Finite Group Databasebeta

← #192916

Subgroups — #192916

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60266C211yesmaximal, characteristic
2#21900C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic, complemented
4#17793C211yesmaximal, characteristic, complemented
5#17473C211yesmaximal, characteristic
6#8905C212yesmaximal, complemented
7#8904C212yesmaximal, complemented
8#8903C212yesmaximal, complemented
9#8902C212yesmaximal, complemented
10#8094C211yesmaximal, characteristic, complemented
11#7897C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2995—22no—
14#2993—22no—
15#2991—21no—
16#2990—21no—
17#2989—22no—
18#2962—21no—
19#2935—21no—
20#2930—22no—
21#2925C2212yes—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25#2906C2212yes—
26C2×#423C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2285C2211yescharacteristic
29#1986—21no—
30#1980—21no—
31#1897—22no—
32#1806C2212yes—
33#1806C2212yes—
34#1574—21no—
35#1574—21no—
36#1571C2212yes—
37#1570C2212yes—
38#1569—22no—
39#1568—22no—
40#1567C2211yescharacteristic
41#1567C2211yescharacteristic
42#1566C2211yescharacteristic
43#1566C2211yescharacteristic
44#1564—22no—
45#1494C412yescomplemented
46#1494C412yescomplemented
47#1493C411yescharacteristic, complemented
48#1493C411yescharacteristic, complemented
49#1492C411yescharacteristic, complemented
50#1492C411yescharacteristic, complemented
51#1476C2212yes—
52#1476C2212yes—
53#1303C2212yes—
54#1303C2212yes—
55#1279C2212yes—
56#1279C2212yes—
57C2×#131—22no—
58C22×M16—21no—
59C23×C8—21no—
60#533C2×C411yescharacteristic
61#533D411yescharacteristic
62#533Q811yescharacteristic
63#532C2×C411yescharacteristic
64#532D411yescharacteristic
65#532Q811yescharacteristic
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529C2×C412yes—
71#529Q812yes—
72#525—21no—
73#525—21no—
74#525D412yes—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41C2312yes—
78C2×#117—21no—
79C22×C42C2311yescharacteristic
80#446—24no—
81#446—22no—
82C8×Q8—22no—
83#436—24no—
84#436—24no—
85#436—22no—
86#435—24no—
87#435—24no—
88#435—24no—
89C8×D4—22no—
90#433—24no—
91#432—21no—
92#432—21no—
93#432—22no—
94#432—21no—
95#432—21no—
96#432C2×C412yes—
97#432D412yes—
98#431—21no—
99#431—21no—
100#431—22no—
101#431—21no—
102#431—21no—
103#431D412yes—
104#431C2312yes—
105#424—22no—
106#424—24no—
107#423—21no—
108#423—21no—
109#423—22no—
110#423C2×C411yescharacteristic
111#423C2×C411yescharacteristic
112#423D412yes—
113#423D411yescharacteristic
114#423D411yescharacteristic
115#423C2311yescharacteristic
116#423C2311yescharacteristic
117#423C2311yescharacteristic
118#423C2311yescharacteristic
119C2×C4⋊C8—21no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8C2312yes—
122C2×#104—22no—
123C2×#104—22no—
124#408—22no—
125#407—21no—
126#407—21no—
127#407—21no—
128#405—22no—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8C2×C411yescharacteristic
136C2×C4×C8C2×C411yescharacteristic
137C2×C4×C8C2×C411yescharacteristic
138C2×C4×C8C2×C411yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141#384—21no—
142#382—21no—
143#380—21no—
144#380—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2312yes—
147#339—22no—
148#339—22no—
149#335—22no—
150#335—22no—
151#335—22no—
152#335—22no—
153#334—22no—
154#334—22no—
155#334—22no—
156#334—22no—
157Q8⋊C8C2×C412yes—
158Q8⋊C8C2×C412yes—
159Q8⋊C8C2×C412yes—
160Q8⋊C8C2×C412yes—
161D4⋊C8C2×C412yes—
162D4⋊C8C2×C412yes—
163D4⋊C8C2×C412yes—
164D4⋊C8C2×C412yes—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C23×C4#4111yescharacteristic
171#131—82no—
172#131—82no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—24no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—81no—
184C2×M16—81no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—24no—
193C22×C8—44no—
194C4⋊Q8C4⋊C411yescharacteristic
195C4⋊Q8C4⋊C411yescharacteristic
196C4⋊Q8C4⋊C411yescharacteristic
197#127C4⋊C411yescharacteristic
198#126—22no—
199#126—22no—
200#125—21no—
201#125—21no—
202#125C4⋊C412yes—
203#124—21no—
204#124—21no—
205#124C4⋊C412yes—
206#123—22no—
207#123—22no—
208#123—22no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8C4⋊C412yes—
214C22⋊Q8C4⋊C412yes—
215#121—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219#121C4⋊C412yes—
220#121C4⋊C412yes—
221#120—22no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C22×C412yes—
227C4×Q8C2×Q812yes—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C22×C412yes—
239C4×D4C2×Q812yes—
240#117—21no—
241#117—21no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—81no—
246#117—81no—
247#117C2×D412yes—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×Q811yescharacteristic
262C2×C42C2×Q811yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—24no—
267C4⋊C8—24no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—22no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—22no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8C22×C412yes—
281C4⋊C8C22×C412yes—
282C4⋊C8C2×D412yes—
283C4⋊C8C2×D412yes—
284C4⋊C8C2×D412yes—
285C4⋊C8C2×D412yes—
286#104—82no—
287#104—82no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296#97—24no—
297#97—22no—
298#97—22no—
299#97—41no—
300C4×C8—24no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—22no—
306C4×C8—21no—
307C4×C8—22no—
308C4×C8—21no—
309C4×C8—41no—
310C4×C8C22×C411yescharacteristic
311C4×C8C22×C411yescharacteristic
312C4×C8C22×C411yescharacteristic
313C4×C8C22×C411yescharacteristic
314#95—22no—
315#95—41no—
316C24#12311yescharacteristic
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—81no—
322#41—81no—
323#41—82no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8C2×C4⋊C412yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4C2×C4⋊C412yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4C2×C4⋊C411yescharacteristic
345C22×C4C2×C4⋊C411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4C22⋊Q811yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4C22⋊Q811yescharacteristic
356C22×C4#12311yescharacteristic
357C22×C4#12311yescharacteristic
358C22×C4#12311yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—81no—
371M16—81no—
372M16—81no—
373M16—82no—
374M16—81no—
375M16—84no—
376C2×C8—24no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—84no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4C2×C4⋊C412yes—
412C4⋊C4C2×C4⋊C412yes—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—82no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—42no—
427C42—42no—
428C42C2×C4⋊C411yescharacteristic
429C42C2×C4⋊C411yescharacteristic
430C42C4×D411yescharacteristic
431C42C4×D411yescharacteristic
432C42C4×D411yescharacteristic
433C42C4×D411yescharacteristic
434C42#12011yescharacteristic
435C42#12011yescharacteristic
436C42C22⋊Q811yescharacteristic
437C42C22⋊Q811yescharacteristic
438C42C22⋊Q811yescharacteristic
439C42C22⋊Q811yescharacteristic
440C42C23×C411yescharacteristic
441C42C22×D411yescharacteristic
442C42C22×D411yescharacteristic
443C42C22×Q811yescharacteristic
444C23—21no—
445C23—21no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23#38511yescharacteristic
450C23#38511yescharacteristic
451C23C2×#12311yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—42no—
455Q8—81no—
456D4—22no—
457D4—22no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—81no—
462D4—82no—
463D4—82no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
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—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#38511yescharacteristic
501C2×C4#38511yescharacteristic
502C2×C4#38511yescharacteristic
503C2×C4#38511yescharacteristic
504C2×C4#38511yescharacteristic
505C2×C4#38511yescharacteristic
506C2×C4C22×C4⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×#12011yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4C2×C22⋊Q811yescharacteristic
512C2×C4#57611yescharacteristic
513C2×C4#57711yescharacteristic
514C2×C4#57711yescharacteristic
515C2×C4#57811yescharacteristic
516C8—22no—
517C8—22no—
518C8—41no—
519C8—41no—
520C8—42no—
521C8—42no—
522C8—42no—
523C8—42no—
524C8—42no—
525C8—42no—
526C8—81no—
527C8—81no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22C2×#38511yescharacteristic
542C22#309611yescharacteristic
543C22#309711yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—162no—
556C4#186112yes—
557C4#291211yescharacteristic
558C4#291311yescharacteristic
559C4#295711yescharacteristic
560C4#295711yescharacteristic
561C4#301211yescharacteristic
562C4#301311yescharacteristic
563C2—21no—
564C2—21no—
565C2—41no—
566C2—41no—
567C2—82no—
568C2#2084911yescharacteristic
569C2C2×#186111yescharacteristic
570C2#2088611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17820C2semidirect
4. #17793C2semidirect
6. #8905C2semidirect
7. #8904C2semidirect
8. #8903C2semidirect
9. #8902C2semidirect
10. #8094C2semidirect
11. #7897C2semidirect
45. #1494C4semidirect
46. #1494C4semidirect
47. #1493C4semidirect
48. #1493C4semidirect
49. #1492C4semidirect
50. #1492C4semidirect

Maximal subgroups

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

subgroupindex
#602662
#219002
#178202
#177932
#174732
#89052
#89042
#89032
#89022
#80942
#78972