Finite Group Databasebeta

← #196434

Subgroups — #196434

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#35010C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic, complemented
4C22×#328C211yesmaximal, characteristic, complemented
5#13960C211yesmaximal, characteristic
6#9112C214yesmaximal, complemented
7#9111C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic, complemented
9C2×#1294C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2972—24no—
13#2972—24no—
14C2×C8×D4—22no—
15C2×C8×D4—22no—
16C2×#431C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2855C2212yes—
21#2848C2212yes—
22#2627—21no—
23#2626—21no—
24#2579—22no—
25#2564—22no—
26#2487C412yescomplemented
27#2417C414yescomplemented
28#2409C412yescomplemented
29#2315—21no—
30#2308—21no—
31C2×#386—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C22×#95C2211yescharacteristic
35#1520—24no—
36C2×#328—24no—
37C2×#328C2214yes—
38C2×#328C2214yes—
39#1441—21no—
40#1439—21no—
41#1434C2212yes—
42C22⋊#95—21no—
43C22⋊#95C2212yes—
44#1294C2214yes—
45#1294C2214yes—
46C23×Q8C2×C411yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—21no—
53C23×C8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8C2×C412yes—
61C2×C22⋊Q8C2×C414yescomplemented
62C2×C22⋊Q8C2312yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×C4×D4D412yes—
70C2×C4×D4D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4D411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42C2311yescharacteristic
83C8×D4—216no—
84C8×D4—216no—
85#431—24no—
86#431—24no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8D414yes—
96C2×C22⋊C8D414yes—
97C2×C22⋊C8C2312yes—
98C2×C22⋊C8C2312yes—
99C2×C4×C8—22no—
100#400—22no—
101#399—22no—
102#398—24no—
103#396—24no—
104#395—22no—
105#393—22no—
106#391C2×C414yescomplemented
107#389C2×C414yescomplemented
108#387—24no—
109#386—24no—
110#386—22no—
111#386—22no—
112#386—44no—
113#385—24no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#381—22no—
119#380—21no—
120#380—21no—
121#380—24no—
122#380—22no—
123#380—22no—
124#380C2×C412yes—
125#380C2×C414yescomplemented
126#380C2312yes—
127#379—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137#328—28no—
138#328—28no—
139C25C2×D411yescharacteristic
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—82no—
145C22×Q8C22×C412yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C22⋊C412yes—
151C22×D4C22⋊C412yes—
152C22×D4SD1614yes—
153C22×D4Q1614yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—82no—
168C23×C4—82no—
169C23×C4—82no—
170C23×C4C22×C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—28no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—82no—
191C22×C8—82no—
192C22×C8—84no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—84no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C414yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C22⋊C412yes—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C22×C412yes—
260C2×C42C2×D412yes—
261C4⋊C8—28no—
262C22⋊C8—28no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8C2×D414yes—
268C22⋊C8C2×D414yes—
269C4×C8—28no—
270#95—22no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—44no—
281#95—42no—
282#95—44no—
283#95—82no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24C2×C22⋊C412yes—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24#12112yes—
296C24#12112yes—
297C24C2×SD1612yes—
298C24C2×SD1612yes—
299C24C2×Q1612yes—
300C24C2×Q1612yes—
301C24#13512yes—
302C24#13512yes—
303C24C22×D411yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—44no—
307C2×Q8—42no—
308C2×Q8—82no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—28no—
318C2×D4—28no—
319C2×D4#10318yes—
320C2×D4#10318yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4#12012yes—
390C22×C4#12012yes—
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4#12312yes—
394C22×C4#12312yes—
395C22×C4C2×SD1612yes—
396C22×C4C2×Q1612yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—84no—
408C2×C8—84no—
409C2×C8—88no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—84no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—88no—
436C22⋊C4—84no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—82no—
471C23#38612yes—
472C23#38612yes—
473C23C2×#10314yes—
474C23C2×#10312yes—
475C23C2×#10312yes—
476C23#41612yes—
477C23#41612yes—
478C23#46514yes—
479C23#46514yes—
480C23#46714yes—
481C23#46714yes—
482C23C22×C22⋊C411yescharacteristic
483C23C2×#12111yescharacteristic
484C23C2×#12111yescharacteristic
485C23C22×SD1611yescharacteristic
486C23C22×Q1611yescharacteristic
487C23C2×#13511yescharacteristic
488C23C2×#13511yescharacteristic
489Q8—82no—
490Q8—82no—
491Q8—84no—
492D4—216no—
493D4—216no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—816no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—164no—
555C2×C4#38612yes—
556C2×C4#38612yes—
557C2×C4C2×#10314yes—
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×#12011yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C8—44no—
563C8—44no—
564C22—21no—
565C22—21no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—24no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22#189214yes—
594C22#189214yes—
595C22C2×#38611yescharacteristic
596C22#288312yes—
597C22C22×#10311yescharacteristic
598C22C2×#41611yescharacteristic
599C22#290112yes—
600C22C2×#46511yescharacteristic
601C22C2×#46511yescharacteristic
602C22C2×#46711yescharacteristic
603C22C2×#46711yescharacteristic
604C4—24no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—42no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4—82no—
617C4—162no—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—24no—
623C2—24no—
624C2—44no—
625C2#1761412yes—
626C2#1761512yes—
627C2#2090311yescharacteristic
628C2C2×#189211yescharacteristic
629C2#2098411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1520C2semidirect
4. C22×#328C2semidirect
6. #9112C2semidirect
7. #9111C2semidirect
8. #8023C2semidirect
9. C2×#1294C2semidirect
26. #2487C4semidirect
27. #2417C4semidirect
28. #2409C4semidirect
61. C2×C22⋊Q8C2×C4semidirect
106. #391C2×C4semidirect
107. #389C2×C4semidirect
125. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#350102
C2×#15202
C22×#3282
#139602
#91122
#91112
#80232
C2×#12942