Finite Group Databasebeta

← #182936

Subgroups — #182936

Order 512.

961 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60204C211yesmaximal, characteristic
2#20476C211yesmaximal, characteristic, complemented
3#17651C211yesmaximal, characteristic
4#17571C211yesmaximal, characteristic
5C2×#1451C211yesmaximal, characteristic, complemented
6#8249C214yesmaximal, complemented
7#8248C214yesmaximal, complemented
8#8090C211yesmaximal, characteristic, complemented
9#8074C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2966—21no—
12#2966—21no—
13#2966C2212yes—
14#2966C2212yes—
15#2965—21no—
16#2965C412yescomplemented
17C8×#41—21no—
18C8×#41C412yescomplemented
19#2960C411yescharacteristic, complemented
20#2960C411yescharacteristic, complemented
21#2931C411yescharacteristic, complemented
22#2931C411yescharacteristic, complemented
23#2929—21no—
24#2929—21no—
25#2922—21no—
26#2920C2211yescharacteristic
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C4×#131—21no—
30C2×C4×M16C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1770—21no—
34C8×C4⋊C4—21no—
35#1752—21no—
36C8×C22⋊C4—21no—
37#1746C2212yes—
38#1726C2212yes—
39#1531C412yescomplemented
40#1530C412yescomplemented
41#1529C414yescomplemented
42#1524C2214yes—
43#1518C412yes—
44#1517C412yes—
45#1516C414yes—
46#1500C2214yes—
47#1451—21no—
48#1451—21no—
49#1451C2212yes—
50#1451C2212yes—
51C2×#322—21no—
52C2×#322—21no—
53#1302C2214yes—
54#1300C2214yes—
55C2×#131—21no—
56C2×#131—21no—
57C22×M16—21no—
58C23×C8D411yescharacteristic
59C23×C8D411yescharacteristic
60#533C2×C411yescharacteristic
61#533C2×C411yescharacteristic
62#533C2×C411yescharacteristic
63#532C2×C411yescharacteristic
64#532C2×C411yescharacteristic
65#532C2×C411yescharacteristic
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529C2×C412yes—
71#529C2312yes—
72#525—21no—
73#525—21no—
74#525C2×C412yes—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41C2×C412yes—
78C22×C42C2311yescharacteristic
79#446—21no—
80#446—21no—
81#446C2×C412yes—
82#446C2×C412yes—
83C8×Q8—21no—
84C8×Q8—21no—
85C8×Q8C2×C412yes—
86C8×Q8C2×C412yes—
87#436—21no—
88#436—21no—
89#436C2×C412yes—
90#436C2×C412yes—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95C8×D4—22no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4C2×C412yes—
102C8×D4C2×C412yes—
103#433—22no—
104#433—22no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432C2×C412yes—
109#432C2×C412yes—
110#432C2312yes—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—22no—
115#431—22no—
116#431C2×C412yes—
117#431C2×C412yes—
118#431C2312yes—
119#424—21no—
120#424—21no—
121#423C2×C411yescharacteristic
122#423C2×C411yescharacteristic
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125#408—21no—
126#408—21no—
127#407—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8C2×C412yes—
130#405—21no—
131#405—21no—
132#405—22no—
133#405—22no—
134C4×M16—21no—
135C4×M16—21no—
136C4×M16—21no—
137C4×M16—22no—
138C4×M16C2×C412yes—
139C4×M16C2312yes—
140C2×#97—22no—
141C2×#97C2×C412yes—
142C2×#97C2312yes—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—22no—
146C2×C4×C8—22no—
147C2×C4×C8C2×C411yescharacteristic
148C2×C4×C8C2×C411yescharacteristic
149C2×C4×C8C2×C411yescharacteristic
150C2×C4×C8C2×C411yescharacteristic
151C2×C4×C8D411yescharacteristic
152C2×C4×C8D411yescharacteristic
153C2×C4×C8D411yescharacteristic
154C2×C4×C8D411yescharacteristic
155C2×C4×C8D411yescharacteristic
156C2×C4×C8D411yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#383—21no—
160#382C2×C412yes—
161#381C2×C412yes—
162#380—21no—
163C4×C4⋊C4—21no—
164C4×C22⋊C4—21no—
165#376C2312yes—
166#336—22no—
167#336—22no—
168#330C2×C418yes—
169#329C2×C418yes—
170#328C2×C418yes—
171#327C2×C418yes—
172#322—22no—
173#322—22no—
174#322—22no—
175#322—22no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41C4212yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182#131—44no—
183#131—44no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—44no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8C22⋊C411yescharacteristic
199C22×C8C22⋊C411yescharacteristic
200C22×C8C22⋊C411yescharacteristic
201C22×C8C22⋊C411yescharacteristic
202C22×C8C22⋊C411yescharacteristic
203C22×C8C22⋊C411yescharacteristic
204C22×C8C22⋊C411yescharacteristic
205C22×C8C22⋊C411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C4⋊Q8C4211yescharacteristic
211C4⋊Q8C4211yescharacteristic
212C4⋊Q8C4211yescharacteristic
213#127C4211yescharacteristic
214#126—22no—
215#126—22no—
216#125—21no—
217#125—21no—
218#125C22×C412yes—
219#124—21no—
220#124—21no—
221#124C22×C412yes—
222#123—22no—
223#123—22no—
224#123—22no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8C4212yes—
230C22⋊Q8C22×C412yes—
231#121—21no—
232#121—21no—
233#121—21no—
234#121—21no—
235#121C4212yes—
236#121C22×C412yes—
237#120—22no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8C4212yes—
243C4×Q8C22×C412yes—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C4212yes—
255C4×D4C22×C412yes—
256#117—21no—
257#117—21no—
258#117—22no—
259#117—22no—
260#117—22no—
261#117C4212yes—
262C2×C4⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42C2411yescharacteristic
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8C22×C412yes—
289C4⋊C8C22×C412yes—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8C22×C414yes—
300#97—22no—
301#97—22no—
302#97—22no—
303#97—22no—
304#97—22no—
305#97—44no—
306#97C22×C414yes—
307#97C22×C414yes—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—44no—
316C4×C8C22⋊C411yescharacteristic
317C4×C8C22⋊C411yescharacteristic
318C4×C8C22⋊C411yescharacteristic
319C4×C8C22⋊C411yescharacteristic
320C4×C8C22⋊C411yescharacteristic
321C4×C8C22⋊C411yescharacteristic
322C4×C8C22⋊C411yescharacteristic
323C4×C8C22⋊C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C22×C411yescharacteristic
326C4×C8C22×C411yescharacteristic
327C4×C8C22×C411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332#95—22no—
333#95—22no—
334#95—22no—
335#95C22×C414yes—
336C24C4×D411yescharacteristic
337#41—42no—
338#41—42no—
339#41—44no—
340#41—44no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8C2×C4212yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4C2×C4212yes—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4C2×C4211yescharacteristic
359C22×C4C2×C4211yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4#11711yescharacteristic
363C22×C4#11711yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376M16—44no—
377M16—44no—
378M16—44no—
379M16—42no—
380M16—42no—
381M16—44no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—44no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8C2×C22⋊C411yescharacteristic
406C2×C8C2×C22⋊C411yescharacteristic
407C2×C8C2×C22⋊C411yescharacteristic
408C2×C8C2×C22⋊C411yescharacteristic
409C2×C8C2×C22⋊C411yescharacteristic
410C2×C8C2×C22⋊C411yescharacteristic
411C2×C8C2×C22⋊C411yescharacteristic
412C2×C8C2×C22⋊C411yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4C2×C4212yes—
425C4⋊C4C2×C4212yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—44no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—42no—
440C42—42no—
441C42C2×C4211yescharacteristic
442C42C2×C4211yescharacteristic
443C42C2×C22⋊C411yescharacteristic
444C42C2×C22⋊C411yescharacteristic
445C42#11711yescharacteristic
446C42#11711yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42C4×D411yescharacteristic
451C42C4×D411yescharacteristic
452C42C4×D411yescharacteristic
453C42C23×C411yescharacteristic
454C42C23×C411yescharacteristic
455C42C22×D411yescharacteristic
456C42C2×#4111yescharacteristic
457C23—22no—
458C23—22no—
459C23—22no—
460C23—41no—
461C23C4×C22⋊C411yescharacteristic
462C23C4×C22⋊C411yescharacteristic
463C23C2×C4×D411yescharacteristic
464Q8—42no—
465Q8—42no—
466Q8—42no—
467D4—42no—
468D4—42no—
469D4—44no—
470D4—44no—
471D4—44no—
472D4—42no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4C4×C22⋊C411yescharacteristic
504C2×C4C4×C22⋊C411yescharacteristic
505C2×C4C4×C22⋊C411yescharacteristic
506C2×C4C4×C22⋊C411yescharacteristic
507C2×C4C4×C22⋊C411yescharacteristic
508C2×C4C4×C22⋊C411yescharacteristic
509C2×C4#44414yes—
510C2×C4C22×C4211yescharacteristic
511C2×C4C22×C22⋊C411yescharacteristic
512C2×C4C2×#11711yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—21no—
522C8—44no—
523C8—44no—
524C8—44no—
525C8—42no—
526C8—42no—
527C8#42112yes—
528C8#42112yes—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22#175712yes—
541C22C2×C4×C22⋊C411yescharacteristic
542C22C2×#44412yes—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—84no—
556C4#176512yes—
557C4#176512yes—
558C4C2×#42111yescharacteristic
559C4C2×#42111yescharacteristic
560C2—22no—
561C2—22no—
562C2—41no—
563C2—82no—
564C2C2×#175711yescharacteristic
565C2C4×#42111yescharacteristic
566C2C2×#176511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C2×C4×C22⋊C4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20476C2semidirect
5. C2×#1451C2semidirect
6. #8249C2semidirect
7. #8248C2semidirect
8. #8090C2semidirect
9. #8074C2semidirect
16. #2965C4semidirect
18. C8×#41C4semidirect
19. #2960C4semidirect
20. #2960C4semidirect
21. #2931C4semidirect
22. #2931C4semidirect
39. #1531C4semidirect
40. #1530C4semidirect
41. #1529C4semidirect

Maximal subgroups

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

subgroupindex
#602042
#204762
#176512
#175712
C2×#14512
#82492
#82482
#80902
#80742