Finite Group Databasebeta

← #363468

Subgroups — #363468

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1983C212yesmaximal
3C2×#1832C211yesmaximal, characteristic, complemented
4#13779C211yesmaximal, characteristic, complemented
5#11052C211yesmaximal, characteristic, complemented
6#11051C211yesmaximal, characteristic
7#11050C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic
9#10849C212yesmaximal, complemented
10#10789C212yesmaximal, complemented
11#10788C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20#2326—24no—
21#2326C418yescomplemented
22C2×#391C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#388—22no—
25C2×#385—22no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×#376C2211yescharacteristic
31#1983—28no—
32#1832—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1436—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124C2×C414yes—
52C2×#124D414yes—
53C2×#124Q814yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—42no—
56C2×C4×Q8—42no—
57C2×C4×Q8C2×C414yes—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102—42no—
92#391—22no—
93#391C2×C414yes—
94#390—22no—
95#390C2×C414yes—
96#388—216no—
97#385—216no—
98#384—24no—
99#384—22no—
100#384C2×C414yes—
101#382—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104#376C2×C414yes—
105#376D414yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95D412yes—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336—24no—
118#336—22no—
119#336—22no—
120C25—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C22×C412yes—
129C22×Q8C2×D412yes—
130C22×Q8C2×Q812yes—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C22×C412yes—
139C22×D4C2×D412yes—
140C22×D4C2×Q812yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
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×C8—22no—
170#124—28no—
171#124—28no—
172#124—28no—
173#124C4⋊C4116yescomplemented
174C4×Q8—48no—
175C4×Q8—48no—
176C4×D4—48no—
177C4×D4—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4#4114yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—44no—
224C2×C42—82no—
225C2×C42—82no—
226C2×C42C22×C412yes—
227C2×C42C22×C412yes—
228C2×C42C22×C412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C4⋊C8—48no—
237#103—48no—
238#103—48no—
239#102—48no—
240#102—48no—
241#95—28no—
242#95—28no—
243#95—44no—
244#95—82no—
245#95—82no—
246#95C22⋊C418yes—
247#95C22⋊C418yes—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8C2×C4⋊C414yes—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4C2×C4⋊C414yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—48no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—88no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—88no—
346C22×C4C2×C22⋊C414yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C414yes—
350C22×C4#12114yes—
351C22×C4C22⋊Q814yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)12yes—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—88no—
390C4⋊C4—88no—
391C22⋊C4—232no—
392C22⋊C4—48no—
393C22⋊C4—88no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—48no—
399C42—88no—
400C42C2×C4⋊C414yes—
401C42#11714yes—
402C42#12114yes—
403C42C22⋊Q814yes—
404C23—24no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—82no—
426C23—82no—
427C23C22×C22⋊C411yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#52511yescharacteristic
431C23#52511yescharacteristic
432C23#53511yescharacteristic
433C23#53511yescharacteristic
434C23#57612yes—
435C23#57712yes—
436C23#57712yes—
437C23#57812yes—
438Q8—28no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442D4—28no—
443D4—44no—
444D4—44no—
445D4—44no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—816no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
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#38018yes—
502C2×C4C22×C4⋊C411yescharacteristic
503C2×C4C2×#11711yescharacteristic
504C2×C4C4×#4112yes—
505C2×C4#51812yes—
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4#53212yes—
509C2×C4#53512yes—
510C2×C4#54612yes—
511C2×C4#54612yes—
512C2×C4#55312yes—
513C2×C4#55312yes—
514C2×C4#55412yes—
515C2×C4#55412yes—
516C2×C4#55912yes—
517C2×C4#55912yes—
518C8—82no—
519C8—82no—
520C22—44no—
521C22—48no—
522C22—82no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22—82no—
529C22—82no—
530C22—84no—
531C22C2×#38011yescharacteristic
532C22#229711yescharacteristic
533C22#230911yescharacteristic
534C22#253511yescharacteristic
535C22#253511yescharacteristic
536C22#258011yescharacteristic
537C22#258011yescharacteristic
538C22#289712yes—
539C22#289812yes—
540C22#304512yes—
541C22#304612yes—
542C22#308312yes—
543C22#308412yes—
544C22#311812yes—
545C22#312112yes—
546C22#312412yes—
547C22#312512yes—
548C22#322011yescharacteristic
549C22#322111yescharacteristic
550C22#322211yescharacteristic
551C22#322311yescharacteristic
552C22#322412yes—
553C22#322512yes—
554C4—28no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—44no—
559C4—84no—
560C4—84no—
561C4—82no—
562C4—82no—
563C4#234912yes—
564C4#235712yes—
565C4#254412yes—
566C4#256612yes—
567C2—84no—
568C2#1394111yescharacteristic
569C2#2065212yes—
570C2#2139712yes—
571C2#2181311yescharacteristic
572C2#2181411yescharacteristic
573C2#2817111yescharacteristic
574C2#2817212yes—
575C2#2817311yescharacteristic
576C2#2926712yes—
577C2#2926811yescharacteristic
578C2#2926911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
3. C2×#1832C2semidirect
4. #13779C2semidirect
5. #11052C2semidirect
7. #11050C2semidirect
9. #10849C2semidirect
10. #10789C2semidirect
21. #2326C4semidirect
173. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19832
C2×#18322
#137792
#110522
#110512
#110502
#110492
#108492
#107892
#107882