Finite Group Databasebeta

← #396571

Subgroups — #396571

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1878C211yesmaximal, characteristic
2C2×#1878C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4#15185C211yesmaximal, characteristic
5Q8×#95C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11255C211yesmaximal, characteristic
9#11190C212yesmaximal
10#11077C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#384C2211yescharacteristic
23#1878—28no—
24#1878—24no—
25#1803—24no—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1445—22no—
35#1442—21no—
36#1442—21no—
37C4⋊#95—21no—
38C4⋊#95—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4×#95—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×Q8C2311yescharacteristic
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16D414yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C42—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88#391—24no—
89#391—22no—
90#391—22no—
91#391—42no—
92#384—22no—
93#384—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—22no—
110#340—24no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—42no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8C2×D412yes—
130C22×Q8C2×D412yes—
131C22×Q8C2×D412yes—
132C22×Q8#4112yes—
133C22×Q8#4114yes—
134C22×Q8#4114yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C2×Q16—28no—
147C2×Q16—216no—
148C2×Q16—44no—
149C2×Q16—44no—
150C2×Q16—48no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8C2×D412yes—
168C4×Q8—28no—
169C4×Q8—44no—
170C4×Q8—48no—
171C4×Q8—48no—
172C4×Q8—48no—
173C4×Q8—48no—
174C4×Q8—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4#4114yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42C2×D412yes—
218C2×C42#4112yes—
219C4⋊C8—44no—
220#103—48no—
221#103—44no—
222#103—48no—
223#103—44no—
224#95—22no—
225#95—42no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—81no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95Q1614yescomplemented
237C24C22×D411yescharacteristic
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8#12114yes—
270C2×Q8#12314yes—
271C2×Q8#12318yes—
272C2×Q8#12414yes—
273C2×Q8#12618yes—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
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—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
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—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
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—41no—
326C22×C4—41no—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4C2×Q1614yes—
330C22×C4C2×Q1612yes—
331C22×C4#13712yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342Q16—416no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—416no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C42—22no—
379C42—22no—
380C42—24no—
381C42—48no—
382C42—416no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C23C2×#12111yescharacteristic
388C23#52512yes—
389C23D4211yescharacteristic
390C23#54612yes—
391C23#54811yescharacteristic
392C23C22×Q1611yescharacteristic
393C23C2×#13711yescharacteristic
394C23#57512yes—
395C23#57711yescharacteristic
396C23#57712yes—
397C23#57811yescharacteristic
398Q8—24no—
399Q8—28no—
400Q8—24no—
401Q8—28no—
402Q8—24no—
403Q8—42no—
404Q8—44no—
405Q8—42no—
406Q8—81no—
407Q8—81no—
408Q8—82no—
409Q8#39718yes—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—81no—
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—81no—
466C2×C4—81no—
467C2×C4#46214yes—
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12312yes—
470C2×C4C2×#12311yescharacteristic
471C2×C4C2×#12411yescharacteristic
472C2×C4C2×#12612yes—
473C2×C4#52912yes—
474C2×C4#53211yescharacteristic
475C2×C4#53411yescharacteristic
476C2×C4#53711yescharacteristic
477C2×C4#53812yes—
478C2×C4#53912yes—
479C2×C4#54612yes—
480C2×C4#54711yescharacteristic
481C2×C4#55112yes—
482C2×C4#55112yes—
483C2×C4#55212yes—
484C2×C4#55311yescharacteristic
485C2×C4#55312yes—
486C8—42no—
487C8—81no—
488C8—82no—
489C22#241211yescharacteristic
490C22#241611yescharacteristic
491C22#241812yes—
492C22#248112yes—
493C22#285011yescharacteristic
494C22C2×#46211yescharacteristic
495C22#304011yescharacteristic
496C22#304311yescharacteristic
497C22#304611yescharacteristic
498C22#310412yes—
499C22#311612yes—
500C22#312112yes—
501C22#312512yes—
502C22#327112yes—
503C22#327811yescharacteristic
504C22D4×Q1611yescharacteristic
505C22#329912yes—
506C22#331612yes—
507C22#331712yes—
508C22#332712yes—
509C22#336311yescharacteristic
510C22#337112yes—
511C22#337411yescharacteristic
512C4—28no—
513C4—24no—
514C4—44no—
515C4—42no—
516C4—82no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4—161no—
521C4—162no—
522C4C2×#39711yescharacteristic
523C4#241912yes—
524C4#249811yescharacteristic
525C4#251012yes—
526C4#254311yescharacteristic
527C4#263911yescharacteristic
528C2#1473511yescharacteristic
529C2#2453211yescharacteristic
530C2#2454311yescharacteristic
531C2#2473211yescharacteristic
532C2#2474711yescharacteristic
533C2#2490612yes—
534C2#2492412yes—
535C2#2755612yes—
536C2#2758012yes—
537C2#3302711yescharacteristic
538C2#3303711yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
236. #95Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#18782
C2×#18032
#151852
Q8×#952
#113712
#113162
#112552
#111902
#110772
#107892