Finite Group Databasebeta

← #368941

Subgroups — #368941

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic
6#11222C214yesmaximal
7#10918C214yesmaximal
8#10854C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12#2502—24no—
13#2502C418yescomplemented
14C2×#398—22no—
15#2368—22no—
16#2360—22no—
17#2345—22no—
18#2345—21no—
19Q8×C22⋊C4—21no—
20#2329—22no—
21#2315C2212yes—
22C4×C22⋊Q8—22no—
23C2×#391—21no—
24C2×#385C2214yes—
25C2×#384—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#2273C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1996—24no—
33C2×#342C2214yes—
34C2×#342C2214yes—
35C2×#342C2212yes—
36C2×#342C2212yes—
37#1445—22no—
38#1445—22no—
39C4⋊#95—22no—
40C4⋊#95—22no—
41C22⋊#95C2212yes—
42C22⋊#95C2214yes—
43C22⋊#95C2212yes—
44C23×Q8D411yescharacteristic
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C414yes—
54C2×C22⋊Q8D414yes—
55C2×C4×Q8—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2311yescharacteristic
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2312yes—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42D412yes—
81#398—216no—
82#391—24no—
83#391—22no—
84#391—28no—
85#389—24no—
86#389—22no—
87#387—24no—
88#387—24no—
89#387—24no—
90#385—24no—
91#385—24no—
92#385—28no—
93#385C2×C4116yes—
94#384—28no—
95#380—22no—
96#380—22no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—22no—
102#380—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4C2×C414yes—
117C2×#95—21no—
118C2×#95—24no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—28no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95D412yes—
125C2×#95D412yes—
126#342—28no—
127#342—24no—
128#342—28no—
129#342Q818yescomplemented
130C25C2411yescharacteristic
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—44no—
135C22×Q8C22⋊C414yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—44no—
146C23×C4—84no—
147C23×C4C22×C414yes—
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D414yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4112yes—
162C23×C4#4112yes—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C4×Q8—84no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—216no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—88no—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22⋊C418yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4C22×C414yes—
233C2×C22⋊C4C22×C414yes—
234C2×C22⋊C4C2×D414yes—
235C2×C22⋊C4C2×Q814yes—
236C2×C22⋊C4C2×Q814yes—
237C2×C22⋊C4C2×Q814yes—
238C2×C22⋊C4#4114yes—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—42no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—216no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—42no—
259#95—42no—
260#95—84no—
261#95—84no—
262#95—88no—
263C24—42no—
264C24—42no—
265C24C23×C411yescharacteristic
266C24C22×D411yescharacteristic
267C24C22×D411yescharacteristic
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C24C22×Q812yes—
271C24C2×#4111yescharacteristic
272C24C2×#4111yescharacteristic
273C24C2×#4112yes—
274C242−1+412yes—
275C242−1+412yes—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—48no—
279C2×Q8—44no—
280C2×Q8—82no—
281C2×Q8—84no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—48no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C414yes—
343C22×C4C2×C22⋊C414yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12312yes—
358C22×C4#12312yes—
359C22×C4C4⋊Q814yes—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—82no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—816no—
373C4⋊C4—84no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4C4×Q818yescomplemented
395C42—48no—
396C42—44no—
397C42—82no—
398C42—84no—
399C42—84no—
400C42—84no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23C22×C22⋊C411yescharacteristic
420C23C2×C4×D411yescharacteristic
421C23C2×C4×D411yescharacteristic
422C23C2×C4×Q812yes—
423C23#51912yes—
424C23C2×#12011yescharacteristic
425C23C2×#12111yescharacteristic
426C23C2×#12111yescharacteristic
427C23C2×#12311yescharacteristic
428C23C2×C4⋊Q811yescharacteristic
429C23#53211yescharacteristic
430C23#53312yes—
431C23#53611yescharacteristic
432C23#53611yescharacteristic
433C23#54112yes—
434C23#54712yes—
435C23#54812yes—
436C23#54812yes—
437C23D4×Q812yes—
438C23D4×Q812yes—
439C23D4×Q812yes—
440C23#55612yes—
441C23#55712yes—
442Q8—48no—
443Q8—84no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—82no—
499C2×C4—88no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—82no—
505C2×C4#38612yes—
506C2×C4#38612yes—
507C2×C4#38612yes—
508C2×C4#38612yes—
509C2×C4#45814yes—
510C22—22no—
511C22—24no—
512C22—22no—
513C22—24no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22C2×#38611yescharacteristic
539C22C4×C4⋊Q811yescharacteristic
540C22#232711yescharacteristic
541C22Q8×C22⋊C412yes—
542C22#236512yes—
543C22#242912yes—
544C22#243211yescharacteristic
545C22#243411yescharacteristic
546C22#244612yes—
547C22#254511yescharacteristic
548C22#254611yescharacteristic
549C22C2×#45812yes—
550C22#302712yes—
551C4—22no—
552C4—22no—
553C4—22no—
554C4—48no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—84no—
559C4—82no—
560C4—84no—
561C4—84no—
562C4—162no—
563C4#189014yes—
564C2—22no—
565C2—22no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—22no—
571C2—21no—
572C2#1402411yescharacteristic
573C2C2×#189011yescharacteristic
574C2#2096111yescharacteristic
575C2#2566412yes—
576C2#2585312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2502C4semidirect
129. #342Q8semidirect
394. C22⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#342122
C2×#19962
#139242
#112222
#109182
#108542
C4×#3422