Finite Group Databasebeta

← #363204

Subgroups — #363204

Order 512.

3014 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C2×#1964C212yesmaximal, complemented
3C2×#1799C211yesmaximal, characteristic, complemented
4C2×C4×#99C211yesmaximal, characteristic, complemented
5#13924C212yesmaximal, complemented
6#10980C214yesmaximal, complemented
7#10761C214yesmaximal, complemented
8C2×#553—28no—
9C2×#553—24no—
10C2×#553—24no—
11C2×#553—24no—
12C2×#553—24no—
13C22×#126C412yescomplemented
14C22×#125C411yescharacteristic, complemented
15C22×#123C412yescomplemented
16C22×#121C411yescharacteristic, complemented
17C22×#121C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C411yescharacteristic, complemented
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C411yescharacteristic, complemented
23C23×C4⋊C4C2211yescharacteristic
24C22×#99C2212yes—
25C22×#99C2218yes—
26C22×#99C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1964—28no—
30#1799—24no—
31C4×#99—24no—
32C2×#342C2212yes—
33C2×#342C2212yes—
34#1445—24no—
35C4⋊#95—24no—
36C22⋊#95C2218yes—
37C23×D4C2×C412yes—
38C23×D4C2×C411yescharacteristic
39C23×D4C2312yes—
40C23×D4C2311yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2×C411yescharacteristic
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45#553—232no—
46#553—232no—
47#553—232no—
48#553—232no—
49C2×#126—28no—
50C2×#126—28no—
51C2×#126—28no—
52C2×#125—24no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#123—28no—
57C2×#123—28no—
58C2×#123—28no—
59C2×#121—24no—
60C2×#121—24no—
61C2×#121—24no—
62C2×#121—28no—
63C2×#121—24no—
64C2×#121—28no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121D418yes—
72C2×C4×D4—24no—
73C2×C4×D4—28no—
74C2×C4×D4—28no—
75C2×C4×D4—22no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4D414yes—
84C2×C4×D4D414yes—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—22no—
88C2×#117—22no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4C2×C412yes—
94C22×C4⋊C4C2×C412yes—
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C4⋊C4C2×C411yescharacteristic
97C22×C4⋊C4C2×C412yes—
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C4⋊C4D414yes—
101C22×C4⋊C4D414yes—
102C22×C22⋊C4C2×C412yes—
103C22×C22⋊C4C2×C412yes—
104C22×C22⋊C4C2×C412yes—
105C22×C22⋊C4C2×C411yescharacteristic
106C22×C22⋊C4C2×C412yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C42—22no—
111C22×C42C2×C412yes—
112C22×C42C2×C411yescharacteristic
113C2×#99—24no—
114C2×#99—28no—
115C2×#99—24no—
116C2×#99—216no—
117C2×#99—216no—
118C2×#99—48no—
119C2×#99—44no—
120#380—24no—
121#380—24no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—24no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—216no—
127#342—24no—
128#342—24no—
129C25C22×C411yescharacteristic
130C25C22×C412yes—
131C25C2411yescharacteristic
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—24no—
138C22×D4—28no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—24no—
149C22×D4—44no—
150C22×D4—84no—
151C22×D4C22⋊C414yes—
152C22×D4C2×D418yes—
153C22×D4C2×D412yes—
154C22×D4C2×D412yes—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—84no—
169C23×C4—84no—
170C23×C4C22⋊C414yes—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C412yes—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D418yes—
181C23×C4C2×D414yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185#126—264no—
186#125—216no—
187#125—216no—
188#123—264no—
189#121—232no—
190#121—232no—
191#121—216no—
192#121—216no—
193#121—28no—
194#121—28no—
195#121—28no—
196#121—28no—
197C4×D4—232no—
198C4×D4—232no—
199C4×D4—216no—
200C4×D4—216no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—28no—
205#117—216no—
206#117—216no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C412yes—
234C2×C4⋊C4C22⋊C418yes—
235C2×C4⋊C4C22⋊C414yes—
236C2×C4⋊C4C22⋊C414yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4C22⋊C414yes—
262C2×C22⋊C4C22⋊C418yes—
263C2×C22⋊C4C2×D412yes—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4#4118yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—48no—
273C2×C42—44no—
274C2×C42C22⋊C414yes—
275#99—216no—
276#99—432no—
277#99—48no—
278#95—44no—
279#95—44no—
280#95—816no—
281C24—21no—
282C24—22no—
283C24—21no—
284C24—22no—
285C24—24no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—44no—
293C24—44no—
294C24—42no—
295C24C2×C22⋊C414yes—
296C24C2×C22⋊C412yes—
297C24C23×C411yescharacteristic
298C24C22×D412yes—
299C24C22×D411yescharacteristic
300C24C22×D411yescharacteristic
301C24C22×D412yes—
302C24C2×#4114yes—
303C24H(5,𝔽2)12yes—
304C242−1+412yes—
305C2×D4—28no—
306C2×D4—24no—
307C2×D4—28no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—216no—
315C2×D4—216no—
316C2×D4—216no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—44no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—48no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—44no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—416no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4#12014yes—
403C22×C4#12014yes—
404C22×C4#12014yes—
405C22×C4#12014yes—
406C4⋊C4—216no—
407C4⋊C4—216no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—216no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—88no—
417C22⋊C4—216no—
418C22⋊C4—216no—
419C22⋊C4—216no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—216no—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C42—216no—
432C42—24no—
433C42—24no—
434C42—88no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—28no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—48no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—84no—
471C23—88no—
472C23—84no—
473C23—88no—
474C23C22×C22⋊C411yescharacteristic
475C23C22×C22⋊C411yescharacteristic
476C23C22×C22⋊C411yescharacteristic
477C23C4×#4112yes—
478C23#52012yes—
479C23C2×#12012yes—
480C23C2×#12012yes—
481C23#52512yes—
482C23#53212yes—
483C23#53511yescharacteristic
484C23#53611yescharacteristic
485C23#53712yes—
486C23#54614yes—
487C23#54714yes—
488C23#54814yes—
489C23#55014yes—
490D4—48no—
491D4—48no—
492D4—416no—
493D4—44no—
494D4—48no—
495D4—44no—
496D4—48no—
497D4—48no—
498D4—816no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—48no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—416no—
538C2×C4—48no—
539C2×C4—88no—
540C2×C4—88no—
541C2×C4—816no—
542C2×C4—84no—
543C2×C4—88no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—816no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4#37912yes—
550C2×C4#37914yes—
551C2×C4#37912yes—
552C2×C4#41014yes—
553C22—28no—
554C22—28no—
555C22—24no—
556C22—24no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—82no—
579C22—88no—
580C22—82no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—88no—
586C22C2×#37911yescharacteristic
587C22#229711yescharacteristic
588C22#231111yescharacteristic
589C22#233912yes—
590C22#234112yes—
591C22#240714yes—
592C22#243112yes—
593C22#243212yes—
594C22C2×#41012yes—
595C22#288312yes—
596C4—22no—
597C4—24no—
598C4—42no—
599C4—44no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—44no—
604C4—88no—
605C4—164no—
606C4#178214yes—
607C2—24no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—21no—
612C2—22no—
613C2—82no—
614C2—84no—
615C2#1393711yescharacteristic
616C2C2×#178211yescharacteristic
617C2#2058611yescharacteristic
618C2#2253514yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#553C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1964C2semidirect
3. C2×#1799C2semidirect
4. C2×C4×#99C2semidirect
5. #13924C2semidirect
6. #10980C2semidirect
7. #10761C2semidirect
13. C22×#126C4semidirect
14. C22×#125C4semidirect
15. C22×#123C4semidirect
16. C22×#121C4semidirect
19. C22×C4×D4C4semidirect
22. C22×#117C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#19642
C2×#17992
C2×C4×#992
#139242
#109802
#107612