Finite Group Databasebeta

← #163970

Subgroups — #163970

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34267C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic, complemented
3#22473C211yesmaximal, characteristic, complemented
4#7647C212yesmaximal, complemented
5#7633C212yesmaximal, complemented
6#7623C212yesmaximal, complemented
7#7617C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#537C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#446—21no—
16C2×#436—21no—
17#2364—22no—
18#2363—21no—
19#2362—21no—
20#2361—22no—
21#2361—21no—
22#2360—21no—
23#2324C412yes—
24#2323C412yes—
25#2311C411yescharacteristic
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28#2297C2211yescharacteristic
29C2×#389C2211yescharacteristic
30#2274C411yescharacteristic
31#1984—21no—
32#1955—22no—
33#1954—21no—
34#1870—22no—
35#1817C2212yes—
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2211yescharacteristic
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1728C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C411yescharacteristic
54C22×#41C2×C411yescharacteristic
55C22×M16—22no—
56#537—24no—
57#537—24no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×#121—21no—
61C2×#121—22no—
62C2×#121C2×C412yes—
63#520—44no—
64C4×#41—44no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C412yes—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D412yes—
74C2×C4×D4D412yes—
75C2×#117—21no—
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C412yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85#446—44no—
86#436—44no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×#97—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#391—21no—
101#391C2×C411yescharacteristic
102#391C2311yescharacteristic
103#390—21no—
104#390—22no—
105#390C2×C411yescharacteristic
106#390C2311yescharacteristic
107#389—22no—
108#389—21no—
109#389—21no—
110#389C2×C412yes—
111#389C2312yes—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—22no—
118#386—22no—
119#386—21no—
120#386—22no—
121#386—21no—
122#386—22no—
123#385—22no—
124#385—22no—
125#380C2×C412yes—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C412yes—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C2×#95—21no—
138#336—22no—
139#336—22no—
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—42no—
153C2×#41M1614yescomplemented
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C22×D4—21no—
157C22×D4—22no—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C412yes—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C2×M16—44no—
172C2×M16—44no—
173C2×M16—82no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22⋊Q8—28no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183#121—216no—
184#121—28no—
185#121—24no—
186#121—24no—
187C4×Q8—24no—
188C4×Q8—42no—
189C4×Q8—44no—
190C4×D4—24no—
191C4×D4—44no—
192C4×D4—42no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—48no—
196C4×D4—82no—
197C4×D4—82no—
198#117—24no—
199#117—44no—
200#117—44no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4M1614yescomplemented
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—42no—
240C2×C42—44no—
241C2×C42C22×C412yes—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2411yescharacteristic
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—44no—
251C22⋊C8—44no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—44no—
255C22⋊C8—81no—
256C22⋊C8—82no—
257C22⋊C8—81no—
258#97—44no—
259C4×C8—42no—
260C4×C8—42no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271C24—21no—
272C24C2×C22⋊C412yes—
273C24C2×C22⋊C411yescharacteristic
274#41—48no—
275#41—48no—
276#41—48no—
277#41—48no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8C2×M1612yes—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—28no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4C2×M1612yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×M1612yes—
347C22×C4C2×M1612yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—88no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—41no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4#9914yes—
385C4⋊C4C2×M1614yes—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—44no—
397C22⋊C4—81no—
398C22⋊C4—81no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—24no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—44no—
409C42—42no—
410C42—82no—
411C42—82no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23#40712yes—
433C23C22×C22⋊C411yescharacteristic
434Q8—42no—
435Q8—42no—
436Q8—44no—
437D4—42no—
438D4—42no—
439D4—48no—
440D4—48no—
441D4—44no—
442D4—88no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—84no—
492C2×C4—164no—
493C2×C4—162no—
494C2×C4#40712yes—
495C2×C4C2×#9914yes—
496C2×C4C2×#9912yes—
497C2×C4#41312yes—
498C2×C4C22×M1611yescharacteristic
499C2×C4C22×M1611yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57811yescharacteristic
504C8—42no—
505C8—42no—
506C8—81no—
507C8—81no—
508C8—82no—
509C8—82no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22C2×#40711yescharacteristic
532C22C22×#9911yescharacteristic
533C22C2×#41311yescharacteristic
534C22#289711yescharacteristic
535C22#289811yescharacteristic
536C22#290111yescharacteristic
537C22#290111yescharacteristic
538C4—21no—
539C4—21no—
540C4—24no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—44no—
546C4—41no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—81no—
552C4—81no—
553C4—82no—
554C4#146514yes—
555C2—42no—
556C2—82no—
557C2—82no—
558C2C2×#146511yescharacteristic
559C2#1745311yescharacteristic
560C2#1745411yescharacteristic
561C2#1760811yescharacteristic
562C2#1761111yescharacteristic
563C2#1768711yescharacteristic
564C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22532C2semidirect
3. #22473C2semidirect
4. #7647C2semidirect
5. #7633C2semidirect
6. #7623C2semidirect
7. #7617C2semidirect
8. #7597C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect
11. #7563C2semidirect
153. C2×#41M16semidirect
217. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#342672
#225322
#224732
#76472
#76332
#76232
#76172
#75972
#75762
#75732
#75632