Finite Group Databasebeta

← #402291

Subgroups — #402291

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C212yesmaximal
2C2×#2056C212yesmaximal
3C2×#1878C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#15240C211yesmaximal, characteristic
6#13817C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic
8#11322C211yesmaximal, characteristic
9#11077C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10850C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25C2×C4×C4⋊C4C2212yes—
26#2074—28no—
27#2056—28no—
28#1878—24no—
29#1803—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1445—21no—
38#1445—21no—
39#1445—22no—
40#1436—22no—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C22×Q16—22no—
48C22×Q16—22no—
49C22×Q16—24no—
50C22×Q16—22no—
51C22×Q16—42no—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4×Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2312yes—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×C4⋊C8—42no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—42no—
85#391—24no—
86#391—24no—
87#391—22no—
88#391—22no—
89#391—24no—
90#391D418yes—
91#384—22no—
92#384—22no—
93#382—24no—
94#382—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—42no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—22no—
116#336—24no—
117#336—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C2×D412yes—
136C22×Q8C2×D414yes—
137C22×Q8#4112yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C2×Q16—28no—
150C2×Q16—48no—
151C2×Q16—48no—
152C2×Q16—44no—
153C2×Q16—44no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8#4112yes—
171C4×Q8—28no—
172C4×Q8—44no—
173C4×Q8—48no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4#4114yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—82no—
212C2×C42C2×D412yes—
213C2×C42C2×D414yes—
214C2×C42#4112yes—
215C2×C42#4114yes—
216C4⋊C8—48no—
217C4⋊C8—44no—
218#103—48no—
219#103—48no—
220#103—44no—
221#103—44no—
222#95—28no—
223#95—28no—
224#95—22no—
225#95—22no—
226#95—42no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—81no—
236C24C22×D411yescharacteristic
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8#12714yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4#12114yes—
326C22×C4#12114yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#13512yes—
330C22×C4#13712yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C2×#4112yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)12yes—
339C22×C42−1+411yescharacteristic
340C22×C42−1+411yescharacteristic
341Q16—416no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4#12314yes—
376C42—22no—
377C42—24no—
378C42—22no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—48no—
384C42—88no—
385C23C2×#12111yescharacteristic
386C23C2×#12112yes—
387C23#54612yes—
388C23#54711yescharacteristic
389C23#54811yescharacteristic
390C23C2×#13511yescharacteristic
391C23C2×#13711yescharacteristic
392C23#57512yes—
393C23#57711yescharacteristic
394C23#57811yescharacteristic
395C23#57812yes—
396Q8—24no—
397Q8—24no—
398Q8—42no—
399Q8—42no—
400Q8—44no—
401Q8—44no—
402Q8—42no—
403Q8—42no—
404Q8—44no—
405Q8—81no—
406Q8—81no—
407Q8—82no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4#46414yes—
466C2×C4C2×#12311yescharacteristic
467C2×C4#52511yescharacteristic
468C2×C4#52912yes—
469C2×C4#52912yes—
470C2×C4C2×#12711yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4#53212yes—
473C2×C4#53411yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53912yes—
476C2×C4#54212yes—
477C2×C4#54612yes—
478C2×C4#54612yes—
479C2×C4#54712yes—
480C2×C4#54712yes—
481C2×C4#55112yes—
482C2×C4#55211yescharacteristic
483C2×C4#55611yescharacteristic
484C8—42no—
485C8—81no—
486C8—82no—
487C22#241211yescharacteristic
488C22#241511yescharacteristic
489C22#242512yes—
490C22#242712yes—
491C22#285611yescharacteristic
492C22C2×#46411yescharacteristic
493C22#303512yes—
494C22#303712yes—
495C22#304011yescharacteristic
496C22#304412yes—
497C22#304411yescharacteristic
498C22#304611yescharacteristic
499C22#304612yes—
500C22#330012yes—
501C22#331212yes—
502C22#331512yes—
503C22#332312yes—
504C22#334211yescharacteristic
505C22#334912yes—
506C22#335611yescharacteristic
507C22#336511yescharacteristic
508C22#337411yescharacteristic
509C22#337712yes—
510C4—24no—
511C4—44no—
512C4—44no—
513C4—42no—
514C4—44no—
515C4—44no—
516C4—84no—
517C4—82no—
518C4—162no—
519C4—161no—
520C4#239511yescharacteristic
521C4#250111yescharacteristic
522C4#252112yes—
523C4#253911yescharacteristic
524C4#255312yes—
525C4#266311yescharacteristic
526C2#1483911yescharacteristic
527C2#2452911yescharacteristic
528C2#2454511yescharacteristic
529C2#2468311yescharacteristic
530C2#2469911yescharacteristic
531C2#2520112yes—
532C2#2522012yes—
533C2#2532812yes—
534C2#2533312yes—
535C2#3316511yescharacteristic
536C2#3316911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#20742
C2×#20562
C2×#18782
C2×#18032
#152402
#138172
#113572
#113222
#110772
#109322
#108502