Finite Group Databasebeta

← #356196

Subgroups — #356196

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3C2×#1770C211yesmaximal, characteristic
4C2×#1763C212yesmaximal
5C2×C4×#103C211yesmaximal, characteristic
6C4⋊C42C211yesmaximal, characteristic
7#10884C212yesmaximal
8#10881C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#97C2212yes—
19C22×C4×C8C2211yescharacteristic
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4C418yescomplemented
22C2×#389—22no—
23C2×#389C2212yes—
24C2×#384—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1803C418yescomplemented
33#1770—24no—
34#1763—28no—
35C4×#103C418yescomplemented
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8C2×C418yes—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8C2×C414yes—
62C2×C4×Q8C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8C2×C414yes—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103C2×C418yes—
91C2×#103C2×C414yes—
92C2×#103C2×C414yes—
93C2×#103D414yes—
94C2×#103Q814yes—
95C2×#97—24no—
96C2×#97—24no—
97C2×#97—42no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8C2×C414yes—
101#389—24no—
102#389—216no—
103#389C2×C418yes—
104#384—28no—
105#384—22no—
106#384—22no—
107#384C2×C414yes—
108#384C2×C414yes—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—42no—
113C4×C4⋊C4—44no—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4D414yes—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D412yes—
123#340—24no—
124#340—24no—
125#340—42no—
126#340—42no—
127#340C2×C418yes—
128#340C2×C414yes—
129#340C2×C414yes—
130#336—24no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8C22×C412yes—
139C22×Q8C22×C412yes—
140C22×Q8C22×C412yes—
141C22×Q8C2×D412yes—
142C22×Q8C2×Q812yes—
143C22×Q8#4112yes—
144C22×Q8#4112yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8C22×C412yes—
173C22×C8C22×C414yes—
174C22×C8C22×C412yes—
175C22×C8C2×D412yes—
176C22×C8C2×Q812yes—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—28no—
180C4×Q8—216no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8—48no—
184C4×Q8C42116yescomplemented
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×Q812yes—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—82no—
258C2×C42C22×C414yes—
259C2×C42C22×C412yes—
260C2×C42C22×C412yes—
261C2×C42C22×C412yes—
262C2×C42C2×D414yes—
263C2×C42C2×D412yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—28no—
267#103—28no—
268#103—216no—
269#103C4⋊C4116yescomplemented
270#97—48no—
271C4×C8—28no—
272#95—28no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—82no—
277#95C22⋊C418yes—
278C24C22×D411yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8C2×C4214yes—
301C2×Q8C2×C4⋊C414yes—
302C2×Q8#11714yes—
303C2×Q8C4×D414yes—
304C2×Q8C4×D414yes—
305C2×Q8C4×Q814yes—
306C2×Q8C4×Q814yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
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—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4C2×C22⋊C414yes—
366C22×C4C2×C22⋊C414yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4C4×D412yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4#13712yes—
375C22×C4#13712yes—
376C22×C4#13712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C23×C411yescharacteristic
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8C2×C4⋊C414yes—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—216no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—416no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4C2×C4214yes—
440C4⋊C4C2×C4⋊C414yes—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—216no—
446C42—416no—
447C42—48no—
448C42—88no—
449C42C2×C4214yes—
450C42#11714yes—
451C42C4×D414yes—
452C42C4×D414yes—
453C23C22×C22⋊C411yescharacteristic
454C23C2×C4×D411yescharacteristic
455C23C2×C4×D411yescharacteristic
456C23D4211yescharacteristic
457C23#54611yescharacteristic
458C23#54811yescharacteristic
459C23D4×Q811yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×#13711yescharacteristic
462C23C2×#13711yescharacteristic
463C23C2×#13711yescharacteristic
464C23#57611yescharacteristic
465C23#57711yescharacteristic
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—24no—
471Q8—24no—
472Q8—24no—
473Q8—24no—
474Q8—24no—
475Q8C4×C4⋊C418yescomplemented
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4C4×C22⋊C414yes—
539C2×C4C4×C22⋊C414yes—
540C2×C4#41714yes—
541C2×C4#41914yes—
542C2×C4#44014yes—
543C2×C4#44014yes—
544C2×C4#44114yes—
545C2×C4#44114yes—
546C2×C4C22×C4211yescharacteristic
547C2×C4C22×C4⋊C411yescharacteristic
548C2×C4C2×#11711yescharacteristic
549C2×C4C2×#11711yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×Q811yescharacteristic
555C2×C4C2×C4×Q811yescharacteristic
556C2×C4#51811yescharacteristic
557C2×C4#51911yescharacteristic
558C2×C4#52011yescharacteristic
559C2×C4#52011yescharacteristic
560C2×C4#52011yescharacteristic
561C2×C4#52011yescharacteristic
562C2×C4D4211yescharacteristic
563C2×C4#54611yescharacteristic
564C2×C4#54711yescharacteristic
565C2×C4#54711yescharacteristic
566C2×C4#54811yescharacteristic
567C2×C4D4×Q811yescharacteristic
568C2×C4#55211yescharacteristic
569C2×C4#55211yescharacteristic
570C2×C4#55311yescharacteristic
571C2×C4#55311yescharacteristic
572C2×C4#55411yescharacteristic
573C2×C4#55411yescharacteristic
574C8—24no—
575C8—42no—
576C8—44no—
577C22C2×C4×C22⋊C411yescharacteristic
578C22D4×C22⋊C411yescharacteristic
579C22Q8×C22⋊C411yescharacteristic
580C22D4×C4⋊C411yescharacteristic
581C22#235211yescharacteristic
582C22#236011yescharacteristic
583C22#236111yescharacteristic
584C22C2×#41911yescharacteristic
585C22C2×#41711yescharacteristic
586C22#289711yescharacteristic
587C22#289811yescharacteristic
588C22C2×#44011yescharacteristic
589C22C2×#44011yescharacteristic
590C22C2×#44111yescharacteristic
591C22C2×#44111yescharacteristic
592C22#294611yescharacteristic
593C22#294711yescharacteristic
594C22#295111yescharacteristic
595C22#295211yescharacteristic
596C22#327411yescharacteristic
597C22#327611yescharacteristic
598C22#327811yescharacteristic
599C22#327811yescharacteristic
600C22#332811yescharacteristic
601C22#332911yescharacteristic
602C22#333211yescharacteristic
603C22#333311yescharacteristic
604C22#336811yescharacteristic
605C22#336911yescharacteristic
606C22#337411yescharacteristic
607C22#337511yescharacteristic
608C22#338511yescharacteristic
609C22#338611yescharacteristic
610C22#338711yescharacteristic
611C22#338711yescharacteristic
612C4—24no—
613C4—28no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—84no—
619C4—84no—
620C4—82no—
621C4#176318yescomplemented
622C4C2×C4×C4⋊C411yescharacteristic
623C4#227411yescharacteristic
624C4#234411yescharacteristic
625C4#234511yescharacteristic
626C4D4×C4⋊C411yescharacteristic
627C4#235211yescharacteristic
628C4#236011yescharacteristic
629C4#236111yescharacteristic
630C2C22⋊C4×C4⋊C411yescharacteristic
631C2C2×#176311yescharacteristic
632C2#2041511yescharacteristic
633C2#2243011yescharacteristic
634C2#2243211yescharacteristic
635C2#2252411yescharacteristic
636C2#2252511yescharacteristic
637C2#2267111yescharacteristic
638C2#2267411yescharacteristic
639C2#2275911yescharacteristic
640C2#2276311yescharacteristic
641C2#2297011yescharacteristic
642C2#2297111yescharacteristic
643C2#2300311yescharacteristic
644C2#2300411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. Q8×C4⋊C4C4semidirect
32. #1803C4semidirect
35. C4×#103C4semidirect
184. C4×Q8C42semidirect
269. #103C4⋊C4semidirect
475. Q8C4×C4⋊C4semidirect
621. C4#1763semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#18032
C2×#17702
C2×#17632
C2×C4×#1032
C4⋊C422
#108842
#108812
#108492
#108412
#107892