Finite Group Databasebeta

← #412746

Subgroups — #412746

Order 512.

1262 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×#2068C212yesmaximal
2C2×#2066C212yesmaximal
3C2×#1981C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1949C211yesmaximal, characteristic
6C2×#1843C211yesmaximal, characteristic
7#14274C211yesmaximal, characteristic
8#13783C211yesmaximal, characteristic
9#11077C212yesmaximal
10#11077C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2068—28no—
31#2066—28no—
32#1981—24no—
33#1981—24no—
34#1949—24no—
35#1843—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40#1445—21no—
41#1445—21no—
42#1445—22no—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×Q8C2312yes—
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⋊Q8Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—22no—
71C2×#106—24no—
72C2×#106—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8D414yes—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97#391—24no—
98#391—24no—
99#391—24no—
100#391—24no—
101#389—24no—
102#389D418yes—
103#382—22no—
104#382—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
109#376—22no—
110#376—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×Q814yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
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×C8C2×D412yes—
170C22×C8C2×D412yes—
171C4⋊Q8—28no—
172C4⋊Q8—28no—
173C4⋊Q8—44no—
174C4⋊Q8—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—42no—
212C2×C42—82no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×Q812yes—
219C2×C42#4114yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223#107—44no—
224#106—44no—
225C4⋊C8—28no—
226C4⋊C8—48no—
227C4⋊C8—48no—
228C4⋊C8—44no—
229#103—48no—
230#103—48no—
231#103—48no—
232#103—48no—
233#95—24no—
234#95—28no—
235#95—22no—
236#95—22no—
237#95—24no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—82no—
246#95—82no—
247C24C22×D411yescharacteristic
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
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—42no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
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—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4#12114yes—
329C22×C4#12114yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+412yes—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
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—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—88no—
381C4⋊C4#12714yes—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—84no—
387C42—84no—
388C42#12114yes—
389C42C22⋊Q814yes—
390C42C22⋊Q814yes—
391C42#12614yes—
392C23C2×#12111yescharacteristic
393C23C2×#12112yes—
394C23C2×#12111yescharacteristic
395C23C2×C22⋊Q811yescharacteristic
396C23#53712yes—
397C23#57711yescharacteristic
398C23#57712yes—
399C23#57711yescharacteristic
400C23#57811yescharacteristic
401C23#57812yes—
402C23#57811yescharacteristic
403Q8—44no—
404Q8—44no—
405Q8—44no—
406Q8—44no—
407Q8—82no—
408Q8—82no—
409Q8—82no—
410Q8—82no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4#39614yes—
470C2×C4#39614yes—
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×C22⋊Q811yescharacteristic
473C2×C4C2×C22⋊Q811yescharacteristic
474C2×C4C2×#12611yescharacteristic
475C2×C4C2×#12711yescharacteristic
476C2×C4#53211yescharacteristic
477C2×C4#53212yes—
478C2×C4#53511yescharacteristic
479C2×C4#53611yescharacteristic
480C2×C4#54212yes—
481C2×C4D4212yes—
482C2×C4#54712yes—
483C2×C4#54812yes—
484C2×C4D4×Q812yes—
485C2×C4#55012yes—
486C2×C4#55312yes—
487C2×C4#55412yes—
488C2×C4#55612yes—
489C8—42no—
490C8—42no—
491C8—82no—
492C22C2×#39611yescharacteristic
493C22#241612yes—
494C22#253812yes—
495C22#262011yescharacteristic
496C22#285611yescharacteristic
497C22#304211yescharacteristic
498C22#304212yes—
499C22#304411yescharacteristic
500C22#304412yes—
501C22#304611yescharacteristic
502C22#304611yescharacteristic
503C22#304612yes—
504C22#304612yes—
505C22#305711yescharacteristic
506C22#305911yescharacteristic
507C22#306312yes—
508C22#308211yescharacteristic
509C22#308211yescharacteristic
510C22#308212yes—
511C22#325812yes—
512C22#325812yes—
513C22#326012yes—
514C22#326112yes—
515C4—24no—
516C4—24no—
517C4—44no—
518C4—44no—
519C4—44no—
520C4—84no—
521C4—82no—
522C4—84no—
523C4—161no—
524C4—161no—
525C4#245212yes—
526C4#251212yes—
527C4#254111yescharacteristic
528C4#254211yescharacteristic
529C4#254511yescharacteristic
530C4#256611yescharacteristic
531C2#1513411yescharacteristic
532C2#2361311yescharacteristic
533C2#2361511yescharacteristic
534C2#2476412yes—
535C2#2476612yes—
536C2#2829112yes—
537C2#2829412yes—
538C2#2983611yescharacteristic
539C2#2983711yescharacteristic
540C2#3316811yescharacteristic
541C2#3316911yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#20682
C2×#20662
C2×#19812
C2×#19492
C2×#18432
#142742
#137832
#110772
#108502