Finite Group Databasebeta

← #206046

Subgroups — #206046

Order 512.

1103 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3191C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#20525C211yesmaximal, characteristic, complemented
4#17542C212yesmaximal, complemented
5#17400C212yesmaximal, complemented
6#9818C211yesmaximal, characteristic, complemented
7#9817C211yesmaximal, characteristic, complemented
8#9816C212yesmaximal, complemented
9#9815C211yesmaximal, characteristic, complemented
10#9814C211yesmaximal, characteristic, complemented
11#9813C212yesmaximal, complemented
12C2×#525C2212yes—
13#3191—22no—
14#3191—22no—
15#3191—21no—
16#3191—21no—
17#3191—21no—
18#3191—21no—
19C2×#485C412yescomplemented
20C2×#482C412yescomplemented
21C2×#467C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×#465C2212yes—
24C2×#452C412yescomplemented
25C2×#449C412yescomplemented
26C2×#407C2212yes—
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1855—22no—
31#1855—22no—
32#1810C2212yes—
33#1810C2211yescharacteristic
34#1810C2211yescharacteristic
35#1753—21no—
36#1753—21no—
37#1752C2211yescharacteristic
38#1752C2211yescharacteristic
39C8×C22⋊C4C2211yescharacteristic
40C8×C22⋊C4C2211yescharacteristic
41#1749—21no—
42C4×C22⋊C8—21no—
43#1508C2212yes—
44#1508C2212yes—
45#1507C2212yes—
46#1507C2212yes—
47#1504—24no—
48#1501—24no—
49#1472C2212yes—
50#1472C2212yes—
51#1471C2212yes—
52#1471C2212yes—
53#1468—24no—
54#1465—24no—
55C22×#41C2×C412yes—
56C24×C4C2311yescharacteristic
57C22×Q16C2×C411yescharacteristic, complemented
58C22×SD16C2×C412yescomplemented
59C22×D8C2×C411yescharacteristic, complemented
60C22×M16—21no—
61C22×M16—22no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64#525—22no—
65#525—22no—
66#525—22no—
67#525—22no—
68#525—22no—
69#525—22no—
70#525—22no—
71#525—22no—
72C2×#123—22no—
73C2×C22⋊Q8C2×C412yes—
74C2×C22⋊Q8C2312yes—
75C2×#121C2×C412yes—
76C2×#121C2312yes—
77C2×#120—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×#117C2×C412yes—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83#485—22no—
84#485—22no—
85#485—22no—
86#485—22no—
87#482—22no—
88#482—22no—
89#482—22no—
90#482—22no—
91#467—21no—
92#467—21no—
93#467—21no—
94#467—21no—
95#466—21no—
96#466—21no—
97#466—21no—
98#466—21no—
99#465—22no—
100#465—22no—
101#465—22no—
102#465—22no—
103#452—22no—
104#452—22no—
105#452—22no—
106#452—22no—
107#449—22no—
108#449—22no—
109#449—22no—
110#449—22no—
111C2×#108—22no—
112C2×#108—21no—
113C2×#108—21no—
114C2×#107C2×C411yescharacteristic, complemented
115C2×#107C2×C411yescharacteristic, complemented
116C2×#106C2×C412yescomplemented
117C2×#103C2×C412yescomplemented
118C2×#103C2×C412yescomplemented
119C2×#102C2×C412yescomplemented
120C2×#102C2×C412yescomplemented
121#407—22no—
122#407—22no—
123#407—22no—
124#407—22no—
125#407D414yes—
126#407C2312yes—
127#407C2312yes—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8C2×C412yes—
132C2×C22⋊C8C2×C412yes—
133C2×C22⋊C8D412yes—
134C2×C22⋊C8D412yes—
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×C22⋊C8D411yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2311yescharacteristic
141C2×#97—21no—
142C2×#97—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4D412yes—
148C4×C22⋊C4C2311yescharacteristic
149C4×C22⋊C4C2311yescharacteristic
150C2×#95D412yes—
151#341—22no—
152#341—21no—
153#341—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#328—24no—
159#328—24no—
160#327—24no—
161#327—24no—
162#324—24no—
163#324—24no—
164#323—24no—
165#323—24no—
166C25#4111yescharacteristic
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—82no—
172C2×#41—82no—
173C22×Q8C22×C412yes—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4C22×C412yes—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—81no—
181C23×C4—81no—
182C23×C4C22×C412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4#4111yescharacteristic
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C2×Q16—21no—
193C2×Q16—21no—
194C2×Q16—81no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—82no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—81no—
201C2×M16—24no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—42no—
205C2×M16—81no—
206C2×M16—81no—
207C2×M16—82no—
208C2×M16—82no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—42no—
212C22×C8—81no—
213C22×C8—81no—
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C412yes—
216C22×C8C22×C411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D412yes—
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224#123—42no—
225#123—42no—
226#123—42no—
227#123—42no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121—22no—
241#121—22no—
242#121—22no—
243#121—22no—
244#120—42no—
245#120—42no—
246#120—42no—
247#120—42no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256#117—22no—
257#117—22no—
258#117—22no—
259#117—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—44no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—81no—
278C2×C42—81no—
279#108—81no—
280#108—81no—
281#108—82no—
282#107—21no—
283#107—21no—
284#107—21no—
285#107—21no—
286#106—22no—
287#106—22no—
288#103—22no—
289#103—22no—
290#103—22no—
291#103—22no—
292#102—22no—
293#102—22no—
294#102—22no—
295#102—22no—
296C22⋊C8—24no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C22⋊C412yes—
307C22⋊C8C22⋊C412yes—
308C22⋊C8C22⋊C412yes—
309C22⋊C8C22⋊C412yes—
310C22⋊C8C22⋊C412yes—
311C22⋊C8C22⋊C412yes—
312C22⋊C8C22⋊C412yes—
313C22⋊C8C2×D412yes—
314C22⋊C8C2×D412yes—
315#97—81no—
316#97—81no—
317C4×C8—81no—
318C4×C8—81no—
319#95—22no—
320#95C2×D412yes—
321#95C2×D412yes—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—81no—
326C24C4×D412yes—
327C24#12111yescharacteristic
328C24#12111yescharacteristic
329C24#12412yes—
330C24C2×#4111yescharacteristic
331#41—82no—
332#41—82no—
333#41—82no—
334#41—82no—
335#41—84no—
336#41—84no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—81no—
374C22×C4—82no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—84no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12412yes—
390C22×C4#12712yes—
391C22×C4#12712yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4111yescharacteristic
398Q16—81no—
399Q16—81no—
400SD16—82no—
401SD16—82no—
402D8—81no—
403D8—81no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—84no—
409M16—84no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—84no—
429C2×C8C2×C22⋊C411yescharacteristic
430C2×C8C2×C22⋊C411yescharacteristic
431C2×C8C2×C22⋊C412yes—
432C2×C8C2×C22⋊C412yes—
433C2×C8C2×C22⋊C411yescharacteristic
434C2×C8C2×C22⋊C411yescharacteristic
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C22⋊C4—24no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—42no—
462C42—82no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—82no—
481C23—82no—
482C23—88no—
483C23#39012yes—
484C23#39012yes—
485C23C2×C4×D411yescharacteristic
486C23C2×#12111yescharacteristic
487C23C2×#12411yescharacteristic
488Q8—42no—
489Q8—42no—
490Q8—82no—
491D4—42no—
492D4—42no—
493D4—82no—
494D4—84no—
495D4—84no—
496D4—84no—
497D4—84no—
498D4—84no—
499D4—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—88no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#39012yes—
539C2×C4#39012yes—
540C2×C4C22×C22⋊C411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C2×C4C2×#12711yescharacteristic
544C8—21no—
545C8—21no—
546C8—22no—
547C8—22no—
548C8—21no—
549C8—21no—
550C8—42no—
551C8—42no—
552C8—82no—
553C8—82no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—21no—
567C22—44no—
568C22—44no—
569C22—82no—
570C22—84no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—164no—
576C22C2×#39011yescharacteristic
577C22#295612yes—
578C22#306712yes—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—21no—
584C4—44no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4#288311yescharacteristic
590C4#288311yescharacteristic
591C2—22no—
592C2—22no—
593C2—41no—
594C2—42no—
595C2—42no—
596C2—162no—
597C2#2121311yescharacteristic
598C2#2124111yescharacteristic
599C2#2124811yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3191C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #20525C2semidirect
4. #17542C2semidirect
5. #17400C2semidirect
6. #9818C2semidirect
7. #9817C2semidirect
8. #9816C2semidirect
9. #9815C2semidirect
10. #9814C2semidirect
11. #9813C2semidirect
19. C2×#485C4semidirect
20. C2×#482C4semidirect
24. C2×#452C4semidirect
25. C2×#449C4semidirect
57. C22×Q16C2×C4semidirect
58. C22×SD16C2×C4semidirect
59. C22×D8C2×C4semidirect
114. C2×#107C2×C4semidirect
115. C2×#107C2×C4semidirect
116. C2×#106C2×C4semidirect
117. C2×#103C2×C4semidirect
118. C2×#103C2×C4semidirect
119. C2×#102C2×C4semidirect
120. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31912
#219482
#205252
#175422
#174002
#98182
#98172
#98162
#98152
#98142
#98132