Finite Group Databasebeta

← #404164

Subgroups — #404164

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2320C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11208C211yesmaximal, characteristic
7#11122C212yesmaximal
8#11052C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10796C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17#2320—24no—
18#2320—24no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#387—21no—
25C2×#383—21no—
26C2×#382—21no—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2058—24no—
31#2024—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1446—21no—
42#1446—21no—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8Q814yescomplemented
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391D414yes—
88#389—22no—
89#389—22no—
90#389—22no—
91#389D414yes—
92#387—28no—
93#387—28no—
94#383—28no—
95#382—28no—
96#382—42no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99#376—22no—
100#376—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—24no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—82no—
138C22×Q8—82no—
139C22×Q8C2×D412yes—
140C22×Q8C2×Q812yes—
141C22×Q8#4112yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
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#4114yes—
169C4⋊Q8—48no—
170C4⋊Q8—48no—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
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—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×Q812yes—
209C2×C4⋊C4#4114yes—
210C2×C4⋊C4#4112yes—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—82no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230#106—48no—
231#103—48no—
232#103—48no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—42no—
245#95—42no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250C24C22×D411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—48no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8C22⋊Q814yes—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
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—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
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—42no—
333C22×C4—42no—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4C22⋊Q812yes—
337C22×C4C22⋊Q812yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4H(5,𝔽2)12yes—
347C22×C42−1+412yes—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—216no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4C22⋊Q814yes—
382C42—216no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—84no—
392C23C2×#12011yescharacteristic
393C23C2×C22⋊Q811yescharacteristic
394C23#52511yescharacteristic
395C23#52511yescharacteristic
396C23#53211yescharacteristic
397C23#53511yescharacteristic
398C23#53611yescharacteristic
399C23#57712yes—
400C23#57712yes—
401C23#57712yes—
402C23#57812yes—
403Q8—28no—
404Q8—44no—
405Q8—44no—
406Q8—44no—
407Q8—816no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#52511yescharacteristic
478C2×C4#52912yes—
479C2×C4#53312yes—
480C2×C4#54212yes—
481C2×C4#54412yes—
482C2×C4#54612yes—
483C2×C4#54612yes—
484C2×C4#54712yes—
485C2×C4#54812yes—
486C2×C4#55112yes—
487C2×C4#55212yes—
488C2×C4#55512yes—
489C2×C4#55612yes—
490C8—44no—
491C8—82no—
492C22#239611yescharacteristic
493C22#240411yescharacteristic
494C22#243211yescharacteristic
495C22#259911yescharacteristic
496C22#261011yescharacteristic
497C22#262811yescharacteristic
498C22#284811yescharacteristic
499C22#300911yescharacteristic
500C22#301011yescharacteristic
501C22#301011yescharacteristic
502C22#301111yescharacteristic
503C22#308212yes—
504C22#308412yes—
505C22#312112yes—
506C22#312112yes—
507C22#312312yes—
508C22#312512yes—
509C22#317312yes—
510C22#317812yes—
511C22#322212yes—
512C22#322512yes—
513C22#324212yes—
514C22#324412yes—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—44no—
521C4—81no—
522C4—82no—
523C4—81no—
524C4—84no—
525C4—82no—
526C4—162no—
527C4#243812yes—
528C4#243912yes—
529C4#247412yes—
530C4#247712yes—
531C2#1488711yescharacteristic
532C2#2378912yes—
533C2#2394211yescharacteristic
534C2#2394611yescharacteristic
535C2#2568011yescharacteristic
536C2#2568611yescharacteristic
537C2#2940312yes—
538C2#2957112yes—
539C2#2999312yes—
540C2#3296411yescharacteristic
541C2#3296511yescharacteristic
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
60. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
C2×#20582
C2×#20242
#142822
#112932
#112082
#111222
#110522
#109082
#107962
#107892