Finite Group Databasebeta

← #193390

Subgroups — #193390

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60280C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic, complemented
4C2×#1574C211yesmaximal, characteristic, complemented
5#17716C211yesmaximal, characteristic
6#8936C212yesmaximal, complemented
7#8936C212yesmaximal, complemented
8#8935C212yesmaximal, complemented
9#8934C212yesmaximal, complemented
10#7870C211yesmaximal, characteristic, complemented
11#7866C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2994—21no—
14#2994—21no—
15#2994—21no—
16#2994—21no—
17#2986C2211yescharacteristic
18#2986C2211yescharacteristic
19#2979C2211yescharacteristic
20#2979C2211yescharacteristic
21#2977—22no—
22#2965—21no—
23#2965—21no—
24C8×#41—21no—
25C8×#41—21no—
26#2922—22no—
27#2920C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1994C2212yes—
34#1962C2212yes—
35#1871—22no—
36#1806—22no—
37#1574—21no—
38#1574—21no—
39#1574C2212yes—
40#1574C2212yes—
41#1574C2212yes—
42#1574C2212yes—
43#1571—21no—
44#1570—21no—
45#1565—21no—
46#1565—21no—
47C2×#335—21no—
48C2×#334—21no—
49#1554C412yescomplemented
50#1553C412yescomplemented
51#1550C411yescharacteristic, complemented
52#1550C411yescharacteristic, complemented
53#1547C411yescharacteristic, complemented
54#1547C411yescharacteristic, complemented
55#1544C2212yes—
56#1543C2212yes—
57#1276C2212yes—
58#1276C2212yes—
59#1276C2212yes—
60#1276C2212yes—
61C22×M16—21no—
62#533C2×C411yescharacteristic
63#533C2×C411yescharacteristic
64#533C2311yescharacteristic
65#532C2×C411yescharacteristic
66#532C2×C411yescharacteristic
67#532C2311yescharacteristic
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72#529Q812yes—
73#529Q812yes—
74#525—21no—
75#525—21no—
76#525D412yes—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41D412yes—
80C2×#117—21no—
81C22×C42C2311yescharacteristic
82#446—21no—
83#446—21no—
84#446—21no—
85#446—21no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89C8×Q8—21no—
90#436—21no—
91#436—21no—
92#436—21no—
93#436—21no—
94#435—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—22no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4—21no—
106#433—22no—
107#433—22no—
108#433—22no—
109#433—22no—
110#432—21no—
111#432C2×C412yes—
112#431—22no—
113#431—22no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431C2×C412yes—
122#431D411yescharacteristic
123#431D411yescharacteristic
124#431D411yescharacteristic
125#431D411yescharacteristic
126#431C2312yes—
127#431C2312yes—
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#424—22no—
133#424—22no—
134#423—21no—
135#423—21no—
136#423—21no—
137#423C2×C411yescharacteristic
138#423C2×C411yescharacteristic
139#423C2312yes—
140C2×C4⋊C8—21no—
141C2×C4⋊C8—21no—
142C2×C4⋊C8C2×C411yescharacteristic
143C2×C4⋊C8C2×C411yescharacteristic
144C2×C4⋊C8D411yescharacteristic
145C2×C4⋊C8D411yescharacteristic
146C2×C4⋊C8D411yescharacteristic
147C2×C4⋊C8D411yescharacteristic
148C2×#104—24no—
149#408—22no—
150#408—22no—
151#407—21no—
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C4×M16—22no—
156C4×M16—21no—
157C2×C4×C8—22no—
158C2×C4×C8—21no—
159C2×C4×C8—21no—
160C2×C4×C8—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#376—21no—
166C43C2312yes—
167#339—24no—
168#335—22no—
169#335—22no—
170#335—22no—
171#335—22no—
172#334—22no—
173#334—22no—
174#334—22no—
175#334—22no—
176#333C2×C412yes—
177#333C2×C412yes—
178#332C2×C412yes—
179#332C2×C412yes—
180#332C2×C412yes—
181#332C2×C412yes—
182#331C2×C412yes—
183#331C2×C412yes—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C23×C4#4111yescharacteristic
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—41no—
196C2×M16—82no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—41no—
206C4⋊Q8C22×C411yescharacteristic
207C4⋊Q8C22×C411yescharacteristic
208C4⋊Q8C22×C411yescharacteristic
209#127C22×C411yescharacteristic
210#126—22no—
211#126—22no—
212#125—21no—
213#125—21no—
214#125C2×Q812yes—
215#124—21no—
216#124—21no—
217#124C2×Q812yes—
218#123—22no—
219#123—22no—
220#123—22no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C4⋊C412yes—
226C22⋊Q8C4⋊C412yes—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231#121C4⋊C412yes—
232#121C4⋊C412yes—
233#120—22no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8C4⋊C412yes—
239C4×Q8C4⋊C412yes—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4C4⋊C412yes—
251C4×D4C4⋊C412yes—
252#117—21no—
253#117—21no—
254#117—24no—
255#117—22no—
256#117—22no—
257#117—22no—
258#117—42no—
259#117C2×D412yes—
260C2×C4⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×Q811yescharacteristic
274C2×C42C2×Q811yescharacteristic
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—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294C4⋊C8C22×C412yes—
295C4⋊C8C22×C412yes—
296C4⋊C8C22×C412yes—
297C4⋊C8C22×C412yes—
298C4⋊C8C2×D411yescharacteristic
299C4⋊C8C2×D411yescharacteristic
300C4⋊C8C2×D411yescharacteristic
301C4⋊C8C2×D411yescharacteristic
302C4⋊C8C2×D411yescharacteristic
303C4⋊C8C2×D411yescharacteristic
304C4⋊C8C2×D411yescharacteristic
305C4⋊C8C2×D411yescharacteristic
306#104—84no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323#97—22no—
324#97—22no—
325#97—22no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—22no—
334C4×C8—22no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339#95—41no—
340#95—41no—
341C24#12311yescharacteristic
342#41—42no—
343#41—42no—
344#41—42no—
345#41—42no—
346#41—42no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8C2×C4⋊C412yes—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4C2×C4⋊C412yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4C2×C4⋊C411yescharacteristic
368C22×C4C2×C4⋊C411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4#12011yescharacteristic
375C22×C4C22⋊Q811yescharacteristic
376C22×C4C22⋊Q811yescharacteristic
377C22×C4C22⋊Q811yescharacteristic
378C22×C4C22⋊Q811yescharacteristic
379C22×C4#12311yescharacteristic
380C22×C4#12311yescharacteristic
381C22×C4#12311yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385M16—42no—
386M16—42no—
387M16—44no—
388M16—82no—
389M16—82no—
390M16—82no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—81no—
407C2×C8—81no—
408C2×C8—81no—
409C2×C8—81no—
410C2×C8—82no—
411C2×C8—82no—
412C2×C8—82no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4C2×C4⋊C412yes—
425C4⋊C4C2×C4⋊C412yes—
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—24no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42C2×C4⋊C411yescharacteristic
446C42C2×C4⋊C411yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42C4×D411yescharacteristic
451C42#12011yescharacteristic
452C42#12011yescharacteristic
453C42C22⋊Q811yescharacteristic
454C42C22⋊Q811yescharacteristic
455C42C22⋊Q811yescharacteristic
456C42C22⋊Q811yescharacteristic
457C42C23×C411yescharacteristic
458C42C22×D411yescharacteristic
459C42C22×D411yescharacteristic
460C42C22×Q811yescharacteristic
461C23—22no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23#38511yescharacteristic
466C23#38511yescharacteristic
467C23C2×#12311yescharacteristic
468Q8—42no—
469Q8—41no—
470Q8—41no—
471Q8—41no—
472Q8—41no—
473D4—42no—
474D4—44no—
475D4—44no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—82no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4#38511yescharacteristic
517C2×C4#38511yescharacteristic
518C2×C4#38511yescharacteristic
519C2×C4#38511yescharacteristic
520C2×C4#38511yescharacteristic
521C2×C4#38511yescharacteristic
522C2×C4#44312yes—
523C2×C4#44412yes—
524C2×C4C22×C4⋊C411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×#12011yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C8—42no—
531C8—42no—
532C8—42no—
533C8—42no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—82no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22#186612yes—
556C22C2×#38511yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—84no—
569C4#186612yes—
570C4C2×#44311yescharacteristic
571C4C2×#44411yescharacteristic
572C4#295511yescharacteristic
573C4#295511yescharacteristic
574C4#302111yescharacteristic
575C4#302211yescharacteristic
576C2—22no—
577C2—41no—
578C2—41no—
579C2—82no—
580C2#2085811yescharacteristic
581C2C2×#186611yescharacteristic
582C2#2088411yescharacteristic
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. #17809C2semidirect
4. C2×#1574C2semidirect
6. #8936C2semidirect
7. #8936C2semidirect
8. #8935C2semidirect
9. #8934C2semidirect
10. #7870C2semidirect
11. #7866C2semidirect
49. #1554C4semidirect
50. #1553C4semidirect
51. #1550C4semidirect
52. #1550C4semidirect
53. #1547C4semidirect
54. #1547C4semidirect

Maximal subgroups

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

subgroupindex
#602802
#218562
#178092
C2×#15742
#177162
#89362
#89352
#89342
#78702
#78662