Finite Group Databasebeta

← #230184

Subgroups — #230184

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23187C211yesmaximal, characteristic, complemented
2#22968C211yesmaximal, characteristic
3#22962C211yesmaximal, characteristic
4#22703C211yesmaximal, characteristic
5C2×#1669C211yesmaximal, characteristic
6#9835C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9573C211yesmaximal, characteristic, complemented
9#9565C211yesmaximal, characteristic, complemented
10#8981C211yesmaximal, characteristic
11#8981C211yesmaximal, characteristic
12#8925C211yesmaximal, characteristic, complemented
13#8923C211yesmaximal, characteristic, complemented
14#8289C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic
16C2×#494C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#441—21no—
20C2×C8×D4C2211yescharacteristic
21#2365C2211yescharacteristic
22#2365C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1981C2211yescharacteristic
30#1981C2211yescharacteristic
31#1971—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1943—21no—
37#1942—21no—
38#1921—21no—
39#1918—21no—
40#1878—21no—
41#1878—21no—
42#1864—21no—
43#1862—21no—
44#1853—21no—
45#1853—21no—
46#1834C2211yescharacteristic
47#1807—21no—
48C4×#106C2211yescharacteristic
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1669C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69C22×Q16—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80#494—22no—
81#494C2×C414yescomplemented
82#474—22no—
83#474—22no—
84#474C2×C414yes—
85#474C2×C414yes—
86#441—44no—
87C8×D4—22no—
88C8×D4C2×C414yes—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106D411yescharacteristic
96C2×#106D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102D411yescharacteristic
108C2×#102D411yescharacteristic
109C2×C22⋊C8—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#391—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#387—41no—
123#387—41no—
124#385—41no—
125#385—41no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—41no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#335C2×C414yescomplemented
155#332C2×C414yescomplemented
156#332C2×C414yescomplemented
157D4⋊C8—22no—
158#322—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—41no—
164C23×C4—41no—
165C2×Q16—24no—
166C2×Q16—24no—
167C2×Q16—42no—
168C2×Q16—42no—
169C2×Q16—81no—
170C2×Q16—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8C22×C412yes—
190C4⋊Q8C22×C412yes—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—81no—
194C4×Q8—81no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—81no—
200C4×D4—81no—
201C4×D4C22×C412yes—
202C4×D4#4112yes—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42C2411yescharacteristic
237#107—24no—
238#107—24no—
239#107—81no—
240#106—22no—
241#106—41no—
242#106—81no—
243#106—81no—
244#106—81no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253#103—42no—
254#103—42no—
255#103—81no—
256#103—81no—
257#102—24no—
258#102—24no—
259#102—22no—
260#102—22no—
261C22⋊C8—24no—
262#97—42no—
263#97—81no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—81no—
269C4×C8C22×C412yes—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280C24—21no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4C4×D412yes—
297C2×D4#13712yes—
298C2×D4#13712yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—161no—
321C22×C4—161no—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325Q16—48no—
326Q16—84no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—84no—
353C2×C8C4×D412yes—
354C2×C8C4×D412yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4C4×D412yes—
390C4⋊C4AGL(1,ℤ/8)12yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—82no—
402C42—82no—
403C42C23×C411yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C422−1+411yescharacteristic
407C23—22no—
408C23—22no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23D4211yescharacteristic
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419Q8—81no—
420Q8—81no—
421Q8—81no—
422Q8—81no—
423D4—22no—
424D4—22no—
425D4—22no—
426D4#44114yescomplemented
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—164no—
467C2×C4—164no—
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4#54811yescharacteristic
473C2×C4#55311yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
475C2×C4C2×#13711yescharacteristic
476C2×C4C2×#13711yescharacteristic
477C2×C4#57511yescharacteristic
478C2×C4#57611yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C8—22no—
483C8—22no—
484C8—42no—
485C8—41no—
486C8—41no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—81no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22#236111yescharacteristic
499C22#326411yescharacteristic
500C22#326611yescharacteristic
501C22#326611yescharacteristic
502C22#326811yescharacteristic
503C22#329011yescharacteristic
504C22#329311yescharacteristic
505C4—21no—
506C4—21no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—82no—
515C4—82no—
516C4—161no—
517C4—161no—
518C4C2×#44111yescharacteristic
519C4C4×AGL(1,ℤ/8)11yescharacteristic
520C4#294811yescharacteristic
521C4#295211yescharacteristic
522C4#334411yescharacteristic
523C4#334411yescharacteristic
524C4#336611yescharacteristic
525C4#337011yescharacteristic
526C2—44no—
527C2#2002511yescharacteristic
528C2#2003111yescharacteristic
529C2#2004311yescharacteristic
530C2#2010511yescharacteristic
531C2#2300811yescharacteristic
532C2#2302811yescharacteristic
533C2#2304511yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23187C2semidirect
6. #9835C2semidirect
7. #9619C2semidirect
8. #9573C2semidirect
9. #9565C2semidirect
12. #8925C2semidirect
13. #8923C2semidirect
14. #8289C2semidirect
56. #1669C4semidirect
81. #494C2×C4semidirect
154. #335C2×C4semidirect
155. #332C2×C4semidirect
156. #332C2×C4semidirect
426. D4#441semidirect

Maximal subgroups

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

subgroupindex
#231872
#229682
#229622
#227032
C2×#16692
#98352
#96192
#95732
#95652
#89812
#89252
#89232
#82892
#82042