Finite Group Databasebeta

← #160657

Subgroups — #160657

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34321C211yesmaximal, characteristic
2#22785C211yesmaximal, characteristic, complemented
3#21996C211yesmaximal, characteristic, complemented
4#7604C211yesmaximal, characteristic, complemented
5#7599C211yesmaximal, characteristic, complemented
6#7598C211yesmaximal, characteristic, complemented
7#7594C211yesmaximal, characteristic, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7586C211yesmaximal, characteristic, complemented
10#7582C211yesmaximal, characteristic, complemented
11#7580C211yesmaximal, characteristic, complemented
12#7579C211yesmaximal, characteristic, complemented
13#7579C211yesmaximal, characteristic, complemented
14#7565C211yesmaximal, characteristic, complemented
15#7564C211yesmaximal, characteristic, complemented
16C2×#555C411yescharacteristic
17#2369C411yescharacteristic
18#2367—21no—
19#2367—21no—
20#2363C411yescharacteristic
21#2363C411yescharacteristic
22#2361—21no—
23#2354C411yescharacteristic
24D4×C4⋊C4C411yescharacteristic
25#2347—21no—
26#2347C411yescharacteristic
27#2347C411yescharacteristic
28#2337—21no—
29#2337—21no—
30#2324—21no—
31C4×C4⋊Q8C2211yescharacteristic
32C4×#121C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#2296C2211yescharacteristic
35#2296C2211yescharacteristic
36#2293C2211yescharacteristic
37#2274—21no—
38C42×D4C2211yescharacteristic
39#1923—21no—
40#1923—21no—
41#1916—21no—
42#1916—21no—
43#1869—21no—
44#1869—21no—
45#1850—21no—
46#1839C2211yescharacteristic
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56#1744C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C23×C8—21no—
77#555—24no—
78#555C818yescomplemented
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#124—21no—
81C2×#124—21no—
82C2×#121—21no—
83C2×#121C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×#117—21no—
95C2×#117—21no—
96C2×#117C2×C411yescharacteristic
97C2×#117C2×C411yescharacteristic
98C2×#117C2×C411yescharacteristic
99C2×#117C2×C411yescharacteristic
100C22×C4⋊C4—21no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#391—21no—
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390C2×C411yescharacteristic
122#390C2311yescharacteristic
123#389—21no—
124#389—21no—
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2311yescharacteristic
131#389C2311yescharacteristic
132#389C2311yescharacteristic
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#386—21no—
146#386—21no—
147#386—21no—
148#386—21no—
149#385—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#384C2×C411yescharacteristic
156#383—21no—
157#383—21no—
158#380—21no—
159#380C2×C411yescharacteristic
160#380C2×C411yescharacteristic
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C43C2311yescharacteristic
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C22×Q8—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C22⋊C411yescharacteristic
212C22×D4C22×C411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—42no—
231C22×C8—42no—
232C22×C8—42no—
233C22×C8—81no—
234C22×C8—81no—
235C22×C8—81no—
236C4⋊Q8—22no—
237C4⋊Q8C2×C814yescomplemented
238#124—28no—
239#124—28no—
240#121—28no—
241#121—22no—
242#121—22no—
243#121C2×C814yescomplemented
244#121C2×C814yescomplemented
245C4×Q8—81no—
246C4×Q8—81no—
247C4×D4—22no—
248C4×D4—41no—
249C4×D4—41no—
250C4×D4—42no—
251C4×D4—81no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4—81no—
255C4×D4—81no—
256C4×D4—81no—
257C4×D4C2×C814yes—
258C4×D4C2×C814yescomplemented
259C4×D4M1614yes—
260#117—22no—
261#117—22no—
262#117—28no—
263#117—42no—
264#117—42no—
265#117—81no—
266#117—81no—
267#117C2×C814yescomplemented
268#117C2×C814yescomplemented
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—81no—
322C2×C42—81no—
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2411yescharacteristic
336C4⋊C8—81no—
337C4⋊C8—81no—
338C4⋊C8—81no—
339C22⋊C8—42no—
340C22⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—81no—
343C22⋊C8—81no—
344C22⋊C8—81no—
345C4×C8—42no—
346C4×C8—81no—
347C4×C8—81no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369C24—21no—
370C24C2×C22⋊C411yescharacteristic
371C24C2×C22⋊C411yescharacteristic
372C2×Q8—24no—
373C2×Q8—81no—
374C2×Q8—81no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4C22⋊C814yes—
387C2×D4C22×C812yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C22×C812yes—
435C22×C4C22×C812yes—
436C22×C4C2×M1612yes—
437C22×C4C2×M1612yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—84no—
466C2×C8—84no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4C22⋊C814yes—
491C4⋊C4C22×C812yes—
492C4⋊C4C22×C812yes—
493C4⋊C4C22×C812yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4C22⋊C814yes—
516C22⋊C4C22⋊C814yes—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—24no—
522C42—24no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—42no—
534C42—81no—
535C42—81no—
536C42—81no—
537C42—81no—
538C42—81no—
539C42—81no—
540C42C22×C812yes—
541C42C2×M1612yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C2×C22⋊C812yes—
562C23C2×C22⋊C812yes—
563C23C22×C22⋊C411yescharacteristic
564Q8—84no—
565D4—44no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—162no—
636C2×C4—162no—
637C2×C4C2×C22⋊C812yes—
638C2×C4C2×C22⋊C812yes—
639C2×C4C2×C22⋊C812yes—
640C2×C4C2×C22⋊C812yes—
641C2×C4#42112yes—
642C2×C4C23×C811yescharacteristic
643C2×C4C22×M1611yescharacteristic
644C2×C4#57511yescharacteristic
645C2×C4#57511yescharacteristic
646C2×C4#57611yescharacteristic
647C2×C4#57711yescharacteristic
648C8—42no—
649C8—81no—
650C8—81no—
651C8—81no—
652C8—81no—
653C8—81no—
654C8—81no—
655C8—81no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22C22×C22⋊C811yescharacteristic
678C22#288311yescharacteristic
679C22#288911yescharacteristic
680C22#289611yescharacteristic
681C22#289711yescharacteristic
682C22C2×#42111yescharacteristic
683C22#290111yescharacteristic
684C4—22no—
685C4—22no—
686C4—21no—
687C4—21no—
688C4—22no—
689C4—21no—
690C4—21no—
691C4—22no—
692C4—22no—
693C4—21no—
694C4—21no—
695C4—41no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—81no—
707C2—42no—
708C2—42no—
709C2—82no—
710C2#1735111yescharacteristic
711C2#1743711yescharacteristic
712C2#1743811yescharacteristic
713C2#1759711yescharacteristic
714C2#1762911yescharacteristic
715C2#1768311yescharacteristic
716C2#1776911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#555C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22785C2semidirect
3. #21996C2semidirect
4. #7604C2semidirect
5. #7599C2semidirect
6. #7598C2semidirect
7. #7594C2semidirect
8. #7588C2semidirect
9. #7586C2semidirect
10. #7582C2semidirect
11. #7580C2semidirect
12. #7579C2semidirect
13. #7579C2semidirect
14. #7565C2semidirect
15. #7564C2semidirect
78. #555C8semidirect
237. C4⋊Q8C2×C8semidirect
243. #121C2×C8semidirect
244. #121C2×C8semidirect
258. C4×D4C2×C8semidirect
267. #117C2×C8semidirect
268. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
#343212
#227852
#219962
#76042
#75992
#75982
#75942
#75882
#75862
#75822
#75802
#75792
#75652
#75642