Finite Group Databasebeta

← #205934

Subgroups — #205934

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic, complemented
3#21800C211yesmaximal, characteristic
4#17817C211yesmaximal, characteristic, complemented
5#17715C211yesmaximal, characteristic, complemented
6#17704C211yesmaximal, characteristic
7#17326C211yesmaximal, characteristic, complemented
8#9812C211yesmaximal, characteristic, complemented
9#9808C211yesmaximal, characteristic, complemented
10#9806C211yesmaximal, characteristic, complemented
11#9804C211yesmaximal, characteristic, complemented
12#9800C211yesmaximal, characteristic, complemented
13#9799C211yesmaximal, characteristic, complemented
14#9796C211yesmaximal, characteristic, complemented
15#9794C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3228—22no—
19#3228—22no—
20#3228—21no—
21#3228—21no—
22#3228—21no—
23#3228—21no—
24C2×#500C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#486C2211yescharacteristic
27C2×#486C2211yescharacteristic
28C2×#482C414yes—
29C2×#478C414yes—
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#423C2211yescharacteristic
35#1997C2211yescharacteristic
36#1997C2211yescharacteristic
37#1967—22no—
38#1966C2211yescharacteristic
39#1966C2211yescharacteristic
40#1966C2211yescharacteristic
41#1966C2211yescharacteristic
42#1965C2211yescharacteristic
43#1965C2211yescharacteristic
44#1901—22no—
45#1801—22no—
46#1785—22no—
47#1574—22no—
48#1573—22no—
49#1567C2211yescharacteristic
50#1567C2211yescharacteristic
51#1566C2211yescharacteristic
52#1566C2211yescharacteristic
53#1552—22no—
54#1549C2211yescharacteristic
55#1549C2211yescharacteristic
56#1547—22no—
57#1546C2211yescharacteristic
58#1546C2211yescharacteristic
59#1546C2211yescharacteristic
60#1546C2211yescharacteristic
61#1545C2211yescharacteristic
62#1545C2211yescharacteristic
63#1544—22no—
64#1540—22no—
65#1454—21no—
66#1454—21no—
67#1452—21no—
68#1451C2211yescharacteristic
69#1451C2211yescharacteristic
70#1450—21no—
71C8×M16C2211yescharacteristic
72C8×M16C2211yescharacteristic
73C22×#41—21no—
74C22×M16—21no—
75#533—21no—
76#533—21no—
77#533—21no—
78#533—21no—
79#533—21no—
80#533—21no—
81#533—21no—
82#533—21no—
83#532—21no—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532—21no—
89#532—21no—
90#532—21no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#125C2×C412yes—
96C2×#124—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×#121—21no—
100C2×#121C2×C412yes—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×D4—21no—
104C2×C4×D4C2×C412yes—
105C2×#117C2×C412yes—
106C22×C42C2311yescharacteristic
107#500—21no—
108#500—21no—
109#500—22no—
110#498—21no—
111#498—21no—
112#498—22no—
113#486—21no—
114#486—21no—
115#486—21no—
116#486—21no—
117#486—22no—
118#486—22no—
119#482—24no—
120#482—24no—
121#482—24no—
122#482—24no—
123#478—24no—
124#478—24no—
125#478—24no—
126#478—24no—
127#432—21no—
128#432—21no—
129#432—21no—
130#432—21no—
131#431—21no—
132#431—21no—
133#431—21no—
134#431—21no—
135#431—21no—
136#431—21no—
137#431D412yes—
138#431D412yes—
139#431D412yes—
140#431D411yescharacteristic
141#431D411yescharacteristic
142#431D411yescharacteristic
143#431D411yescharacteristic
144#431C2311yescharacteristic
145#431C2311yescharacteristic
146C2×#107C2×C414yes—
147C2×#106C2×C414yes—
148#423—21no—
149#423—21no—
150#423—21no—
151#423D412yes—
152#423C2311yescharacteristic
153#423C2311yescharacteristic
154#423C2311yescharacteristic
155#423C2311yescharacteristic
156C2×C4⋊C8—21no—
157C2×C4⋊C8—21no—
158C2×C4⋊C8C2×C412yes—
159C2×C4⋊C8D412yes—
160C2×#102C2×C414yes—
161C2×#102C2×C414yes—
162C2×#101—22no—
163C2×#101—22no—
164C2×#101—22no—
165C2×#101—22no—
166C2×#100—22no—
167C2×#100—22no—
168#407—21no—
169#407D412yes—
170C2×C22⋊C8—21no—
171C2×C22⋊C8C2×C412yes—
172C4×M16—21no—
173C4×M16C2311yescharacteristic
174C4×M16C2311yescharacteristic
175C2×C4×C8—21no—
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178#343—22no—
179#343—22no—
180#335—22no—
181#335—22no—
182#334—22no—
183#334—22no—
184#332—22no—
185#332—22no—
186#332—22no—
187#332—22no—
188#332—22no—
189#332—22no—
190#331—22no—
191#331—22no—
192#322—21no—
193#322—21no—
194#322—21no—
195#322—21no—
196#322—21no—
197#322—21no—
198C82—21no—
199C82—21no—
200C2×#41—41no—
201C2×#41—41no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—82no—
205C2×#41—82no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×C412yes—
213C23×C4—22no—
214C23×C4C22×C411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4#4111yescharacteristic
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—41no—
222C2×M16—82no—
223C2×M16C2×D412yes—
224C2×M16C2×D412yes—
225C22×C8—22no—
226C22×C8—41no—
227C22×C8C22×C412yes—
228C22×C8C22×C412yes—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241#127—21no—
242#127—21no—
243#127—21no—
244#127—21no—
245#125—22no—
246#125—22no—
247#125—22no—
248#125—22no—
249#124—41no—
250#124—41no—
251#124—41no—
252#124—41no—
253C22⋊Q8—41no—
254C22⋊Q8—41no—
255C22⋊Q8—41no—
256C22⋊Q8—41no—
257C22⋊Q8—41no—
258C22⋊Q8—41no—
259C22⋊Q8—41no—
260C22⋊Q8—41no—
261#121—22no—
262#121—22no—
263#121—22no—
264#121—22no—
265#121—41no—
266#121—41no—
267#121—41no—
268#121—41no—
269C4×Q8—41no—
270C4×Q8—41no—
271C4×Q8—41no—
272C4×Q8—41no—
273C4×Q8—41no—
274C4×Q8—41no—
275C4×Q8—41no—
276C4×Q8—41no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—41no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C22×C412yes—
295C2×C4⋊C4C22×C412yes—
296C2×C4⋊C4C22×C412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C42—22no—
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42#4111yescharacteristic
312C2×C42C2411yescharacteristic
313#107—24no—
314#107—24no—
315#106—24no—
316#106—24no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—41no—
329C4⋊C8—41no—
330C4⋊C8C22⋊C412yes—
331C4⋊C8C22⋊C412yes—
332C4⋊C8C22⋊C412yes—
333C4⋊C8C22⋊C412yes—
334C4⋊C8C2×D412yes—
335C4⋊C8C2×D412yes—
336#102—24no—
337#102—24no—
338#102—24no—
339#102—24no—
340#101—82no—
341#101—82no—
342#101—82no—
343#101—82no—
344#100—82no—
345#100—82no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—22no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—41no—
352C22⋊C8—41no—
353C22⋊C8C22⋊C412yes—
354C22⋊C8C22⋊C412yes—
355C22⋊C8C22⋊C412yes—
356C22⋊C8C22⋊C412yes—
357#97—21no—
358#97—21no—
359#97—41no—
360C4×C8—22no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—41no—
366C4×C8C2×D411yescharacteristic
367C4×C8C2×D411yescharacteristic
368C4×C8C2×D411yescharacteristic
369C4×C8C2×D411yescharacteristic
370C4×C8C2×D412yes—
371C4×C8C2×D412yes—
372C24—22no—
373C24C4×D411yescharacteristic
374#41—82no—
375#41—82no—
376#41—82no—
377#41—82no—
378#41—84no—
379#41—84no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—82no—
403C2×D4—84no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—84no—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4#12111yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12111yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12411yescharacteristic
426C22×C4#12411yescharacteristic
427C22×C4#12711yescharacteristic
428C22×C4#12711yescharacteristic
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4111yescharacteristic
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—42no—
437M16—42no—
438M16—42no—
439M16—42no—
440M16—82no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—44no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—82no—
453C2×C8C2×C22⋊C412yes—
454C2×C8C2×C22⋊C412yes—
455C2×C8C2×C22⋊C412yes—
456C2×C8C2×C22⋊C412yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C42—42no—
480C42—42no—
481C42C2×C22⋊C411yescharacteristic
482C42C2×C22⋊C411yescharacteristic
483C42C4×D411yescharacteristic
484C42C4×D411yescharacteristic
485C42#12111yescharacteristic
486C42#12111yescharacteristic
487C42#12111yescharacteristic
488C42#12111yescharacteristic
489C42#12411yescharacteristic
490C42#12411yescharacteristic
491C42#12711yescharacteristic
492C42#12711yescharacteristic
493C42C22×D411yescharacteristic
494C42C22×D411yescharacteristic
495C42C22×D411yescharacteristic
496C42C2×#4111yescharacteristic
497C23—22no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—44no—
502C23—82no—
503C23—82no—
504C23#39011yescharacteristic
505C23#39011yescharacteristic
506C23C2×C4×D411yescharacteristic
507Q8—82no—
508Q8—81no—
509Q8—81no—
510Q8—81no—
511Q8—81no—
512Q8—82no—
513Q8—82no—
514Q8—82no—
515D4—44no—
516D4—82no—
517D4—82no—
518D4—84no—
519D4—84no—
520D4—81no—
521D4—81no—
522D4—82no—
523D4—82no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—82no—
551C2×C4#39011yescharacteristic
552C2×C4#39011yescharacteristic
553C2×C4#39011yescharacteristic
554C2×C4#39011yescharacteristic
555C2×C4#39011yescharacteristic
556C2×C4#39011yescharacteristic
557C2×C4C22×C22⋊C411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×#12111yescharacteristic
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12411yescharacteristic
562C2×C4C2×#12711yescharacteristic
563C2×C4#57612yes—
564C2×C4#57712yes—
565C8—22no—
566C8—22no—
567C8—22no—
568C8—22no—
569C8—42no—
570C8—42no—
571C8—42no—
572C8—42no—
573C8—81no—
574C8—81no—
575C8—81no—
576C8—81no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—164no—
590C22C2×#39011yescharacteristic
591C22#294612yes—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4#288912yes—
604C4#306212yes—
605C4#306712yes—
606C4#315411yescharacteristic
607C4#315511yescharacteristic
608C2—22no—
609C2—41no—
610C2—41no—
611C2—162no—
612C2#2122011yescharacteristic
613C2#2123511yescharacteristic
614C2#2124811yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21879C2semidirect
4. #17817C2semidirect
5. #17715C2semidirect
7. #17326C2semidirect
8. #9812C2semidirect
9. #9808C2semidirect
10. #9806C2semidirect
11. #9804C2semidirect
12. #9800C2semidirect
13. #9799C2semidirect
14. #9796C2semidirect
15. #9794C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#218792
#218002
#178172
#177152
#177042
#173262
#98122
#98082
#98062
#98042
#98002
#97992
#97962
#97942