Finite Group Databasebeta

← #414888

Subgroups — #414888

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1987C211yesmaximal, characteristic
6#13783C211yesmaximal, characteristic
7#11268C212yesmaximal
8#11267C212yesmaximal
9#11208C212yesmaximal
10#11207C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2660—24no—
17#2660—24no—
18C2×#398—21no—
19C2×#398—21no—
20C2×#395—22no—
21C2×#389C2212yes—
22C2×#384C2212yes—
23C2×#382C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1989—24no—
28#1989—24no—
29#1988—24no—
30#1987—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8D414yes—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#398—28no—
86#398—28no—
87#395—216no—
88#389—24no—
89#389Q818yescomplemented
90#384—24no—
91#384—24no—
92#382—24no—
93#382—24no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4D414yes—
98#376—22no—
99#376—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D412yes—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—24no—
120#336—24no—
121#336—24no—
122C22×Q8—24no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—81no—
126C22×Q8—81no—
127C22×Q8—81no—
128C22×Q8—81no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—28no—
158C4⋊Q8—28no—
159C4⋊Q8—44no—
160C4⋊Q8—44no—
161#125—28no—
162#125—28no—
163#125—44no—
164#125—44no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
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—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—42no—
221C2×C42—82no—
222C2×C42—81no—
223C2×C42—81no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×Q814yes—
228C2×C42#4114yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C4⋊C8—44no—
233C4⋊C8—44no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—28no—
241#95—42no—
242#95—82no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—48no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—84no—
249C2×Q8—84no—
250C2×Q8—84no—
251C2×Q8—84no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4#12012yes—
307C22×C4#12012yes—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4C22⋊Q814yes—
311C22×C4C22⋊Q814yes—
312C22×C4#12412yes—
313C22×C4#12412yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q812yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C42−1+412yes—
323C22×C42−1+412yes—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C4⋊C4C4⋊Q814yes—
369C42—24no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—84no—
374C42—84no—
375C42#12114yes—
376C42#12114yes—
377C42#12314yes—
378C42#12314yes—
379C23C2×#12011yescharacteristic
380C23C2×#12111yescharacteristic
381C23C2×C22⋊Q812yes—
382C23C2×#12411yescharacteristic
383C23#53612yes—
384C23#57611yescharacteristic
385C23#57611yescharacteristic
386C23#57712yes—
387C23#57711yescharacteristic
388C23#57711yescharacteristic
389C23#57812yes—
390Q8—816no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#39414yes—
452C2×C4#39414yes—
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#12311yescharacteristic
457C2×C4C2×C4⋊Q811yescharacteristic
458C2×C4#53211yescharacteristic
459C2×C4#53312yes—
460C2×C4#53611yescharacteristic
461C2×C4#53611yescharacteristic
462C2×C4#54112yes—
463C2×C4#54812yes—
464C2×C4#54812yes—
465C2×C4D4×Q812yes—
466C2×C4D4×Q812yes—
467C2×C4#55612yes—
468C2×C4#55612yes—
469C2×C4#55712yes—
470C2×C4#55712yes—
471C8—82no—
472C8—81no—
473C8—81no—
474C22C2×#39411yescharacteristic
475C22#240912yes—
476C22#256812yes—
477C22#262111yescharacteristic
478C22#285711yescharacteristic
479C22#300811yescharacteristic
480C22#300911yescharacteristic
481C22#301312yes—
482C22#304211yescharacteristic
483C22#304411yescharacteristic
484C22#304511yescharacteristic
485C22#304511yescharacteristic
486C22#308112yes—
487C22#308212yes—
488C22#308312yes—
489C22#308412yes—
490C22#313611yescharacteristic
491C22#313611yescharacteristic
492C22#313712yes—
493C22#324112yes—
494C22#324212yes—
495C22#324312yes—
496C22#324412yes—
497C4—24no—
498C4—24no—
499C4—44no—
500C4—44no—
501C4—44no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4—84no—
507C4#244612yes—
508C4#244612yes—
509C4#254511yescharacteristic
510C4#254511yescharacteristic
511C4#254611yescharacteristic
512C4#254611yescharacteristic
513C2#1518011yescharacteristic
514C2#2350911yescharacteristic
515C2#2351211yescharacteristic
516C2#2430812yes—
517C2#2431012yes—
518C2#2922812yes—
519C2#2922912yes—
520C2#2987411yescharacteristic
521C2#2987511yescharacteristic
522C2#3320511yescharacteristic
523C2#3320611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
89. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#19892
C2×#19882
C2×#19872
#137832
#112682
#112672
#112082
#112072
#108332