Finite Group Databasebeta

← #464400

Subgroups — #464400

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3C2×#1916C211yesmaximal, characteristic
4#15180C211yesmaximal, characteristic
5#15128C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11122C212yesmaximal
9#11067C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#487—28no—
15C2×#487—24no—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#395—21no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2066—28no—
29#1916—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1446—21no—
38#1444—21no—
39#1442—21no—
40#1442—21no—
41#1441—21no—
42#1441—21no—
43#1441—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#487—232no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—42no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103D418yes—
91C2×C4×C8—24no—
92C2×C4×C8D414yes—
93#395—44no—
94#391—24no—
95#391—42no—
96#391—42no—
97#391—42no—
98#389—24no—
99#389—42no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—44no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—82no—
140C22×Q8C2×D414yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D414yes—
172C4⋊Q8—28no—
173C4⋊Q8—28no—
174C4⋊Q8—48no—
175C4⋊Q8—48no—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
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—22no—
191C2×C4⋊C4—22no—
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—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4SD1614yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4#4114yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2×D412yes—
224C2×C42#4112yes—
225#106—48no—
226C4⋊C8—44no—
227#103—216no—
228#103—216no—
229#103—48no—
230#103—48no—
231C4×C8—28no—
232#95—22no—
233#95—22no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—82no—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—48no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
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×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—88no—
276C2×Q8—88no—
277C2×Q8—84no—
278C2×Q8—84no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
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#12112yes—
332C22×C4#12112yes—
333C22×C4C2×SD1612yes—
334C22×C4C2×SD1612yes—
335C22×C4C2×SD1612yes—
336C22×C4#13512yes—
337C22×C4#13712yes—
338C22×C4#13712yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+411yescharacteristic
347C22×C42−1+412yes—
348C22×C42−1+411yescharacteristic
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8#12014yes—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—88no—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—42no—
397C23C2×#12111yescharacteristic
398C23#53511yescharacteristic
399C23#53611yescharacteristic
400C23D4212yes—
401C23#54812yes—
402C23C22×SD1611yescharacteristic
403C23C2×#13511yescharacteristic
404C23C2×#13711yescharacteristic
405C23C2×#13711yescharacteristic
406C23#57712yes—
407C23#57812yes—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—816no—
413Q8—82no—
414Q8—82no—
415Q8—82no—
416Q8—82no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
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×C4—82no—
475C2×C4#46514yes—
476C2×C4#47014yes—
477C2×C4C2×#12011yescharacteristic
478C2×C4#53211yescharacteristic
479C2×C4#53511yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#53612yes—
482C2×C4#53711yescharacteristic
483C2×C4#53711yescharacteristic
484C2×C4#53712yes—
485C2×C4#53812yes—
486C2×C4#54212yes—
487C2×C4#54211yescharacteristic
488C2×C4#54211yescharacteristic
489C2×C4#54612yes—
490C2×C4#54612yes—
491C2×C4#54712yes—
492C2×C4#55612yes—
493C2×C4#56212yes—
494C2×C4#56312yes—
495C8—24no—
496C8—42no—
497C8—81no—
498C8—81no—
499C22#240811yescharacteristic
500C22#254511yescharacteristic
501C22#262011yescharacteristic
502C22#267412yes—
503C22#267612yes—
504C22C2×#46511yescharacteristic
505C22C2×#47011yescharacteristic
506C22#305711yescharacteristic
507C22#305911yescharacteristic
508C22#320611yescharacteristic
509C22#321011yescharacteristic
510C22#322512yes—
511C22#322911yescharacteristic
512C22#323111yescharacteristic
513C22#324212yes—
514C22#327012yes—
515C22#327012yes—
516C22D4×SD1612yes—
517C22#328212yes—
518C22#335912yes—
519C22#336412yes—
520C22#336912yes—
521C22#337212yes—
522C4—24no—
523C4—24no—
524C4—42no—
525C4—42no—
526C4—84no—
527C4—81no—
528C4—84no—
529C4—84no—
530C4—81no—
531C4—162no—
532C4#243011yescharacteristic
533C4#261611yescharacteristic
534C4#266211yescharacteristic
535C4#270012yes—
536C4#271512yes—
537C4#282011yescharacteristic
538C2#1650511yescharacteristic
539C2#2426411yescharacteristic
540C2#2427311yescharacteristic
541C2#2863911yescharacteristic
542C2#2864711yescharacteristic
543C2#2984511yescharacteristic
544C2#2984811yescharacteristic
545C2#3080612yes—
546C2#3080812yes—
547C2#3087812yes—
548C2#3088112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4872
C2×#20662
C2×#19162
#151802
#151282
#113842
#111902
#111222
#110672
#110102
#108612