Finite Group Databasebeta

← #363194

Subgroups — #363194

Order 512.

2838 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C2×#1998C212yesmaximal, complemented
3#21190C212yesmaximal, complemented
4C2×#1800C211yesmaximal, characteristic, complemented
5C2×C4×#100C211yesmaximal, characteristic, complemented
6#10967C212yesmaximal, complemented
7#10967C212yesmaximal, complemented
8#10728C212yesmaximal, complemented
9#10727C212yesmaximal, complemented
10C2×#553—28no—
11C2×#553—24no—
12C2×#553—24no—
13C2×#553—24no—
14C2×#553—24no—
15C22×#126C412yes—
16C22×#125C2211yescharacteristic
17C22×#123C412yes—
18C22×#121C412yes—
19C22×#121C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C412yes—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C2×#423C2212yes—
27C22×#100C2212yes—
28C22×#100C2212yes—
29C2×#407C2214yes—
30C2×#407C2214yes—
31C2×C4×M16C2212yes—
32#1998—28no—
33#1955—24no—
34#1954—24no—
35#1800—24no—
36C4×#100—24no—
37C2×#343C2214yes—
38C2×#343C2214yes—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C23×D4C2×C412yes—
42C23×D4C2×C412yes—
43C23×D4C2×C411yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16C2314yes—
50C22×M16C2314yes—
51#553—232no—
52#553—232no—
53#553—232no—
54#553—232no—
55C2×#126—28no—
56C2×#126—28no—
57C2×#126—28no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125D418yes—
63C2×#123—28no—
64C2×#123—28no—
65C2×#123—24no—
66C2×#123—24no—
67C2×#121—28no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—24no—
73C2×#121—24no—
74C2×#121—28no—
75C2×#121—24no—
76C2×#121—24no—
77C2×#121—28no—
78C2×C4×D4—28no—
79C2×C4×D4—28no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—24no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4D414yes—
89C2×C4×D4D414yes—
90C2×C4×D4D414yes—
91C2×C4×D4D414yes—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—22no—
95C2×#117—22no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4C2×C412yes—
101C22×C4⋊C4C2×C412yes—
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C412yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2×C412yes—
108C22×C22⋊C4C2×C412yes—
109C22×C22⋊C4C2×C412yes—
110C22×C22⋊C4C2×C411yescharacteristic
111C22×C22⋊C4C2×C412yes—
112C22×C22⋊C4C2×C411yescharacteristic
113C22×C42C2×C412yes—
114C22×C42C2311yescharacteristic
115#423—24no—
116#423—24no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×#100—24no—
120C2×#100—28no—
121C2×#100—24no—
122C2×#100—44no—
123#407—28no—
124#407—28no—
125#407—44no—
126#407—44no—
127C2×C22⋊C8—28no—
128C2×C22⋊C8—28no—
129C4×M16—24no—
130C4×M16—24no—
131C2×#97—22no—
132C2×C4×C8—22no—
133#343—24no—
134#343—28no—
135#343—28no—
136#343—24no—
137C25C22×C411yescharacteristic
138C25C22×C412yes—
139C25C22×C411yescharacteristic
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—24no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—44no—
154C22×D4—48no—
155C22×D4—84no—
156C22×D4C22⋊C414yes—
157C22×D4C2×D412yes—
158C22×D4C2×D412yes—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—84no—
178C23×C4—84no—
179C23×C4C22⋊C418yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2411yescharacteristic
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16#4118yes—
202C22×C8—24no—
203C22×C8—24no—
204#126—264no—
205#125—28no—
206#125—28no—
207#125—28no—
208#125—28no—
209#123—232no—
210#123—232no—
211#121—264no—
212#121—216no—
213#121—216no—
214#121—216no—
215#121—216no—
216C4×D4—232no—
217C4×D4—232no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—28no—
226#117—216no—
227#117—216no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C22⋊C418yes—
249C2×C4⋊C4C22⋊C414yes—
250C2×C4⋊C4C22⋊C418yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4C22⋊C412yes—
271C2×C22⋊C4C22⋊C412yes—
272C2×C22⋊C4C22⋊C418yes—
273C2×C22⋊C4C22⋊C412yes—
274C2×C22⋊C4C22⋊C412yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42C2×D414yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C4⋊C8—44no—
284C4⋊C8—44no—
285#100—216no—
286#100—48no—
287C22⋊C8—416no—
288C22⋊C8—416no—
289#97—44no—
290C4×C8—44no—
291C24—21no—
292C24—21no—
293C24—24no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—22no—
303C24—24no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—44no—
308C24—44no—
309C24C2×C22⋊C412yes—
310C24C2×C22⋊C412yes—
311C24C2×C22⋊C414yes—
312C24C23×C411yescharacteristic
313C2×D4—216no—
314C2×D4—28no—
315C2×D4—216no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—28no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—416no—
328C2×D4—48no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—88no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
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—216no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C414yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C2×#4114yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408M16—28no—
409M16—28no—
410M16—44no—
411M16—44no—
412M16—44no—
413M16—44no—
414M16—44no—
415M16—44no—
416C2×C8—28no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C4⋊C4—216no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—216no—
433C22⋊C4—216no—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—216no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C42—216no—
442C42#12014yes—
443C42#12014yes—
444C42#12014yes—
445C42#12014yes—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—48no—
462C23—48no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—41no—
483C23—84no—
484C23—84no—
485C23—88no—
486C23—88no—
487C23#41314yes—
488C23C22×C22⋊C411yescharacteristic
489C23C22×C22⋊C411yescharacteristic
490C23C22×C22⋊C411yescharacteristic
491C23C4×#4112yes—
492C23#52012yes—
493D4—416no—
494D4—432no—
495D4—48no—
496D4—44no—
497D4—44no—
498D4—816no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—88no—
535C2×C4—88no—
536C2×C4—816no—
537C2×C4#37914yes—
538C2×C4#37912yes—
539C2×C4#37912yes—
540C2×C4C2×#12012yes—
541C2×C4C2×#12012yes—
542C2×C4#52512yes—
543C2×C4#53212yes—
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53712yes—
547C2×C4#54614yes—
548C2×C4#54714yes—
549C2×C4#54814yes—
550C2×C4#55014yes—
551C8—48no—
552C8—84no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—28no—
561C22—44no—
562C22—48no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—48no—
574C22—84no—
575C22—84no—
576C22—82no—
577C22—84no—
578C22—88no—
579C22—82no—
580C22—88no—
581C22C2×#37911yescharacteristic
582C22#229711yescharacteristic
583C22#231111yescharacteristic
584C22#233912yes—
585C22#234112yes—
586C22C2×#41312yes—
587C22#288812yes—
588C4—24no—
589C4—22no—
590C4—22no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4#240714yes—
597C4#243112yes—
598C4#243212yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—44no—
603C2—82no—
604C2—84no—
605C2#1393711yescharacteristic
606C2#2059111yescharacteristic
607C2#2059211yescharacteristic
608C2#2248214yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#553C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1998C2semidirect
3. #21190C2semidirect
4. C2×#1800C2semidirect
5. C2×C4×#100C2semidirect
6. #10967C2semidirect
7. #10967C2semidirect
8. #10728C2semidirect
9. #10727C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#19982
#211902
C2×#18002
C2×C4×#1002
#109672
#107282
#107272