Finite Group Databasebeta

← #196728

Subgroups — #196728

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic
2#35297C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4C2×#1554C211yesmaximal, characteristic, complemented
5C2×#1542C211yesmaximal, characteristic, complemented
6#9130C212yesmaximal, complemented
7#9129C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#9127C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#8038C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2988—24no—
15#2988—24no—
16C2×C8×Q8—21no—
17C2×C8×D4—21no—
18C2×C8×D4—22no—
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2212yes—
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2625—21no—
28#2616—21no—
29#2584—21no—
30#2581C412yes—
31#2562—21no—
32#2536C412yes—
33#2485—22no—
34#2415C414yes—
35#2323—22no—
36#2285C2211yescharacteristic
37#1818—21no—
38#1797C2212yes—
39#1797C2212yes—
40#1780—21no—
41#1750—22no—
42#1554—24no—
43#1542—24no—
44C2×#333C2212yes—
45C2×#333C2211yescharacteristic
46C2×#333C2211yescharacteristic
47C2×#332C2212yes—
48C2×#332C2211yescharacteristic
49C2×#332C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41D411yescharacteristic
55C23×C8—22no—
56C23×C8—21no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C414yes—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117—21no—
81C2×#117C2×C412yes—
82C2×#117D412yes—
83C2×#117D411yescharacteristic
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C8×Q8—28no—
87C8×D4—216no—
88C8×D4—28no—
89#432—22no—
90#432—22no—
91#431—22no—
92#431—24no—
93#431—24no—
94#431—22no—
95#431—22no—
96#431D414yes—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8D414yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—22no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119#400—22no—
120#398—22no—
121#397—24no—
122#395—22no—
123#394—22no—
124#394—24no—
125#391—22no—
126#391C2×C414yes—
127#389—24no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389C2×C414yes—
132#388—24no—
133#388—24no—
134#387—24no—
135#384C2312yes—
136#382C2×C412yes—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—21no—
141#380—21no—
142#380C2×C414yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145#336—22no—
146#336—22no—
147#333—24no—
148#333—22no—
149#333—22no—
150#332—24no—
151#332—22no—
152#332—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—24no—
156C2×#41—24no—
157C22×Q8—21no—
158C22×Q8C22⋊C411yescharacteristic
159C22×Q8C22×C412yes—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22⋊C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—28no—
172C22×C8—24no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—82no—
177C22×C8—82no—
178C22×C8—82no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187#125—42no—
188#125—42no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C22⋊Q8—84no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212#117—22no—
213#117—22no—
214#117—24no—
215#117—24no—
216#117—42no—
217#117—42no—
218#117—82no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C414yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258C4⋊C8—28no—
259C4⋊C8—24no—
260C4⋊C8—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8C2×D412yes—
270C4⋊C8C2×D414yes—
271C4⋊C8C2×D412yes—
272C22⋊C8—28no—
273C22⋊C8—24no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278#97—42no—
279C4×C8—28no—
280C4×C8—24no—
281C4×C8—42no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C412yes—
290#41—216no—
291#41—28no—
292#41—28no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4#12012yes—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12312yes—
343C22×C4#13514yes—
344C22×C4#13514yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—22no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—84no—
363C2×C8—88no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—84no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—44no—
389C22⋊C4—48no—
390C22⋊C4—82no—
391C22⋊C4—84no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—44no—
399C42—82no—
400C42C4×D412yes—
401C42C4×D412yes—
402C42#12012yes—
403C42#12112yes—
404C42#12112yes—
405C42#12312yes—
406C42C22×D411yescharacteristic
407C42C22×D411yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C23—21no—
411C23—22no—
412C23—21no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23#38612yes—
421C23#41614yes—
422C23C22×C22⋊C411yescharacteristic
423Q8—28no—
424Q8—82no—
425Q8—82no—
426Q8—82no—
427Q8—81no—
428Q8—81no—
429D4—28no—
430D4—216no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#38612yes—
472C2×C4#38612yes—
473C2×C4#38612yes—
474C2×C4#41614yes—
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×#12011yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4C2×#13512yes—
482C2×C4C2×#13512yes—
483C2×C4#57712yes—
484C2×C4#57812yes—
485C8—44no—
486C8—42no—
487C8—42no—
488C8—82no—
489C8—81no—
490C8—81no—
491C22—22no—
492C22—24no—
493C22—24no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22C2×#38611yescharacteristic
504C22#288811yescharacteristic
505C22#288911yescharacteristic
506C22C2×#41612yes—
507C22#289812yes—
508C4—21no—
509C4—21no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—24no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—84no—
518C4—82no—
519C4#300911yescharacteristic
520C4#301111yescharacteristic
521C4#305012yes—
522C4#305011yescharacteristic
523C4#305011yescharacteristic
524C4#309712yes—
525C2—42no—
526C2—44no—
527C2#1772512yes—
528C2#1772612yes—
529C2#2091311yescharacteristic
530C2#2096711yescharacteristic
531C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1554C2semidirect
5. C2×#1542C2semidirect
6. #9130C2semidirect
7. #9129C2semidirect
8. #9128C2semidirect
9. #9127C2semidirect
10. #8044C2semidirect
11. #8038C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
#352972
#207212
C2×#15542
C2×#15422
#91302
#91292
#91282
#91272
#80442
#80382