Finite Group Databasebeta

← #203898

Subgroups — #203898

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21987C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21962C211yesmaximal, characteristic
4#20545C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9618C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9616C211yesmaximal, characteristic, complemented
12#9612C211yesmaximal, characteristic, complemented
13#9611C211yesmaximal, characteristic, complemented
14#8044C211yesmaximal, characteristic
15C2×#1296C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2283C2211yescharacteristic
23#1949—21no—
24#1949—21no—
25#1948—21no—
26#1948—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1853—21no—
36#1853—21no—
37#1853—21no—
38#1852—21no—
39#1847—21no—
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1836—21no—
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1808—21no—
46#1807—21no—
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1768—21no—
52#1768—21no—
53#1752—21no—
54C8×C22⋊C4—21no—
55#1573C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432C2×C414yes—
77#431C2×C414yes—
78#431Q814yes—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yes—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8Q814yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8D411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#384—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#383—21no—
123#382—21no—
124#382D411yescharacteristic
125#382D411yescharacteristic
126#381—21no—
127#380—21no—
128#380—21no—
129#380D411yescharacteristic
130#380D411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136#376—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—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yescomplemented
154#335C2×C414yescomplemented
155#334C2×C414yescomplemented
156#334C2×C414yescomplemented
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8—81no—
172C22×C8—81no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177#117—82no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—41no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—41no—
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q812yes—
212C2×C42C2×Q812yes—
213C2×C42#4112yes—
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—81no—
217#107—81no—
218#107—81no—
219#107—81no—
220#106—81no—
221#106—81no—
222#106—81no—
223#106—81no—
224#106—81no—
225#106—81no—
226#106—81no—
227#106—81no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8C4⋊C414yes—
237C4⋊C8C4⋊C414yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240C4⋊C8C22×C412yes—
241C4⋊C8C22×C412yes—
242C4⋊C8C2×Q812yes—
243C4⋊C8C2×Q812yes—
244C22⋊C8—24no—
245C22⋊C8C4⋊C414yes—
246C22⋊C8C4⋊C414yes—
247#97—24no—
248#97—81no—
249#97—81no—
250C4×C8—81no—
251C4×C8—81no—
252C4×C8C22×C412yes—
253C4×C8C22×C412yes—
254C4×C8C2×Q812yes—
255C4×C8C2×Q812yes—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95—41no—
266C24#12111yescharacteristic
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C2×C4⋊C412yes—
289C22×C4C4×D412yes—
290C22×C4C4×Q812yes—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22⋊Q812yes—
295C22×C4#12512yes—
296C22×C4C4⋊Q812yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—84no—
327C2×C8—84no—
328C2×C8C2×C4⋊C412yes—
329C2×C8C2×C4⋊C412yes—
330C2×C8C2×C4⋊C412yes—
331C2×C8C2×C4⋊C412yes—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—82no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—82no—
371C42C2×C4⋊C412yes—
372C42C4×D412yes—
373C42C4×Q812yes—
374C42C22⋊Q812yes—
375C42#12512yes—
376C42C4⋊Q812yes—
377C42C23×C411yescharacteristic
378C42C22×Q811yescharacteristic
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23#38912yes—
386C23#47112yes—
387C23#47112yes—
388C23C2×#12111yescharacteristic
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—164no—
432C2×C4#38912yes—
433C2×C4#38912yes—
434C2×C4#38912yes—
435C2×C4C22×C4⋊C411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C4×Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×#12511yescharacteristic
440C2×C4C2×C4⋊Q811yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57811yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—22no—
451C8—22no—
452C8—22no—
453C8—42no—
454C8—42no—
455C8—42no—
456C8—42no—
457C8—42no—
458C8—42no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22#195114yes—
466C22C2×#38911yescharacteristic
467C22#304511yescharacteristic
468C22#304611yescharacteristic
469C22#306211yescharacteristic
470C22#306311yescharacteristic
471C22C2×#47111yescharacteristic
472C22C2×#47111yescharacteristic
473C4—22no—
474C4—22no—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4#291211yescharacteristic
487C4#291311yescharacteristic
488C4#294611yescharacteristic
489C4#294711yescharacteristic
490C4#308111yescharacteristic
491C4#308211yescharacteristic
492C4#316311yescharacteristic
493C4#316411yescharacteristic
494C2—24no—
495C2#1958011yescharacteristic
496C2#1958111yescharacteristic
497C2#1958811yescharacteristic
498C2#1958911yescharacteristic
499C2#2116011yescharacteristic
500C2#2117511yescharacteristic
501C2C2×#195111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9622C2semidirect
7. #9622C2semidirect
8. #9618C2semidirect
9. #9617C2semidirect
10. #9616C2semidirect
11. #9616C2semidirect
12. #9612C2semidirect
13. #9611C2semidirect
55. #1573C4semidirect
153. #335C2×C4semidirect
154. #335C2×C4semidirect
155. #334C2×C4semidirect
156. #334C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219872
#219812
#219622
#205452
C2×#15732
#96222
#96182
#96172
#96162
#96122
#96112
#80442
C2×#12962