Finite Group Databasebeta

← #188535

Subgroups — #188535

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34223C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3C2×#1513C211yesmaximal, characteristic
4C2×#1504C211yesmaximal, characteristic
5#13966C211yesmaximal, characteristic, complemented
6#8576C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8574C212yesmaximal, complemented
9#8573C212yesmaximal, complemented
10#8020C211yesmaximal, characteristic, complemented
11#8016C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C2×#407C2211yescharacteristic
15#2374—22no—
16#2363C412yescomplemented
17#2362C412yescomplemented
18#2357—21no—
19#2353—21no—
20#2347C411yescharacteristic, complemented
21#2346C411yescharacteristic, complemented
22#2341C411yescharacteristic, complemented
23#2339C411yescharacteristic, complemented
24#2321—22no—
25#2318—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28#2280C2211yescharacteristic
29C2×#380C2211yescharacteristic
30#2275—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1851—21no—
34#1851—21no—
35#1851—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1513—24no—
39#1504—24no—
40C2×#328C2211yescharacteristic
41C2×#328C2212yes—
42C2×#328C2211yescharacteristic
43C2×#327C2211yescharacteristic
44C2×#327C2212yes—
45C2×#327C2211yescharacteristic
46#1442—21no—
47#1442—21no—
48#1434C2212yes—
49#1434C2212yes—
50C22⋊#95—22no—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C22×#41C2×C411yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C22×M16—21no—
60#525—22no—
61#525—22no—
62#525D414yes—
63#525Q814yes—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2311yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#121C2311yescharacteristic
69C2×#120—21no—
70C2×C4×Q8C2×C412yescomplemented
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yescomplemented
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C412yes—
77C22×C4⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42C2312yes—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83#407—22no—
84#407—22no—
85#407—22no—
86#407—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8C2311yescharacteristic
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2311yescharacteristic
92#391C2×C412yescomplemented
93#390—22no—
94#390—22no—
95#390C2×C412yescomplemented
96#389C2×C412yescomplemented
97#389C2×C412yescomplemented
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#383—21no—
111#382—21no—
112#381—21no—
113#381—21no—
114#380—22no—
115#380—22no—
116#380—21no—
117#380C2×C411yescharacteristic
118#380C2×C411yescharacteristic
119#380C2312yes—
120#379—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yescomplemented
125C4×C4⋊C4C2×C412yescomplemented
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4C2×C411yescharacteristic
133C4×C22⋊C4C2×C411yescharacteristic
134C4×C22⋊C4C2×C412yes—
135C4×C22⋊C4D411yescharacteristic
136C4×C22⋊C4D411yescharacteristic
137C4×C22⋊C4D412yes—
138#376—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152#336—22no—
153#336—22no—
154#328—22no—
155#328—22no—
156#328—24no—
157#327—22no—
158#327—22no—
159#327—24no—
160C25C2×D411yescharacteristic
161C2×#41—22no—
162C2×#41—82no—
163C2×#41C4⋊C414yes—
164C22×Q8C22×C411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×Q812yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—82no—
195C2×M16—82no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199#123—24no—
200#123—24no—
201C22⋊Q8—22no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8C4⋊C414yes—
205C22⋊Q8C2×D412yes—
206C22⋊Q8C2×Q812yes—
207#121—22no—
208#121—21no—
209#121—21no—
210#121C4⋊C414yes—
211#121C2×D412yes—
212#121C2×Q812yes—
213#120—24no—
214#120—24no—
215C4×Q8—82no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—82no—
223#117—22no—
224#117—82no—
225#117C4⋊C414yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C22⋊C4C22⋊C412yes—
256C2×C22⋊C4C22⋊C412yes—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—82no—
263C2×C42C22×C412yes—
264C2×C42C22×C412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C4⋊C8—82no—
268C4⋊C8—82no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C22⋊C8—22no—
272C22⋊C8—21no—
273C22⋊C8—22no—
274C22⋊C8—21no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277#95—21no—
278#95—22no—
279#95—21no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—82no—
296#95—82no—
297#95C22⋊C411yescharacteristic
298#95C22⋊C412yes—
299#95C22⋊C411yescharacteristic
300#95C22⋊C411yescharacteristic
301#95C22⋊C411yescharacteristic
302#95C22⋊C412yes—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—41no—
307C24—41no—
308C24—42no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24#12112yes—
312C24C22⋊Q812yes—
313C24C22×D411yescharacteristic
314#41—84no—
315#41—84no—
316C2×Q8—21no—
317C2×Q8—42no—
318C2×Q8C2×C4⋊C412yes—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—21no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—81no—
328C2×D4—81no—
329C2×D4—82no—
330C2×D4C2×C4⋊C412yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—48no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4#11712yes—
390C22×C4#11712yes—
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4#12112yes—
394C22×C4C22⋊Q812yes—
395C22×C4C22⋊Q812yes—
396C22×C4C22⋊Q812yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403M16—84no—
404M16—84no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4C2×C4⋊C412yes—
424C4⋊C4C2×C4⋊C412yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—82no—
444C22⋊C4—84no—
445C42—24no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—44no—
451C42—82no—
452C42—82no—
453C42—82no—
454C23—22no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23#38012yes—
482C23#38012yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×#12111yescharacteristic
485C23C2×C22⋊Q811yescharacteristic
486C23#57512yes—
487C23#57512yes—
488Q8—42no—
489Q8—42no—
490D4—42no—
491D4—42no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—84no—
496D4—84no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—816no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—164no—
564C2×C4#38012yes—
565C2×C4#38012yes—
566C2×C4C22×C4⋊C411yescharacteristic
567C2×C4C2×#11711yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C8—81no—
571C8—81no—
572C8—82no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—82no—
598C22—82no—
599C22C2×#38011yescharacteristic
600C22#288312yes—
601C22#289612yes—
602C22#290111yescharacteristic
603C22#290111yescharacteristic
604C22#304011yescharacteristic
605C22#304011yescharacteristic
606C22#308012yes—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—22no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—162no—
625C2—21no—
626C2—21no—
627C2—22no—
628C2—42no—
629C2—42no—
630C2—82no—
631C2#1762312yes—
632C2#1762912yes—
633C2#2063311yescharacteristic
634C2#2064911yescharacteristic
635C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21934C2semidirect
5. #13966C2semidirect
6. #8576C2semidirect
7. #8575C2semidirect
8. #8574C2semidirect
9. #8573C2semidirect
10. #8020C2semidirect
11. #8016C2semidirect
16. #2363C4semidirect
17. #2362C4semidirect
20. #2347C4semidirect
21. #2346C4semidirect
22. #2341C4semidirect
23. #2339C4semidirect
70. C2×C4×Q8C2×C4semidirect
74. C2×C4×D4C2×C4semidirect
92. #391C2×C4semidirect
95. #390C2×C4semidirect
96. #389C2×C4semidirect
97. #389C2×C4semidirect
124. C4×C4⋊C4C2×C4semidirect
125. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342232
#219342
C2×#15132
C2×#15042
#139662
#85762
#85752
#85742
#85732
#80202
#80162