Finite Group Databasebeta

← #358203

Subgroups — #358203

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13786C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11190C212yesmaximal
8#11067C212yesmaximal
9#11024C212yesmaximal
10#10703C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#441—24no—
16C2×#441—24no—
17C2×#441C418yescomplemented
18C22×#106C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2320—24no—
26#2320—24no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#383—21no—
32C2×#382—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×#376C2211yescharacteristic
36C4×#103—24no—
37C4×#103—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2212yes—
47#1434—22no—
48C23×Q8C2311yescharacteristic
49C23×Q8C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16C2×C414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C414yes—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8D414yes—
69C2×C4×Q8D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#441—232no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106C2×C414yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103C2×C414yes—
98C2×#103C2×C414yes—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97C2×C414yes—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105#391—24no—
106#391—24no—
107#391—24no—
108#391—24no—
109#389—24no—
110#389—24no—
111#387—216no—
112#383—28no—
113#382—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4D414yes—
118#376—22no—
119#376—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D412yes—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137#336—24no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C22×C412yes—
154C22×Q8C22×C412yes—
155C22×Q8C2×D412yes—
156C22×Q8C2×D412yes—
157C22×Q8#4114yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×Q16—216no—
170C2×Q16—216no—
171C2×Q16—28no—
172C2×Q16—28no—
173C2×Q16—28no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C22×C412yes—
188C22×C8C22×C412yes—
189C22×C8#4114yes—
190C22×C8#4114yes—
191C4×Q8—28no—
192C4×Q8—28no—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×Q8—28no—
196C4×Q8—44no—
197C4×Q8—44no—
198C4×Q8C22⋊C4116yes—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4114yes—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—82no—
247C2×C42—82no—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254#106—28no—
255#103—216no—
256#103—216no—
257#103—28no—
258#103—28no—
259#97—28no—
260C4×C8—44no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—42no—
271#95—82no—
272#95—82no—
273C24C22×D411yescharacteristic
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8C2×C22⋊C414yes—
309C2×Q8#11714yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378Q16—232no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8#11714yes—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—216no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—48no—
413C4⋊C4—88no—
414C4⋊C4C2×C22⋊C414yes—
415C42—216no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—48no—
421C42—84no—
422C42—84no—
423C42C2×C22⋊C414yes—
424C42C4×D414yes—
425C42#12114yes—
426C42#12314yes—
427C23C2×C4×D411yescharacteristic
428C23C2×#12011yescharacteristic
429C23C2×#12111yescharacteristic
430C23#52512yes—
431C23#53212yes—
432C23#57712yes—
433C23#57712yes—
434C23#57812yes—
435C23#57812yes—
436Q8—24no—
437Q8—24no—
438Q8—24no—
439Q8—24no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4#38614yes—
505C2×C4#38614yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C2×#11711yescharacteristic
508C2×C4C2×#11711yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C4×#4111yescharacteristic
511C2×C4C4×#4111yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C2×C4#52512yes—
515C2×C4#52512yes—
516C2×C4#52512yes—
517C2×C4#52512yes—
518C2×C4#52912yes—
519C2×C4#52912yes—
520C2×C4#52912yes—
521C2×C4#52912yes—
522C2×C4#53212yes—
523C2×C4#53212yes—
524C8—24no—
525C8—44no—
526C8—42no—
527C22C2×#38611yescharacteristic
528C22#229011yescharacteristic
529C22#229311yescharacteristic
530C22#238812yes—
531C22#239212yes—
532C22#294712yes—
533C22#294712yes—
534C22#300911yescharacteristic
535C22#301111yescharacteristic
536C22#301312yes—
537C22#304311yescharacteristic
538C22#304411yescharacteristic
539C22#304612yes—
540C22#312012yes—
541C22#312112yes—
542C22#312512yes—
543C22#312712yes—
544C22#317312yes—
545C22#317712yes—
546C22#317812yes—
547C22#317812yes—
548C4—24no—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4#228011yescharacteristic
559C4#229011yescharacteristic
560C4#229311yescharacteristic
561C4#229711yescharacteristic
562C4#239212yes—
563C4#239512yes—
564C2#1383911yescharacteristic
565C2#2093011yescharacteristic
566C2#2093111yescharacteristic
567C2#2112712yes—
568C2#2128912yes—
569C2#2358112yes—
570C2#2358612yes—
571C2#2369112yes—
572C2#2369412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#441C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4412
C2×#23202
C2×C4×#1032
#137862
#112202
#111902
#110672
#110242
#107032
#106942