Finite Group Databasebeta

← #164095

Subgroups — #164095

Order 512.

1060 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#35254C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic, complemented
3#22469C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7694C212yesmaximal, complemented
6#7678C212yesmaximal, complemented
7#7654C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7572C211yesmaximal, characteristic, complemented
10#7572C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#535C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×C8×D4—21no—
16#2854C2212yes—
17#2854C2212yes—
18#2624C412yes—
19#2624C412yes—
20#2448—21no—
21#2448—21no—
22#2445—21no—
23#2445—21no—
24#2418—22no—
25#2418—22no—
26#2316C411yescharacteristic
27#2311C411yescharacteristic
28#2309C411yescharacteristic
29#2296C2211yescharacteristic
30C2×#390C2211yescharacteristic
31#1984—21no—
32#1984—21no—
33#1870—22no—
34#1869—22no—
35#1817C2212yescomplemented
36#1817C2212yescomplemented
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2211yescharacteristic
41#1728C2212yescomplemented
42C2×#336C2212yescomplemented
43#1271C2211yescharacteristic, complemented
44#1271C2212yescomplemented
45#1271C2211yescharacteristic, complemented
46#1271C2212yescomplemented
47#1271C2211yescharacteristic, complemented
48#1271C2211yescharacteristic, complemented
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57#535—44no—
58C2×#127—21no—
59C2×#124—21no—
60C2×#123—22no—
61C2×#123—22no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#120—22no—
65C4×#41—44no—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D412yes—
71C2×C4×D4D412yes—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C412yes—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2312yes—
81#436—44no—
82C8×D4—44no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#399—22no—
94#399—22no—
95#397—22no—
96#397—22no—
97#394—22no—
98#394—22no—
99#394—22no—
100#394—22no—
101#391—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390C2×C412yes—
108#390C2×C412yes—
109#390C2×C412yes—
110#390C2312yes—
111#389—21no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#389C2311yescharacteristic
118#389C2311yescharacteristic
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#385—21no—
127#385—21no—
128#380—22no—
129#380C2×C412yes—
130#380C2×C412yes—
131#379—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C25—21no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C22⋊C412yes—
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22×C412yes—
158C23×C4—21no—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—44no—
166C2×M16—44no—
167C2×M16—82no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8—82no—
177#127—44no—
178#124—44no—
179#123—82no—
180#123—82no—
181#123—82no—
182#123—82no—
183#121—44no—
184#121—44no—
185#120—48no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—44no—
191C4×D4—44no—
192#117—24no—
193#117—44no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—81no—
228C2×C22⋊C4—81no—
229C2×C22⋊C4—81no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—42no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—81no—
252C22⋊C8—81no—
253C4×C8—42no—
254C4×C8—42no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—81no—
268C24—81no—
269C24—81no—
270C24—81no—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C412yes—
273#41—416no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—88no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4#9914yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4#13114yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335M16—88no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—88no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—22no—
393C42—22no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—21no—
412C23—21no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—84no—
421C23—84no—
422C23—84no—
423C23—162no—
424C23—162no—
425C23—162no—
426C23—162no—
427C23#40812yes—
428C23C2×#9914yes—
429C23C22×C22⋊C411yescharacteristic
430Q8—44no—
431D4—28no—
432D4—48no—
433D4—84no—
434D4—84no—
435D4—164no—
436D4—164no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—162no—
484C2×C4#40812yes—
485C2×C4C2×#9912yes—
486C2×C4#41312yes—
487C2×C4C2×#13112yes—
488C2×C4#57611yescharacteristic
489C2×C4#57611yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C8—42no—
493C8—42no—
494C8—81no—
495C8—81no—
496C8—82no—
497C8—82no—
498C22—22no—
499C22—22no—
500C22—24no—
501C22—24no—
502C22—22no—
503C22—24no—
504C22—42no—
505C22—41no—
506C22—42no—
507C22—42no—
508C22—41no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—168no—
520C22C2×#40811yescharacteristic
521C22C22×#9911yescharacteristic
522C22C2×#41311yescharacteristic
523C22#289711yescharacteristic
524C22#289711yescharacteristic
525C22#290111yescharacteristic
526C22#290111yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—41no—
535C4—41no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4—82no—
540C4—81no—
541C4—81no—
542C4#146814yes—
543C2—44no—
544C2—42no—
545C2—162no—
546C2C2×#146811yescharacteristic
547C2#1746811yescharacteristic
548C2#1748311yescharacteristic
549C2#1760311yescharacteristic
550C2#1760311yescharacteristic
551C2#1769111yescharacteristic
552C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
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. #22473C2semidirect
3. #22469C2semidirect
4. #7718C2semidirect
5. #7694C2semidirect
6. #7678C2semidirect
7. #7654C2semidirect
8. #7598C2semidirect
9. #7572C2semidirect
10. #7572C2semidirect
11. #7564C2semidirect
35. #1817C22semidirect
36. #1817C22semidirect
41. #1728C22semidirect
42. C2×#336C22semidirect
43. #1271C22semidirect
44. #1271C22semidirect
45. #1271C22semidirect
46. #1271C22semidirect
47. #1271C22semidirect
48. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#224732
#224692
#77182
#76942
#76782
#76542
#75982
#75722
#75642