Finite Group Databasebeta

← #189169

Subgroups — #189169

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21890C211yesmaximal, characteristic
4#21862C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic, complemented
6#8656C211yesmaximal, characteristic, complemented
7#8655C211yesmaximal, characteristic, complemented
8#8653C211yesmaximal, characteristic, complemented
9#8650C211yesmaximal, characteristic, complemented
10#8649C211yesmaximal, characteristic, complemented
11#8646C211yesmaximal, characteristic, complemented
12#8644C211yesmaximal, characteristic, complemented
13#8643C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic, complemented
15#8039C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994C418yescomplemented
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4—21no—
24C2×C8×D4C2211yescharacteristic
25C2×#433—21no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#424—21no—
29C2×#423C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#1991—21no—
34#1989—21no—
35#1983—21no—
36#1958—21no—
37#1949—21no—
38#1949—21no—
39#1895—21no—
40#1894—21no—
41#1875—21no—
42#1873—21no—
43#1854—21no—
44#1854—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1802C2211yescharacteristic
54#1802C2211yescharacteristic
55#1802C2211yescharacteristic
56#1802C2211yescharacteristic
57#1574—24no—
58C2×#335C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#334C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74#529—22no—
75#529—22no—
76#529C2×C414yes—
77#529Q814yes—
78C2×#126—21no—
79C2×#125D411yescharacteristic
80C2×#124D411yescharacteristic
81C2×#123—21no—
82C2×C22⋊Q8D411yescharacteristic
83C2×#121D411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117—21no—
89C2×#117—21no—
90C22×C42C2311yescharacteristic
91#446—22no—
92#446C2×C414yes—
93C8×Q8—22no—
94C8×Q8C2×C414yes—
95#436—22no—
96#436C2×C414yes—
97#435—28no—
98C8×D4—28no—
99C8×D4—22no—
100C8×D4C2×C414yes—
101#433—28no—
102#431—22no—
103#431—22no—
104#431C2×C414yes—
105#431D414yes—
106C2×#107—21no—
107C2×#107—21no—
108C2×#106—21no—
109C2×#106—21no—
110#424—28no—
111#423—22no—
112#423—22no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8C2×C414yes—
119C2×C4⋊C8D411yescharacteristic
120C2×C4⋊C8D411yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#102—21no—
132C2×#102—21no—
133C2×#102—21no—
134C2×#102—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8D411yescharacteristic
139C2×C22⋊C8D411yescharacteristic
140C2×#97—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382—21no—
150#382—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#335—22no—
172#335—22no—
173#334—22no—
174#334—22no—
175C22×Q8C2×D411yescharacteristic
176C22×D4—21no—
177C22×D4C2×D411yescharacteristic
178C23×C4—21no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×M16—24no—
183C2×M16—42no—
184C2×M16—81no—
185C2×M16—81no—
186C2×M16—81no—
187C2×M16—81no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—81no—
197C22×C8—81no—
198C22×C8—81no—
199C22×C8—81no—
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204#126—24no—
205#126—24no—
206#125—22no—
207#125C4⋊C414yes—
208#124—22no—
209#124C4⋊C414yes—
210#123—24no—
211#123—24no—
212C22⋊Q8—22no—
213C22⋊Q8C4⋊C414yes—
214#121—22no—
215#121C4⋊C414yes—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8C22×C412yes—
219C4×Q8C2×Q812yes—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4C22×C412yes—
225C4×D4C2×Q812yes—
226#117—24no—
227#117—24no—
228#117—81no—
229#117—81no—
230#117—81no—
231#117—81no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C42—21no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C22×C412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×Q812yes—
265C2×C42#4111yescharacteristic
266C2×C42#4112yes—
267C2×C42C2411yescharacteristic
268#107—81no—
269#107—81no—
270#106—81no—
271#106—81no—
272C4⋊C8—24no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—42no—
286C4⋊C8C22⋊C414yes—
287C4⋊C8C22⋊C414yes—
288C4⋊C8C22×C412yes—
289C4⋊C8C22×C412yes—
290#103—81no—
291#103—81no—
292#103—81no—
293#103—81no—
294#102—81no—
295#102—81no—
296#102—81no—
297#102—81no—
298C22⋊C8—24no—
299C22⋊C8—24no—
300C22⋊C8—42no—
301C22⋊C8C22⋊C414yes—
302C22⋊C8C22⋊C414yes—
303#97—24no—
304C4×C8—24no—
305C4×C8—42no—
306C4×C8C22×C412yes—
307C4×C8C22×C412yes—
308C4×C8C2×D412yes—
309C4×C8C2×D412yes—
310#95—41no—
311#95—41no—
312#95—41no—
313#95—41no—
314C24—21no—
315C24#12111yescharacteristic
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8C2×C4⋊C412yes—
322C2×D4—24no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4C2×C4⋊C412yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C4⋊C412yes—
356C22×C4#11712yes—
357C22×C4#12112yes—
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4C22⋊Q812yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366M16—84no—
367M16—84no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—84no—
395C2×C8—84no—
396C2×C8C2×C22⋊C412yes—
397C2×C8C2×C22⋊C412yes—
398C2×C8C2×C22⋊C412yes—
399C2×C8C2×C22⋊C412yes—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4C2×C4⋊C412yes—
428C4⋊C4C2×C4⋊C412yes—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C42—22no—
438C42—22no—
439C42—82no—
440C42—82no—
441C42C2×C22⋊C412yes—
442C42C2×C4⋊C412yes—
443C42#11712yes—
444C42#12112yes—
445C42C22⋊Q812yes—
446C42C22⋊Q812yes—
447C42C23×C411yescharacteristic
448C42C22×D411yescharacteristic
449C42C22×Q811yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23#38012yes—
465C23C2×#12111yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—22no—
472D4—48no—
473D4—41no—
474D4—41no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4#38012yes—
519C2×C4#38012yes—
520C2×C4#38012yes—
521C2×C4C22×C22⋊C411yescharacteristic
522C2×C4C22×C4⋊C411yescharacteristic
523C2×C4C2×#11711yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4#57611yescharacteristic
528C2×C4#57611yescharacteristic
529C2×C4#57711yescharacteristic
530C2×C4#57711yescharacteristic
531C2×C4#57711yescharacteristic
532C2×C4#57711yescharacteristic
533C2×C4#57811yescharacteristic
534C2×C4#57811yescharacteristic
535C8—22no—
536C8—22no—
537C8—22no—
538C8—22no—
539C8—42no—
540C8—42no—
541C8—81no—
542C8—81no—
543C8—81no—
544C8—81no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#38011yescharacteristic
560C22#304511yescharacteristic
561C22#304611yescharacteristic
562C22#306211yescharacteristic
563C22#306311yescharacteristic
564C22#306711yescharacteristic
565C22#306711yescharacteristic
566C4—22no—
567C4—22no—
568C4—21no—
569C4—21no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—161no—
575C4—161no—
576C4—161no—
577C4—161no—
578C4#289711yescharacteristic
579C4#289811yescharacteristic
580C4#291211yescharacteristic
581C4#291311yescharacteristic
582C4#306211yescharacteristic
583C4#306311yescharacteristic
584C4#308311yescharacteristic
585C4#308411yescharacteristic
586C2—42no—
587C2—82no—
588C2#1957811yescharacteristic
589C2#1957911yescharacteristic
590C2#1958611yescharacteristic
591C2#1958711yescharacteristic
592C2#2065211yescharacteristic
593C2#2066411yescharacteristic
594C2#2067811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21984C2semidirect
5. C2×#1574C2semidirect
6. #8656C2semidirect
7. #8655C2semidirect
8. #8653C2semidirect
9. #8650C2semidirect
10. #8649C2semidirect
11. #8646C2semidirect
12. #8644C2semidirect
13. #8643C2semidirect
14. #8045C2semidirect
15. #8039C2semidirect
18. #2994C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#219842
#218902
#218622
C2×#15742
#86562
#86552
#86532
#86502
#86492
#86462
#86442
#86432
#80452
#80392