Finite Group Databasebeta

← #364836

Subgroups — #364836

Order 512.

1462 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5#13786C211yesmaximal, characteristic
6#11219C212yesmaximal
7#11067C212yesmaximal
8#10932C212yesmaximal
9#10907C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—24no—
15C2×#477—24no—
16C2×#477C418yescomplemented
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22#2331—24no—
23#2331—24no—
24C2×#391—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#389C2212yes—
28C2×#387—22no—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1988—24no—
34#1805—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1436—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—22no—
46C23×Q8—21no—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8D414yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#477—232no—
77C2×#107—24no—
78C2×#107—24no—
79C2×#107C2×C418yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8C2×C414yes—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103C2×C418yes—
92#391—216no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391Q814yescomplemented
97#389—24no—
98#389—24no—
99#387—216no—
100#382—22no—
101#382D414yes—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106#376—22no—
107#376—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#336—24no—
125#336—42no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—82no—
141C22×Q8—82no—
142C22×Q8C22×C412yes—
143C22×Q8C2×Q812yes—
144C22×Q8#4112yes—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C22×C414yes—
170C4⋊Q8—28no—
171C4⋊Q8—28no—
172C4⋊Q8—28no—
173C4⋊Q8C22⋊C4116yes—
174C4×Q8—28no—
175C4×Q8—28no—
176C4×Q8—44no—
177C4×Q8—44no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×Q812yes—
216C2×C4⋊C4#4112yes—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—24no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—28no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—82no—
233C2×C42—82no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×Q812yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243#107—216no—
244C4⋊C8—28no—
245C4⋊C8—44no—
246#103—216no—
247#103—48no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—28no—
256#95—42no—
257#95—82no—
258#95—82no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—216no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—88no—
289C2×Q8—88no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8C4×Q814yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
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—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4C22⋊Q814yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366C22×C42−1+411yescharacteristic
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—216no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—48no—
402C4⋊C4—88no—
403C4⋊C4C2×C22⋊C418yes—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—48no—
413C42—84no—
414C42—88no—
415C42C2×C22⋊C414yes—
416C42C4×D414yes—
417C42#12114yes—
418C42#12314yes—
419C23C2×C4×D411yescharacteristic
420C23C2×#12011yescharacteristic
421C23C2×#12111yescharacteristic
422C23C2×C22⋊Q811yescharacteristic
423C23#52511yescharacteristic
424C23#53611yescharacteristic
425C23#53711yescharacteristic
426C23#57612yes—
427C23#57712yes—
428C23#57812yes—
429C23#57812yes—
430Q8—24no—
431Q8—24no—
432Q8—42no—
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—816no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4#38614yes—
498C2×C4#38614yes—
499C2×C4C22×C22⋊C411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×C4×Q811yescharacteristic
502C2×C4C4×#4111yescharacteristic
503C2×C4#51911yescharacteristic
504C2×C4#52011yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×#12311yescharacteristic
507C2×C4#52911yescharacteristic
508C2×C4#53311yescharacteristic
509C2×C4#54111yescharacteristic
510C2×C4#54211yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#54711yescharacteristic
513C2×C4#54811yescharacteristic
514C2×C4#54811yescharacteristic
515C2×C4#54811yescharacteristic
516C2×C4#54811yescharacteristic
517C2×C4#54811yescharacteristic
518C2×C4D4×Q811yescharacteristic
519C2×C4D4×Q811yescharacteristic
520C2×C4D4×Q811yescharacteristic
521C2×C4#55011yescharacteristic
522C2×C4#55011yescharacteristic
523C2×C4#55211yescharacteristic
524C2×C4#55211yescharacteristic
525C2×C4#55411yescharacteristic
526C2×C4#55611yescharacteristic
527C8—44no—
528C8—82no—
529C22C2×#38611yescharacteristic
530C22C4×C22⋊Q811yescharacteristic
531C22#232311yescharacteristic
532C22#240711yescharacteristic
533C22#240911yescharacteristic
534C22#253611yescharacteristic
535C22#253811yescharacteristic
536C22#294612yes—
537C22#295012yes—
538C22#301112yes—
539C22#301212yes—
540C22#304111yescharacteristic
541C22#304211yescharacteristic
542C22#304411yescharacteristic
543C22#304411yescharacteristic
544C22#308212yes—
545C22#308312yes—
546C22#312112yes—
547C22#312412yes—
548C22#324212yes—
549C22#324312yes—
550C22#325812yes—
551C22#326212yes—
552C4—24no—
553C4—24no—
554C4—44no—
555C4—48no—
556C4—44no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—82no—
562C4Q8×C22⋊C411yescharacteristic
563C4#234111yescharacteristic
564C4#236011yescharacteristic
565C4#236211yescharacteristic
566C4#242111yescharacteristic
567C4#242311yescharacteristic
568C4#242611yescharacteristic
569C4#242911yescharacteristic
570C2#1396911yescharacteristic
571C2#2092911yescharacteristic
572C2#2093511yescharacteristic
573C2#2162512yes—
574C2#2212512yes—
575C2#2416812yes—
576C2#2431112yes—
577C2#2821111yescharacteristic
578C2#2821311yescharacteristic
579C2#2828811yescharacteristic
580C2#2829111yescharacteristic
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
16. C2×#477C4semidirect
96. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#23312
C2×#19882
C2×#18052
#137862
#112192
#110672
#109322
#109072
#108502
#108332