Finite Group Databasebeta

← #164069

Subgroups — #164069

Order 512.

1101 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35308C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic, complemented
3#22469C211yesmaximal, characteristic, complemented
4#7730C212yesmaximal, complemented
5#7707C212yesmaximal, complemented
6#7690C212yesmaximal, complemented
7#7667C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic, complemented
9#7575C211yesmaximal, characteristic, complemented
10#7575C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#537C411yescharacteristic
13C2×#536C411yescharacteristic
14C2×#536C411yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C2×#446—21no—
17C2×C8×D4—21no—
18#2857C2212yes—
19#2850C2212yes—
20#2621C412yes—
21#2620C412yes—
22#2429—21no—
23#2429—21no—
24#2427—21no—
25#2427—21no—
26#2425—22no—
27#2425—22no—
28#2317C411yescharacteristic
29#2297C2211yescharacteristic
30C2×#391C2211yescharacteristic
31#1984—21no—
32#1955—22no—
33#1954—21no—
34#1869—22no—
35#1817C2212yes—
36#1797C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2212yes—
47#1272C2212yes—
48#1272C2212yes—
49#1272C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C22×#41C2×C411yescharacteristic
55C23×Q8C2×C411yescharacteristic
56C23×C8—22no—
57#537—44no—
58#536—44no—
59#536—44no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C412yes—
62C2×#124—21no—
63C2×#124C2×C412yes—
64C2×#123—22no—
65C2×#123—22no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8C2×C412yes—
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8C2×C412yes—
72C2×#121—22no—
73C2×#121—21no—
74C2×#121C2×C412yes—
75C4×#41—44no—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×C4×D4D411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117C2×C411yescharacteristic
84C2×#117C2×C412yes—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88#446—44no—
89C8×D4—44no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—21no—
100C2×#97—22no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103#398—22no—
104#398—22no—
105#397—22no—
106#397—22no—
107#396—22no—
108#396—22no—
109#394—22no—
110#394—22no—
111#391—21no—
112#391—21no—
113#391C2×C412yes—
114#391C2×C411yescharacteristic
115#391C2312yes—
116#391C2311yescharacteristic
117#390C2×C411yescharacteristic
118#390C2311yescharacteristic
119#389—21no—
120#389—21no—
121#389—22no—
122#389—22no—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#386—21no—
128#386—21no—
129#385—22no—
130#384C2×C412yes—
131#382C2×C412yes—
132#380—22no—
133#380—22no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C2×#95—21no—
137C2×#95—21no—
138#336—22no—
139#336—22no—
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—24no—
148C2×#41—24no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C22×Q8—21no—
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22×C412yes—
158C22×Q8C22×C411yescharacteristic
159C22×D4—21no—
160C22×D4—22no—
161C22×D4C22⋊C412yes—
162C22×D4C22⋊C411yescharacteristic
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—44no—
180C22×C8—44no—
181C22×C8—82no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184#124—44no—
185#124—44no—
186#123—48no—
187#123—48no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194#121—44no—
195#121—48no—
196#121—44no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×D4—24no—
200C4×D4—44no—
201C4×D4—42no—
202C4×D4—44no—
203#117—24no—
204#117—44no—
205#117—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C42—21no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42C22×C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2411yescharacteristic
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—44no—
249C22⋊C8—44no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—44no—
253C22⋊C8—82no—
254C22⋊C8—81no—
255C22⋊C8—81no—
256#97—44no—
257C4×C8—42no—
258C4×C8—42no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—21no—
269C24—21no—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C412yes—
272#41—416no—
273#41—88no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—88no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4#10014yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4#13114yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—88no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4#9914yes—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—44no—
395C22⋊C4—81no—
396C22⋊C4—81no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—24no—
401C42—41no—
402C42—41no—
403C42—42no—
404C42—42no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23#40812yes—
425C23C2×#10014yes—
426C23C22×C22⋊C411yescharacteristic
427Q8—44no—
428Q8—84no—
429Q8—82no—
430Q8—84no—
431D4—28no—
432D4—48no—
433D4—82no—
434D4—84no—
435D4—164no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—162no—
484C2×C4#40812yes—
485C2×C4C2×#9914yes—
486C2×C4C2×#9912yes—
487C2×C4C2×#10012yes—
488C2×C4C2×#13112yes—
489C2×C4#57711yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57811yescharacteristic
492C2×C4#57811yescharacteristic
493C8—42no—
494C8—42no—
495C8—81no—
496C8—82no—
497C8—81no—
498C8—82no—
499C22—24no—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—22no—
504C22—24no—
505C22—41no—
506C22—42no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—42no—
511C22—42no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22C2×#40811yescharacteristic
521C22C22×#9911yescharacteristic
522C22C22×#10011yescharacteristic
523C22#289811yescharacteristic
524C22#289811yescharacteristic
525C22#290111yescharacteristic
526C22#290111yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—24no—
532C4—22no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—42no—
537C4—81no—
538C4—82no—
539C4—81no—
540C4—81no—
541C4—82no—
542C4#147114yes—
543C2—44no—
544C2—42no—
545C2—161no—
546C2C2×#147111yescharacteristic
547C2#1746911yescharacteristic
548C2#1748411yescharacteristic
549C2#1760611yescharacteristic
550C2#1760811yescharacteristic
551C2#1768511yescharacteristic
552C2#1777911yescharacteristic
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. #22469C2semidirect
4. #7730C2semidirect
5. #7707C2semidirect
6. #7690C2semidirect
7. #7667C2semidirect
8. #7597C2semidirect
9. #7575C2semidirect
10. #7575C2semidirect
11. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#353082
#225322
#224692
#77302
#77072
#76902
#76672
#75972
#75752
#75632