Finite Group Databasebeta

← #416345

Subgroups — #416345

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1959C211yesmaximal, characteristic
2C2×#1958C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic, complemented
4C2×#1768C211yesmaximal, characteristic, complemented
5#14276C211yesmaximal, characteristic, complemented
6#14273C211yesmaximal, characteristic
7#11077C212yesmaximal, complemented
8#11076C212yesmaximal
9#11067C212yesmaximal, complemented
10#11066C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1959—24no—
30#1958—24no—
31#1834—24no—
32#1768—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1441—22no—
39#1441—22no—
40#1439—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#124—22no—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124Q814yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8D414yes—
90#391—24no—
91#391—42no—
92#390—24no—
93#390—42no—
94#389—42no—
95#389—42no—
96#384—22no—
97#384—22no—
98#384—22no—
99#384D414yes—
100#382—22no—
101#382—22no—
102#376—22no—
103#376—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#336—24no—
118C25—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C2×Q812yes—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C2×Q812yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×D414yes—
162C22×C8#4114yes—
163#124—28no—
164#124—28no—
165#124—44no—
166#124—44no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×Q812yes—
220C2×C42#4114yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—216no—
225#103—48no—
226#103—48no—
227#102—48no—
228#102—48no—
229#97—44no—
230C4×C8—28no—
231#95—24no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—44no—
238#95—82no—
239#95—82no—
240C24—22no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
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—42no—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4C22⋊Q812yes—
337C22×C4C22⋊Q812yes—
338C22×C4#12414yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4112yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)12yes—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—88no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C42—24no—
385C42—24no—
386C42—42no—
387C42—42no—
388C42#12114yes—
389C42C22⋊Q814yes—
390C42C22⋊Q814yes—
391C42#12614yes—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23C2×#12111yescharacteristic
416C23C2×#12111yescharacteristic
417C23C2×C22⋊Q811yescharacteristic
418C23C2×#12411yescharacteristic
419C23#53211yescharacteristic
420C23#53511yescharacteristic
421C23#53511yescharacteristic
422C23#57612yes—
423C23#57712yes—
424C23#57712yes—
425C23#57812yes—
426Q8—44no—
427Q8—44no—
428Q8—82no—
429Q8—82no—
430D4—44no—
431D4—44no—
432D4—82no—
433D4—82no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#39614yes—
488C2×C4#39614yes—
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4#52512yes—
493C2×C4C2×#12611yescharacteristic
494C2×C4#52912yes—
495C2×C4#53512yes—
496C2×C4#54012yes—
497C2×C4#54612yes—
498C2×C4#55012yes—
499C2×C4#55312yes—
500C2×C4#55412yes—
501C2×C4#55412yes—
502C2×C4#55512yes—
503C2×C4#55912yes—
504C2×C4#55912yes—
505C8—44no—
506C8—44no—
507C22—84no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—162no—
513C22—161no—
514C22—161no—
515C22—162no—
516C22—161no—
517C22—161no—
518C22C2×#39611yescharacteristic
519C22#254411yescharacteristic
520C22#254811yescharacteristic
521C22#256311yescharacteristic
522C22#256611yescharacteristic
523C22#261911yescharacteristic
524C22#285011yescharacteristic
525C22#304111yescharacteristic
526C22#304211yescharacteristic
527C22#304311yescharacteristic
528C22#304411yescharacteristic
529C22#306012yes—
530C22#306112yes—
531C22#308312yes—
532C22#308412yes—
533C22#313612yes—
534C22#313712yes—
535C22#317812yes—
536C22#317912yes—
537C22#322412yes—
538C22#322412yes—
539C22#322512yes—
540C22#322512yes—
541C4—24no—
542C4—24no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—84no—
547C4—161no—
548C4—161no—
549C4—162no—
550C4#241412yes—
551C4#245412yes—
552C4#251312yes—
553C4#258012yes—
554C2—162no—
555C2#1522211yescharacteristic
556C2#2361011yescharacteristic
557C2#2361111yescharacteristic
558C2#2862711yescharacteristic
559C2#2862811yescharacteristic
560C2#2875811yescharacteristic
561C2#2875911yescharacteristic
562C2#2915112yes—
563C2#2920312yes—
564C2#2981312yes—
565C2#3304712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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×#1834C2semidirect
4. C2×#1768C2semidirect
5. #14276C2semidirect
7. #11077C2semidirect
9. #11067C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19592
C2×#19582
C2×#18342
C2×#17682
#142762
#142732
#110772
#110762
#110672
#110662
#108432