Finite Group Databasebeta

← #374419

Subgroups — #374419

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C22×#427C211yesmaximal, characteristic, complemented
3C2×#1973C212yesmaximal, complemented
4C2×#1875C211yesmaximal, characteristic
5C2×#1873C211yesmaximal, characteristic, complemented
6C2×#1752C211yesmaximal, characteristic, complemented
7#14138C211yesmaximal, characteristic, complemented
8#14136C211yesmaximal, characteristic
9#11122C212yesmaximal, complemented
10#11121C212yesmaximal, complemented
11#10863C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C2×#427—28no—
22C2×#427—24no—
23C22×#107C2212yes—
24C22×#106C2212yes—
25C22×#106C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×#102C2212yes—
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×#97C2212yes—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1973—28no—
36#1875—24no—
37#1873—24no—
38#1752—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1440—22no—
43#1438—22no—
44#1437—22no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×SD16—24no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—24no—
56C22×SD16C2×C414yes—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8C2×C414yes—
60C23×C8D414yes—
61C23×C8D414yes—
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2×C414yes—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121C2×C414yes—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—42no—
84C2×#117—42no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4D412yes—
96C22×C42—21no—
97C22×C42—21no—
98C22×C42C2312yes—
99#465—232no—
100#427—232no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#107D418yescomplemented
104C2×#106—24no—
105C2×#106—22no—
106C2×#106—22no—
107C2×#106—24no—
108C2×#106—24no—
109C2×#106C2×C414yes—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—42no—
115C2×#103C2×C414yes—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—42no—
121C2×#102C2×C414yes—
122C2×C22⋊C8—22no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8D414yes—
125C2×#97—24no—
126C2×#97—24no—
127C2×#97—24no—
128#391—24no—
129#391—42no—
130#390—24no—
131#390—42no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95D412yes—
138#340—24no—
139#340—24no—
140#336—24no—
141C25—21no—
142C25C2×D411yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22×C412yes—
153C22×D4—24no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—84no—
166C22×D4C22×C412yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4C22×C412yes—
177C23×C4C2×D414yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×SD16—28no—
186C2×SD16—216no—
187C2×SD16—44no—
188C2×SD16—48no—
189C2×SD16—44no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—216no—
193C22×C8—28no—
194C22×C8—216no—
195C22×C8—28no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8C22⋊C4116yescomplemented
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8C2×D414yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D414yes—
211C22×C8#4114yes—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216#121—28no—
217#121—28no—
218#121—44no—
219#121—44no—
220#117—28no—
221#117—28no—
222#117—48no—
223#117—48no—
224#117—44no—
225#117—44no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—84no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278#107—216no—
279#106—216no—
280#106—28no—
281#103—28no—
282#103—48no—
283#102—28no—
284#102—48no—
285C22⋊C8—28no—
286#97—48no—
287#95—24no—
288#95—48no—
289#95—48no—
290#95—84no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—44no—
301C24—44no—
302C24C4×D412yes—
303C24#12112yes—
304C24#12712yes—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—48no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4C2×C22⋊C414yes—
393C22×C4C4×D414yes—
394C22×C4C4×D412yes—
395C22×C4#12114yes—
396C22×C4#12112yes—
397C22×C4#12414yes—
398C22×C4#12712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409SD16—416no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—232no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8C2×C22⋊C414yes—
423C2×C8C2×C22⋊C414yes—
424C2×C8C4×D418yes—
425C2×C8#12014yes—
426C2×C8#12118yes—
427C2×C8#12314yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—816no—
451C42—48no—
452C42—48no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—84no—
476C23—84no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—81no—
481C23—84no—
482C23—81no—
483C23—84no—
484C23—82no—
485C23—82no—
486C23#39014yes—
487C23C2×C4×D411yescharacteristic
488C23C2×#12111yescharacteristic
489C23C2×#12711yescharacteristic
490C23D4212yes—
491C23#54712yes—
492C23#57514yes—
493C23#57714yes—
494Q8—42no—
495Q8—44no—
496Q8—42no—
497Q8—82no—
498Q8—81no—
499Q8—81no—
500D4—42no—
501D4—44no—
502D4—42no—
503D4—816no—
504D4—82no—
505D4—81no—
506D4—81no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#39014yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×D412yes—
551C2×C4#52012yes—
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12112yes—
555C2×C4C2×#12311yescharacteristic
556C2×C4C2×#12411yescharacteristic
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53712yes—
560C2×C4D4212yes—
561C2×C4#54712yes—
562C2×C4#54812yes—
563C2×C4#54812yes—
564C2×C4#55312yes—
565C2×C4#55312yes—
566C8—24no—
567C8—44no—
568C8#38618yes—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—44no—
573C22—44no—
574C22—48no—
575C22—48no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—88no—
580C22—162no—
581C22—162no—
582C22—164no—
583C22C2×#39011yescharacteristic
584C22#236112yes—
585C22#241612yes—
586C22#243311yescharacteristic
587C22#250111yescharacteristic
588C22#294312yes—
589C22#294912yes—
590C22#305412yes—
591C22#305712yes—
592C22#315012yes—
593C22#315412yes—
594C22#326714yes—
595C22#327114yes—
596C22#334814yes—
597C22#335014yes—
598C4—24no—
599C4—82no—
600C4—84no—
601C4—84no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4C2×#38611yescharacteristic
606C4#231111yescharacteristic
607C4#236112yes—
608C4#241612yes—
609C4#243611yescharacteristic
610C4#250411yescharacteristic
611C2—44no—
612C2—162no—
613C2#1413711yescharacteristic
614C2#2122811yescharacteristic
615C2#2123111yescharacteristic
616C2#2303314yes—
617C2#2478614yes—
618C2#2578712yes—
619C2#2793712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#427C2semidirect
3. C2×#1973C2semidirect
5. C2×#1873C2semidirect
6. C2×#1752C2semidirect
7. #14138C2semidirect
9. #11122C2semidirect
10. #11121C2semidirect
11. #10863C2semidirect
20. C2×#465C4semidirect
103. C2×#107D4semidirect
202. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C22×#4272
C2×#19732
C2×#18752
C2×#18732
C2×#17522
#141382
#141362
#111222
#111212
#108632