Finite Group Databasebeta

← #205691

Subgroups — #205691

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17747C211yesmaximal, characteristic, complemented
4#17745C211yesmaximal, characteristic
5#17714C211yesmaximal, characteristic
6#17708C211yesmaximal, characteristic, complemented
7#17325C211yesmaximal, characteristic, complemented
8#9778C211yesmaximal, characteristic, complemented
9#9776C211yesmaximal, characteristic, complemented
10#9774C211yesmaximal, characteristic, complemented
11#9774C211yesmaximal, characteristic, complemented
12#9771C211yesmaximal, characteristic, complemented
13#9769C211yesmaximal, characteristic, complemented
14#9769C211yesmaximal, characteristic, complemented
15#9767C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3228—21no—
19#3228—21no—
20#3228—21no—
21#3228—21no—
22#3228C414yes—
23#3228C414yes—
24C2×#500C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#486C2211yescharacteristic
27C2×#486C2211yescharacteristic
28C2×#482—22no—
29C2×#478—22no—
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#423C2211yescharacteristic
35#1950—22no—
36#1855—22no—
37#1848C2211yescharacteristic
38#1848C2211yescharacteristic
39#1837C2211yescharacteristic
40#1837C2211yescharacteristic
41#1559—22no—
42#1557—22no—
43#1554—22no—
44#1551—22no—
45#1550—22no—
46#1549C2211yescharacteristic
47#1549C2211yescharacteristic
48#1549C2211yescharacteristic
49#1549C2211yescharacteristic
50#1549C2211yescharacteristic
51#1549C2211yescharacteristic
52#1548C2211yescharacteristic
53#1548C2211yescharacteristic
54#1548C2211yescharacteristic
55#1548C2211yescharacteristic
56#1548C2211yescharacteristic
57#1548C2211yescharacteristic
58#1547—22no—
59#1545C2211yescharacteristic
60#1545C2211yescharacteristic
61#1545C2211yescharacteristic
62#1545C2211yescharacteristic
63#1543—22no—
64#1542—22no—
65#1453—21no—
66#1453—21no—
67#1452—21no—
68#1452C2211yescharacteristic
69#1452C2211yescharacteristic
70#1450—21no—
71#1450C2211yescharacteristic
72#1450C2211yescharacteristic
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#533C2×C412yes—
82#533C2×C412yes—
83#532—21no—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532—21no—
89#532C2×C412yes—
90#532C2×C412yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#125—21no—
96C2×#124—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×#121—21no—
100C2×#121—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×#117—21no—
106C22×C42C2311yescharacteristic
107#500—21no—
108#500—21no—
109#500C2×C414yes—
110#498—21no—
111#498—21no—
112#498C2×C414yes—
113#486—21no—
114#486—21no—
115#486—21no—
116#486—21no—
117#486C2×C414yes—
118#486C2×C414yes—
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#432D412yes—
131#431—21no—
132#431—21no—
133#431—21no—
134#431—21no—
135#431—21no—
136#431C2×C412yes—
137#431C2×C412yes—
138#431D412yes—
139#431D411yescharacteristic
140#431D411yescharacteristic
141#431D411yescharacteristic
142#431D411yescharacteristic
143#431C2311yescharacteristic
144#431C2311yescharacteristic
145#431C2311yescharacteristic
146#431C2311yescharacteristic
147C2×#108—22no—
148C2×#108—22no—
149C2×#107—22no—
150C2×#106—22no—
151#423—21no—
152#423—21no—
153#423—21no—
154#423D412yes—
155#423C2311yescharacteristic
156#423C2311yescharacteristic
157#423C2311yescharacteristic
158#423C2311yescharacteristic
159C2×C4⋊C8—21no—
160C2×C4⋊C8—21no—
161C2×C4⋊C8D412yes—
162C2×C4⋊C8D412yes—
163C2×#102—22no—
164C2×#102—22no—
165#407—21no—
166#407—21no—
167C2×C22⋊C8—21no—
168C2×C22⋊C8D412yes—
169C4×M16—21no—
170C4×M16—21no—
171C2×C4×C8—21no—
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174#341—22no—
175#341—22no—
176#333—22no—
177#333—22no—
178#333—22no—
179#333—22no—
180#333—22no—
181#333—22no—
182#332—22no—
183#332—22no—
184#332—22no—
185#332—22no—
186#332—22no—
187#332—22no—
188#331—22no—
189#331—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—22no—
201C2×#41—22no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—84no—
205C22×Q8—21no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—41no—
212C23×C4—41no—
213C23×C4C2×D411yescharacteristic
214C23×C4#4111yescharacteristic
215C23×C4#4111yescharacteristic
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—41no—
222C2×M16—82no—
223C22×C8—22no—
224C22×C8—41no—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8C22×C412yes—
237C4⋊Q8C22×C412yes—
238C4⋊Q8C22×C412yes—
239#127—21no—
240#127—21no—
241#127—21no—
242#127C22×C412yes—
243#125—22no—
244#125—22no—
245#125—41no—
246#125—41no—
247#124—22no—
248#124—22no—
249#124—41no—
250#124—41no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—41no—
256C22⋊Q8—41no—
257C22⋊Q8—41no—
258C22⋊Q8—41no—
259#121—22no—
260#121—22no—
261#121—22no—
262#121—22no—
263#121—41no—
264#121—41no—
265#121—41no—
266#121—41no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—41no—
272C4×Q8—41no—
273C4×Q8—41no—
274C4×Q8—41no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—41no—
283#117—22no—
284#117—22no—
285#117—41no—
286#117—41no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C42—41no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42C2411yescharacteristic
311#108—82no—
312#108—82no—
313#107—24no—
314#107—24no—
315#106—24no—
316#106—24no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329C4⋊C8C22⋊C412yes—
330C4⋊C8C22⋊C412yes—
331C4⋊C8C22⋊C412yes—
332C4⋊C8C22⋊C412yes—
333C4⋊C8C2×D412yes—
334C4⋊C8C2×D412yes—
335C4⋊C8C2×D412yes—
336C4⋊C8C2×D412yes—
337#102—24no—
338#102—24no—
339#102—24no—
340#102—24no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—41no—
348C22⋊C8—41no—
349C22⋊C8C22⋊C412yes—
350C22⋊C8C22⋊C412yes—
351C22⋊C8C22⋊C412yes—
352C22⋊C8C22⋊C412yes—
353#97—22no—
354#97—41no—
355C4×C8—22no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—41no—
361C4×C8C22×C412yes—
362C4×C8C22×C412yes—
363C4×C8C2×D411yescharacteristic
364C4×C8C2×D411yescharacteristic
365C4×C8C2×D411yescharacteristic
366C4×C8C2×D411yescharacteristic
367C24—41no—
368C24#12411yescharacteristic
369#41—84no—
370#41—84no—
371#41—84no—
372#41—84no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—84no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—84no—
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12411yescharacteristic
416C22×C4#12411yescharacteristic
417C22×C4#12411yescharacteristic
418C22×C4#12711yescharacteristic
419C22×C4#12711yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423M16—42no—
424M16—42no—
425M16—42no—
426M16—42no—
427M16—82no—
428M16—82no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—44no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—82no—
443C2×C8C2×C22⋊C412yes—
444C2×C8C2×C22⋊C412yes—
445C2×C8C2×C22⋊C412yes—
446C2×C8C2×C22⋊C412yes—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C42—42no—
470C42—42no—
471C42C2×C22⋊C411yescharacteristic
472C42C2×C22⋊C411yescharacteristic
473C42C4×D411yescharacteristic
474C42C4×D411yescharacteristic
475C42C4×D411yescharacteristic
476C42C4×D411yescharacteristic
477C42#12111yescharacteristic
478C42#12111yescharacteristic
479C42#12111yescharacteristic
480C42#12111yescharacteristic
481C42#12711yescharacteristic
482C42#12711yescharacteristic
483C42C23×C411yescharacteristic
484C42C22×D411yescharacteristic
485C42C22×D411yescharacteristic
486C42C22×D411yescharacteristic
487C23—22no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23#39011yescharacteristic
495C23#39011yescharacteristic
496C23C2×#12411yescharacteristic
497Q8—82no—
498Q8—82no—
499Q8—82no—
500Q8—82no—
501Q8—82no—
502Q8—82no—
503D4—44no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—84no—
508D4—81no—
509D4—81no—
510D4—82no—
511D4—82no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4#39011yescharacteristic
540C2×C4#39011yescharacteristic
541C2×C4#39011yescharacteristic
542C2×C4#39011yescharacteristic
543C2×C4#39011yescharacteristic
544C2×C4#39011yescharacteristic
545C2×C4C22×C22⋊C411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12711yescharacteristic
551C2×C4#57712yes—
552C2×C4#57812yes—
553C8—22no—
554C8—22no—
555C8—22no—
556C8—22no—
557C8—42no—
558C8—42no—
559C8—42no—
560C8—42no—
561C8—81no—
562C8—81no—
563C8—81no—
564C8—81no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—164no—
578C22C2×#39011yescharacteristic
579C22#313712yes—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4#289812yes—
592C4#306712yes—
593C4#306812yes—
594C4#315411yescharacteristic
595C4#315611yescharacteristic
596C2—22no—
597C2—41no—
598C2—41no—
599C2—162no—
600C2#2120511yescharacteristic
601C2#2124811yescharacteristic
602C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ 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. #21993C2semidirect
3. #17747C2semidirect
6. #17708C2semidirect
7. #17325C2semidirect
8. #9778C2semidirect
9. #9776C2semidirect
10. #9774C2semidirect
11. #9774C2semidirect
12. #9771C2semidirect
13. #9769C2semidirect
14. #9769C2semidirect
15. #9767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#219932
#177472
#177452
#177142
#177082
#173252
#97782
#97762
#97742
#97712
#97692
#97672