Finite Group Databasebeta

← #363174

Subgroups — #363174

Order 512.

2806 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#1998C212yesmaximal, complemented
3C2×#1967C212yesmaximal, complemented
4#20626C211yesmaximal, characteristic, complemented
5C4×#407C211yesmaximal, characteristic, complemented
6#10968C212yesmaximal, complemented
7#10967C212yesmaximal
8#10763C212yesmaximal, complemented
9#10762C212yesmaximal, complemented
10C2×#529—22no—
11C2×#529—22no—
12C2×#529—22no—
13C2×#529—22no—
14C2×#529—28no—
15C2×#529—28no—
16C22×#126C412yes—
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#123C412yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C412yes—
24C22×C4×D4C2211yescharacteristic
25C22×#117C412yes—
26C23×C42C2211yescharacteristic
27C2×#423C2212yes—
28C22×#101C2214yes—
29C22×#100C2214yes—
30C2×#407C2212yes—
31C2×#407C2212yes—
32C2×C4×M16C2212yes—
33#1998—28no—
34#1967—28no—
35#1797—22no—
36#1797—22no—
37#1749—22no—
38C4×C22⋊C8—22no—
39C2×#343C2214yes—
40C2×#343C2214yes—
41#1728C2212yes—
42#1728C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2×C412yes—
45C23×D4C2311yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16C2314yes—
51C22×M16C2314yes—
52#529—232no—
53#529—232no—
54#529—232no—
55#529—232no—
56C2×#126—28no—
57C2×#126—28no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125—21no—
64C2×#125—24no—
65C2×#125D418yes—
66C2×#124—21no—
67C2×#124—21no—
68C2×#124—21no—
69C2×#124—21no—
70C2×#124—24no—
71C2×#124D418yes—
72C2×#123—24no—
73C2×#123—24no—
74C2×#123—28no—
75C2×#123—28no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—24no—
80C2×#121—24no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×C4×D4—28no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—24no—
99C2×#117—24no—
100C22×C4⋊C4C2×C412yes—
101C22×C4⋊C4C2×C412yes—
102C22×C4⋊C4C2×C412yes—
103C22×C4⋊C4C2×C412yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2×C412yes—
107C22×C22⋊C4C2×C412yes—
108C22×C22⋊C4C2×C412yes—
109C22×C22⋊C4C2×C412yes—
110C22×C22⋊C4C2×C412yes—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42C2×C412yes—
114C22×C42C2×C411yescharacteristic
115C22×C42C2×C411yescharacteristic
116C22×C42D412yes—
117C22×C42D412yes—
118C22×C42D412yes—
119C22×C42D412yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#423—24no—
123#423—24no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—22no—
126C2×#101—28no—
127C2×#101—28no—
128C2×#101—44no—
129C2×#100—28no—
130C2×#100—28no—
131C2×#100—44no—
132#407—28no—
133#407—44no—
134C2×C22⋊C8—24no—
135C2×C22⋊C8—24no—
136C4×M16—24no—
137C4×M16—24no—
138C2×#97—22no—
139C2×C4×C8—22no—
140#343—28no—
141#343—28no—
142#336—24no—
143#336—24no—
144C25C22×C411yescharacteristic
145C25C22×C412yes—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C2×D414yes—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—48no—
163C22×D4C2×D414yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—48no—
179C23×C4C22⋊C412yes—
180C23×C4C22⋊C412yes—
181C23×C4C22⋊C412yes—
182C23×C4C22⋊C412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C412yes—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2411yescharacteristic
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16#4118yes—
206C22×C8—24no—
207C22×C8—24no—
208#126—232no—
209#126—232no—
210#125—28no—
211#125—28no—
212#125—28no—
213#125—28no—
214#124—28no—
215#124—28no—
216#124—28no—
217#124—28no—
218#123—232no—
219#123—232no—
220C22⋊Q8—216no—
221C22⋊Q8—216no—
222#121—216no—
223#121—216no—
224C4×Q8—216no—
225C4×Q8—216no—
226C4×D4—232no—
227C4×D4—232no—
228C4×D4—216no—
229C4×D4—216no—
230#117—232no—
231#117—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C22⋊C418yes—
251C2×C4⋊C4C22⋊C418yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
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—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C22⋊C418yes—
270C2×C22⋊C4C22⋊C418yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—28no—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42C22⋊C412yes—
285C2×C42C22⋊C412yes—
286C2×C42C22⋊C412yes—
287C2×C42C22⋊C412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C4⋊C8—44no—
297C4⋊C8—44no—
298#101—416no—
299#100—416no—
300C22⋊C8—216no—
301C22⋊C8—48no—
302#97—44no—
303C4×C8—44no—
304C24—21no—
305C24—21no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—42no—
314C24—44no—
315C24—44no—
316C24C2×C22⋊C412yes—
317C24C2×C22⋊C414yes—
318C24C23×C411yescharacteristic
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—28no—
322C2×Q8—28no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×D4—216no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—416no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—44no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—216no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C414yes—
409C22×C4C2×C22⋊C414yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C2×#4114yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+412yes—
421M16—28no—
422M16—28no—
423M16—44no—
424M16—44no—
425M16—44no—
426M16—44no—
427M16—44no—
428M16—44no—
429C2×C8—28no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—44no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C42—216no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42#12014yes—
460C42#12014yes—
461C42#12014yes—
462C42#12014yes—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—28no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—88no—
491C23—88no—
492C23#41314yes—
493C23C22×C22⋊C411yescharacteristic
494C23C22×C22⋊C411yescharacteristic
495C23C22×C22⋊C411yescharacteristic
496C23C4×#4112yes—
497C23#52012yes—
498Q8—48no—
499Q8—44no—
500Q8—44no—
501D4—432no—
502D4—48no—
503D4—44no—
504D4—44no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—432no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4#37912yes—
549C2×C4#37912yes—
550C2×C4#37912yes—
551C2×C4#37912yes—
552C2×C4#41314yes—
553C2×C4C2×#12012yes—
554C2×C4C2×#12012yes—
555C2×C4#52512yes—
556C2×C4#53212yes—
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53712yes—
560C2×C4#54614yes—
561C2×C4#54714yes—
562C2×C4#54814yes—
563C2×C4#55014yes—
564C8—48no—
565C8—84no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—48no—
576C22—48no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—88no—
585C22—88no—
586C22—84no—
587C22—84no—
588C22C2×#37911yescharacteristic
589C22#229711yescharacteristic
590C22#231111yescharacteristic
591C22#233912yes—
592C22#234112yes—
593C22C2×#41312yes—
594C22C2×#41312yes—
595C4—24no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—44no—
602C4—44no—
603C4—42no—
604C4—44no—
605C4#240714yes—
606C4#243112yes—
607C4#243212yes—
608C2—22no—
609C2—22no—
610C2—44no—
611C2—84no—
612C2#1393711yescharacteristic
613C2#2059111yescharacteristic
614C2#2059111yescharacteristic
615C2#2142612yes—
616C2#2188012yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1998C2semidirect
3. C2×#1967C2semidirect
4. #20626C2semidirect
5. C4×#407C2semidirect
6. #10968C2semidirect
8. #10763C2semidirect
9. #10762C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#19982
C2×#19672
#206262
C4×#4072
#109682
#109672
#107632
#107622