Finite Group Databasebeta

← #363117

Subgroups — #363117

Order 512.

1650 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
1C2×#2360C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3#13907C212yesmaximal
4#11203C214yesmaximal
5#10846C211yesmaximal, characteristic
6#10846C212yesmaximal
7#10801C212yesmaximal
8C4×#342C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic, complemented
11C23×C4⋊C4C2211yescharacteristic
12#2360—22no—
13#2360—22no—
14#2360—24no—
15C2×#391C411yescharacteristic, complemented
16C2×#389—22no—
17C2×#389C411yescharacteristic, complemented
18C2×#389C412yescomplemented
19C2×#387C412yescomplemented
20C2×#385C2212yes—
21C2×#380C2212yes—
22C2×#380C2214yes—
23C2×C4×C4⋊C4C411yescharacteristic, complemented
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1996—28no—
28C2×#342C2214yes—
29C2×#342C2212yes—
30C2×#342C2214yes—
31C2×#342C2211yescharacteristic
32C2×#342C2212yes—
33C2×#342C2212yes—
34C2×#342C2211yescharacteristic
35#1441—24no—
36C4⋊#95—22no—
37C22⋊#95C2214yes—
38C23×Q8C2×C411yescharacteristic
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8D414yes—
45C2×C22⋊Q8Q814yes—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—21no—
49C2×C4×Q8—44no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—44no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D414yes—
63C22×C22⋊C4C2314yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—22no—
70C22×C42C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C412yes—
74#391—22no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—24no—
79#389—24no—
80#389—48no—
81#387—24no—
82#387—24no—
83#385—24no—
84#385—24no—
85#380—24no—
86#380—24no—
87#380—28no—
88#380—44no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—24no—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109#342—28no—
110#342—28no—
111#342—22no—
112#342—24no—
113#342—24no—
114#342—42no—
115#342Q814yescomplemented
116C25C2411yescharacteristic
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8C4⋊C414yescomplemented
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—44no—
132C23×C4—82no—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D414yes—
144C23×C4C2×D412yes—
145C23×C4C2×D414yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×Q812yes—
149C23×C4#4114yes—
150C22⋊Q8—28no—
151C22⋊Q8—28no—
152C22⋊Q8—28no—
153C22⋊Q8—28no—
154C4×Q8—44no—
155C4×Q8—48no—
156C4×Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C414yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C414yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C4⋊C414yescomplemented
187C2×C4⋊C4C4⋊C418yescomplemented
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C2×D414yes—
213C2×C22⋊C4C2×Q814yes—
214C2×C22⋊C4C2×Q814yes—
215C2×C22⋊C4C2×Q812yes—
216C2×C22⋊C4#4114yes—
217C2×C22⋊C4#4112yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—82no—
233#95—22no—
234#95—24no—
235#95—22no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—48no—
241#95—48no—
242#95—44no—
243#95—44no—
244#95—84no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24C23×C411yescharacteristic
250C24C22×D412yes—
251C24C22×D411yescharacteristic
252C24C22×Q811yescharacteristic
253C24C22×Q811yescharacteristic
254C24C2×#4112yes—
255C24C2×#4111yescharacteristic
256C24C2×#4112yes—
257C24H(5,𝔽2)11yescharacteristic
258C242−1+412yes—
259C242−1+411yescharacteristic
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—48no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—48no—
266C2×Q8—42no—
267C2×Q8—42no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—48no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—84no—
343C22×C4—88no—
344C22×C4—84no—
345C22×C4C2×C22⋊C414yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C4⋊C412yes—
352C22×C4C2×C4⋊C414yes—
353C22×C4C2×C4⋊C414yes—
354C22×C4C2×C4⋊C412yes—
355C22×C4#11712yes—
356C22×C4#11712yes—
357C22×C4C4×Q812yes—
358C22×C4C4×Q812yes—
359C22×C4#12114yes—
360C22×C4#12114yes—
361C22×C4C22⋊Q814yes—
362C22×C4C22⋊Q814yes—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—216no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—84no—
375C4⋊C4—88no—
376C4⋊C4—816no—
377C4⋊C4#9914yes—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C42—42no—
397C42—42no—
398C42—44no—
399C42—44no—
400C42—42no—
401C42—42no—
402C42—82no—
403C42—82no—
404C42—88no—
405C23—21no—
406C23—22no—
407C23—21no—
408C23—22no—
409C23—21no—
410C23—22no—
411C23—21no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23C22×C22⋊C411yescharacteristic
432C23C22×C4⋊C411yescharacteristic
433C23C2×#11711yescharacteristic
434C23C2×C4×Q811yescharacteristic
435C23C4×#4111yescharacteristic
436C23#51911yescharacteristic
437C23#52011yescharacteristic
438C23C2×#12112yes—
439C23C2×C22⋊Q812yes—
440C23#52912yes—
441C23#53312yes—
442C23#54112yes—
443C23#54212yes—
444C23#54612yes—
445C23#54712yes—
446C23#54812yes—
447C23#54812yes—
448C23D4×Q812yes—
449C23D4×Q811yescharacteristic
450C23#55011yescharacteristic
451C23#55711yescharacteristic
452C23#55712yes—
453C23Q8211yescharacteristic
454Q8—48no—
455Q8—48no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—84no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4#38014yes—
536C2×C4#38012yes—
537C2×C4#38012yes—
538C2×C4C2×#9914yes—
539C2×C4C2×#9912yes—
540C2×C4#41012yes—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—41no—
575C22C2×#38011yescharacteristic
576C22#229511yescharacteristic
577C22#232011yescharacteristic
578C22#233911yescharacteristic
579C22Q8×C22⋊C411yescharacteristic
580C22Q8×C4⋊C411yescharacteristic
581C22#235111yescharacteristic
582C22#242612yes—
583C22#242912yes—
584C22#245112yes—
585C22#245312yes—
586C22C22×#9911yescharacteristic
587C22C2×#41011yescharacteristic
588C22#288311yescharacteristic
589C22#288411yescharacteristic
590C4—24no—
591C4—22no—
592C4—24no—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—84no—
600C4—84no—
601C4—81no—
602C4—81no—
603C4—164no—
604C4#179914yes—
605C2—22no—
606C2—22no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—21no—
613C2#1393511yescharacteristic
614C2C2×#179911yescharacteristic
615C2#2063411yescharacteristic
616C2#2247611yescharacteristic
617C2#2247811yescharacteristic
618C2Q8×#9911yescharacteristic
619C2#2251311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C22×C4×Q8C4semidirect
15. C2×#391C4semidirect
17. C2×#389C4semidirect
18. C2×#389C4semidirect
19. C2×#387C4semidirect
23. C2×C4×C4⋊C4C4semidirect
115. #342Q8semidirect
122. C22×Q8C4⋊C4semidirect
186. C2×C4⋊C4C4⋊C4semidirect
187. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#19962
#139072
#112032
#108462
#108012
C4×#3422