Finite Group Databasebeta

← #224521

Subgroups — #224521

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2448C211yesmaximal, characteristic
2#22510C212yesmaximal, complemented
3#9653C212yesmaximal
4#9652C212yesmaximal, complemented
5#9651C212yesmaximal, complemented
6#8561C212yesmaximal, complemented
7#8560C211yesmaximal, characteristic, complemented
8#8155C212yesmaximal, complemented
9#8153C211yesmaximal, characteristic, complemented
10C22×#124C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#446—22no—
13#2448—22no—
14#2448—22no—
15#2448—22no—
16#2448—22no—
17C2×#399C412yes—
18C2×#397C412yes—
19C2×#391C411yescharacteristic
20C2×#390C411yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#388C2212yes—
24C2×#388C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1985—22no—
27#1985—22no—
28#1984—22no—
29#1797C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2211yescharacteristic
37C2×#324C2211yescharacteristic
38C2×#323C2212yes—
39C2×#323C2212yes—
40C2×#323C2212yes—
41C2×#323C2212yes—
42#1280C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45C23×Q8C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C2×#124—22no—
48C2×#124—21no—
49C2×#124—21no—
50C2×#124—44no—
51C2×C4×Q8—44no—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2312yes—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#446—48no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8C2312yes—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×#97—22no—
81C2×#97—22no—
82C2×C4×C8—22no—
83#399—22no—
84#399—22no—
85#399—22no—
86#399—22no—
87#397—22no—
88#397—22no—
89#397—22no—
90#397—22no—
91#391—22no—
92#391—21no—
93#391—21no—
94#390—22no—
95#390—21no—
96#390—21no—
97#390—22no—
98#390—22no—
99#389—22no—
100#389—21no—
101#389—21no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#324—24no—
124#324—41no—
125#324—41no—
126#323—24no—
127#323—24no—
128#323—42no—
129#323—42no—
130C25C22×C412yes—
131C22×Q8—24no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—42no—
135C22×Q8C22⋊C412yes—
136C22×Q8C22⋊C411yescharacteristic
137C22×Q8C22⋊C411yescharacteristic
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—41no—
141C22×D4—41no—
142C22×D4—42no—
143C22×D4—42no—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158#124—48no—
159#124—44no—
160#124—44no—
161C4×Q8—28no—
162C4×Q8—44no—
163C4×Q8—44no—
164C4×Q8—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—42no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C4⋊C8—44no—
225C4⋊C8—44no—
226C4⋊C8—44no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8#4114yes—
235#97—44no—
236#97—44no—
237C4×C8—44no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24C23×C411yescharacteristic
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8#9914yes—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—84no—
276C2×D4—84no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4#13112yes—
329C22×C4C22×D412yes—
330C22×C4C2×#4112yes—
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C42−1+411yescharacteristic
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23C22×C22⋊C411yescharacteristic
411C23C4×#4111yescharacteristic
412C23#52011yescharacteristic
413C23C2×#13111yescharacteristic
414C23#56811yescharacteristic
415C23#56811yescharacteristic
416C23#56811yescharacteristic
417Q8—44no—
418Q8—48no—
419Q8—44no—
420D4—84no—
421D4—84no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4C2×#9912yes—
481C2×C4C2×#9914yes—
482C2×C4#41312yes—
483C2×C4#54812yes—
484C2×C4#55012yes—
485C8—44no—
486C8—82no—
487C8—82no—
488C8—82no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—162no—
509C22—162no—
510C22—162no—
511C22—162no—
512C22#234111yescharacteristic
513C22#287911yescharacteristic
514C22#288011yescharacteristic
515C22C22×#9911yescharacteristic
516C22#288411yescharacteristic
517C22C2×#41311yescharacteristic
518C22#288811yescharacteristic
519C22#296711yescharacteristic
520C22#296811yescharacteristic
521C22#299211yescharacteristic
522C22#299511yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—24no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—82no—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—21no—
541C2—162no—
542C2#1739211yescharacteristic
543C2#1740911yescharacteristic
544C2#1741911yescharacteristic
545C2#1742811yescharacteristic
546C2#2253311yescharacteristic
547C2#2253411yescharacteristic
548C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ 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. #22510C2semidirect
4. #9652C2semidirect
5. #9651C2semidirect
6. #8561C2semidirect
7. #8560C2semidirect
8. #8155C2semidirect
9. #8153C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24482
#225102
#96532
#96522
#96512
#85612
#85602
#81552
#81532