C2×C4C2×#9912yes—
643C2×C4C2×#9912yes—
644C2×C4C2×#9912yes—
645C2×C4#41012yes—
646C2×C4#41012yes—
647C2×C4#41012yes—
648C22—22no—
649C22—21no—
650C22—22no—
651C22—21no—
652C22—22no—
653C22—21no—
654C22—22no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—22no—
659C22—22no—
660C22—21no—
661C22—22no—
662C22—22no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22C2×#38011yescharacteristic
685C22C2×#39111yescharacteristic
686C22#231111yescharacteristic
687C22Q8×C4⋊C412yes—
688C22#235512yes—
689C22#253812yes—
690C22#254611yescharacteristic
691C22#254811yescharacteristic
692C22#256711yescharacteristic
693C22#256811yescharacteristic
694C22#258212yes—
695C22C22×#9911yescharacteristic
696C22C2×#41011yescharacteristic
697C22C2×#41011yescharacteristic
698C22C2×#41011yescharacteristic
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—41no—
708C4—41no—
709C4—44no—
710C4—44no—
711C4—88no—
712C4—82no—
713C4—82no—
714C4—164no—
715C4#179914yes—
716C4#199614yescomplemented
717C2—22no—
718C2—21no—
719C2—21no—
720C2—21no—
721C2—22no—
722C2—22no—
723C2—22no—
724C2—21no—
725C2#1392611yescharacteristic
726C2C2×#179911yescharacteristic
727C2#2063211yescharacteristic
728C2C2×#199611yescharacteristic
729C2#2133211yescharacteristic
730C2#2187411yescharacteristic
731C2#2187511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×C4×Q8C4semidirect
19. C2×#391C4semidirect
20. C2×#389C4semidirect
21. C2×#387C4semidirect
30. C2×C4×C4⋊C4C4semidirect
31. C2×C4×C4⋊C4C4semidirect
117. #380Q8semidirect
153. C22×Q8C4⋊C4semidirect
234. C2×C4⋊C4C4⋊C4semidirect
235. C2×C4⋊C4C4⋊C4semidirect
716. C4#1996semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#19962
#146862
C4×#3802
#112032
#108542
#108462