Finite Group Databasebeta

← #163930

Subgroups — #163930

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34250C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#22532C211yesmaximal, characteristic
4#7641C212yesmaximal, complemented
5#7627C212yesmaximal, complemented
6#7613C212yesmaximal, complemented
7#7613C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#520C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#446—21no—
16C2×#446—21no—
17#2351—21no—
18Q8×C4⋊C4—21no—
19#2349—21no—
20#2349—22no—
21D4×C4⋊C4—21no—
22D4×C4⋊C4—22no—
23#2317C412yes—
24#2317C411yescharacteristic
25#2316C412yes—
26#2316C411yescharacteristic
27#2296C2211yescharacteristic
28#2296C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31#1955—22no—
32#1955—22no—
33#1954—21no—
34#1954—21no—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2212yes—
39#1749C2212yes—
40#1749C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C22×#41C2×C411yescharacteristic
54#520—24no—
55#520—24no—
56#520—44no—
57C4×#41—44no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D412yes—
70C2×#117C2×C412yes—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C412yes—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C412yes—
75C2×#117C2×C411yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2312yes—
84#446—44no—
85#446—44no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—22no—
99C2×#97—22no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102#389—22no—
103#389—21no—
104#389—22no—
105#389—21no—
106#389—21no—
107#389—22no—
108#389—21no—
109#389—22no—
110#389C2×C411yescharacteristic
111#389C2×C412yes—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C411yescharacteristic
116#389C2×C412yes—
117#389C2312yes—
118#389C2311yescharacteristic
119#389C2311yescharacteristic
120#385—21no—
121#385—22no—
122#385—21no—
123#385—22no—
124#385—22no—
125#385—22no—
126#385—22no—
127#384—22no—
128#384—21no—
129#382—22no—
130#382—21no—
131#380—22no—
132#380—22no—
133#380—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—22no—
138C2×#95—21no—
139C2×#95—22no—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2312yes—
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—24no—
148C2×#41—22no—
149C2×#41—22no—
150C22×Q8C22⋊C411yescharacteristic
151C22×D4C22⋊C412yes—
152C22×D4C22⋊C411yescharacteristic
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C4×Q8—28no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—44no—
173C4×D4—216no—
174C4×D4—28no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—48no—
180C4×D4—44no—
181#117—22no—
182#117—24no—
183#117—24no—
184#117—44no—
185#117—42no—
186#117—44no—
187#117—42no—
188#117M1614yes—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4M1614yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2411yescharacteristic
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—44no—
246C4⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—82no—
252C22⋊C8—81no—
253C22⋊C8—82no—
254C22⋊C8—81no—
255#97—44no—
256#97—44no—
257C4×C8—42no—
258C4×C8—42no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95—21no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—21no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C412yes—
271#41—216no—
272#41—216no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8#9914yes—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4#10114yes—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—81no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4—82no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—81no—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×M1612yes—
329C22×C4C2×M1612yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C4⋊C4—28no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4C2×M1612yes—
379C4⋊C4C2×M1614yes—
380C4⋊C4C2×M1612yes—
381C22⋊C4—28no—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C42—28no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—24no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—44no—
398C42—42no—
399C23—22no—
400C23—21no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23#40712yes—
412C23C2×#10114yes—
413C23C22×C22⋊C411yescharacteristic
414Q8—28no—
415D4—216no—
416D4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—88no—
469C2×C4—164no—
470C2×C4—162no—
471C2×C4#40712yes—
472C2×C4C2×#9914yes—
473C2×C4C2×#9912yes—
474C2×C4C2×#10112yes—
475C2×C4C22×M1611yescharacteristic
476C2×C4C22×M1611yescharacteristic
477C2×C4#57611yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57811yescharacteristic
481C8—42no—
482C8—42no—
483C8—81no—
484C8—82no—
485C8—81no—
486C8—82no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—22no—
492C22—24no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—42no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22C2×#40711yescharacteristic
502C22C22×#9911yescharacteristic
503C22C22×#10111yescharacteristic
504C22#289711yescharacteristic
505C22#289811yescharacteristic
506C22#290111yescharacteristic
507C22#290111yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—24no—
513C4—21no—
514C4—21no—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—81no—
523C4—82no—
524C4—81no—
525C4#146314yes—
526C2—44no—
527C2—42no—
528C2C2×#146311yescharacteristic
529C2#1745311yescharacteristic
530C2#1745411yescharacteristic
531C2#1760611yescharacteristic
532C2#1761111yescharacteristic
533C2#1768911yescharacteristic
534C2#1777411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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
4. #7641C2semidirect
5. #7627C2semidirect
6. #7613C2semidirect
7. #7613C2semidirect
8. #7598C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect
11. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#342502
#225322
#76412
#76272
#76132
#75982
#75762
#75732
#75642