Finite Group Databasebeta

← #196940

Subgroups — #196940

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35247C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic, complemented
3#21820C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic
5C2×#1500C211yesmaximal, characteristic
6#9157C212yesmaximal
7#9156C212yesmaximal
8#9153C212yesmaximal, complemented
9#9152C212yesmaximal, complemented
10#8022C211yesmaximal, characteristic, complemented
11#8014C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2855C2212yes—
17#2854C2212yes—
18#2637—22no—
19#2592C412yescomplemented
20#2584C412yescomplemented
21#2582C412yescomplemented
22#2538C412yescomplemented
23#2486—21no—
24#2413—22no—
25#2409—21no—
26#2323—22no—
27#2320—24no—
28#2320—24no—
29C2×#391—21no—
30C2×#389—21no—
31C2×#387—22no—
32#2285C2211yescharacteristic
33C2×#383C2212yes—
34C2×#382C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1985—22no—
38#1984—22no—
39#1817C2212yes—
40#1817C2212yes—
41#1509—24no—
42#1500—24no—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#327C2211yescharacteristic
47C2×#327C2211yescharacteristic
48C2×#327C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#529—22no—
55#529—22no—
56#529—22no—
57#529—22no—
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—21no—
62C2×#123—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8D412yes—
74C2×C4×Q8D412yes—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4D412yes—
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95C2×C22⋊C8C2312yes—
96C2×#97—21no—
97C2×C4×C8—21no—
98#398—24no—
99#398—22no—
100#397—22no—
101#396—24no—
102#395—22no—
103#393—22no—
104#391—28no—
105#391—22no—
106#391C2×C414yes—
107#390C2×C414yes—
108#389—28no—
109#389—22no—
110#389—24no—
111#389C2×C414yes—
112#389C2×C414yes—
113#388—24no—
114#388—24no—
115#387—216no—
116#387—24no—
117#386—22no—
118#385—22no—
119#384C2×C412yes—
120#383—24no—
121#383—24no—
122#382—21no—
123#382—24no—
124#382—24no—
125#381—22no—
126#380—21no—
127#380—22no—
128#380C2312yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4D414yes—
133#376—22no—
134#376D414yes—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142#336—24no—
143#336—24no—
144#328—22no—
145#328—22no—
146#328—24no—
147#327—22no—
148#327—22no—
149#327—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—44no—
153C22×Q8C22⋊C412yes—
154C22×Q8C22⋊C412yes—
155C22×Q8C22×C411yescharacteristic
156C22×D4—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170#126—42no—
171#126—42no—
172#125—22no—
173#125—22no—
174#124—22no—
175#124—22no—
176#123—42no—
177#123—42no—
178#123—82no—
179#123—82no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—48no—
185#121—21no—
186#121—21no—
187#121—21no—
188#121—21no—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—42no—
202C4×D4—42no—
203#117—42no—
204#117—42no—
205#117—81no—
206#117—81no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C414yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—44no—
248C2×C42—81no—
249C2×C42—81no—
250C2×C42C22×C414yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D414yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4112yes—
259C2×C42#4112yes—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—22no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—22no—
268#97—42no—
269C4×C8—42no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—22no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95C2×D412yes—
284#95C2×D412yes—
285#95C2×D414yes—
286C24—21no—
287C24C22×D411yescharacteristic
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—44no—
295C2×Q8—48no—
296C2×Q8—48no—
297C2×Q8—42no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
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—21no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4C2×C22⋊C414yes—
344C22×C4C2×C22⋊C414yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—44no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—82no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C22⋊C4—48no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—84no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—48no—
401C42—42no—
402C42—84no—
403C42—82no—
404C42C4×D412yes—
405C42C4×D412yes—
406C42C4×D412yes—
407C42C4×D412yes—
408C42#12112yes—
409C42#12112yes—
410C42#12112yes—
411C42#12112yes—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—41no—
419C23C22×C22⋊C411yescharacteristic
420C23C2×#12011yescharacteristic
421C23C2×#12311yescharacteristic
422C23#57611yescharacteristic
423C23#57711yescharacteristic
424C23#57711yescharacteristic
425C23#57811yescharacteristic
426Q8—48no—
427Q8—48no—
428Q8—81no—
429Q8—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—81no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4#38612yes—
492C2×C4#38612yes—
493C2×C4#38612yes—
494C2×C4#38612yes—
495C2×C4#42114yes—
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C8—81no—
501C8—81no—
502C8—82no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22C2×#38611yescharacteristic
517C22#288412yes—
518C22#289711yescharacteristic
519C22#289811yescharacteristic
520C22C2×#42112yes—
521C22#301211yescharacteristic
522C22#301311yescharacteristic
523C22#309811yescharacteristic
524C22#309911yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—44no—
533C4—44no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—81no—
538C4—84no—
539C4—81no—
540C4—81no—
541C4#189714yes—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—161no—
547C2#1760112yes—
548C2#1760212yes—
549C2#2091511yescharacteristic
550C2C2×#189711yescharacteristic
551C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2320C2semidirect
3. #21820C2semidirect
8. #9153C2semidirect
9. #9152C2semidirect
10. #8022C2semidirect
11. #8014C2semidirect
19. #2592C4semidirect
20. #2584C4semidirect
21. #2582C4semidirect
22. #2538C4semidirect

Maximal subgroups

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

subgroupindex
#352472
C2×#23202
#218202
C2×#15092
C2×#15002
#91572
#91562
#91532
#91522
#80222
#80142