Finite Group Databasebeta

← #286609

Subgroups — #286609

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27960C211yesmaximal, characteristic
2#27681C211yesmaximal, characteristic
3#24660C211yesmaximal, characteristic
4#23055C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10623C211yesmaximal, characteristic, complemented
7#10590C211yesmaximal, characteristic, complemented
8#10478C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9566C211yesmaximal, characteristic, complemented
11#9226C211yesmaximal, characteristic, complemented
12#9138C211yesmaximal, characteristic
13#8834C211yesmaximal, characteristic
14#8657C211yesmaximal, characteristic, complemented
15#8040C211yesmaximal, characteristic
16C2×#462C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×C22⋊Q8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2100—21no—
24#2099—21no—
25#2099—21no—
26#2095—21no—
27#2035—21no—
28#2001—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1984C2211yescharacteristic
32#1981C2211yescharacteristic
33#1971—21no—
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1935—21no—
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1872—21no—
43#1852—21no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1823—21no—
47#1823—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770—21no—
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#462—22no—
77#462—41no—
78#446—22no—
79#446—41no—
80#432—22no—
81#432—22no—
82#432—41no—
83#432—41no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#391D411yescharacteristic
114#389—21no—
115#389—21no—
116#389D411yescharacteristic
117#387—21no—
118#385—21no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#380—21no—
126#380—21no—
127#380D411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
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#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—22no—
153#334—41no—
154#332—22no—
155#332—41no—
156C22×Q8—21no—
157C22×Q8C2×D411yescharacteristic
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C2×Q16—42no—
163C2×Q16—42no—
164C2×Q16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—81no—
182C22⋊Q8—81no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—81no—
227#107—81no—
228#106—81no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8#4112yes—
242C4⋊C8#4112yes—
243#103—42no—
244#103—42no—
245#103—81no—
246#103—81no—
247#103—81no—
248#103—81no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—42no—
255#97—42no—
256#97—42no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269C24#12111yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315Q16—84no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4#13712yes—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—82no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42H(5,𝔽2)11yescharacteristic
395C422−1+411yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#47112yes—
403C23C2×#12111yescharacteristic
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—81no—
409Q8—81no—
410Q8—81no—
411Q8—81no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—164no—
458C2×C4#52911yescharacteristic
459C2×C4#53911yescharacteristic
460C2×C4#54611yescharacteristic
461C2×C4#54611yescharacteristic
462C2×C4#54711yescharacteristic
463C2×C4#54711yescharacteristic
464C2×C4C2×#13711yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57511yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57611yescharacteristic
469C2×C4#57611yescharacteristic
470C2×C4#57811yescharacteristic
471C2×C4#57811yescharacteristic
472C8—41no—
473C8—41no—
474C8—41no—
475C8—42no—
476C8—42no—
477C8—41no—
478C8—81no—
479C8—81no—
480C8—81no—
481C8—81no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22#242511yescharacteristic
490C22#304011yescharacteristic
491C22#304411yescharacteristic
492C22#305611yescharacteristic
493C22#305811yescharacteristic
494C22C2×#47111yescharacteristic
495C22#306911yescharacteristic
496C4—21no—
497C4—21no—
498C4—42no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—161no—
506C4—161no—
507C4—161no—
508C4—161no—
509C4#331511yescharacteristic
510C4#332211yescharacteristic
511C4#332711yescharacteristic
512C4#333311yescharacteristic
513C4#334911yescharacteristic
514C4#335211yescharacteristic
515C4#335511yescharacteristic
516C4#335711yescharacteristic
517C2—42no—
518C2#1931711yescharacteristic
519C2#1936411yescharacteristic
520C2#1947511yescharacteristic
521C2#1953011yescharacteristic
522C2#2522911yescharacteristic
523C2#2525811yescharacteristic
524C2#2526811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
5. #10656C2semidirect
6. #10623C2semidirect
7. #10590C2semidirect
8. #10478C2semidirect
9. #9616C2semidirect
10. #9566C2semidirect
11. #9226C2semidirect
14. #8657C2semidirect

Maximal subgroups

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

subgroupindex
#279602
#276812
#246602
#230552
#106562
#106232
#105902
#104782
#96162
#95662
#92262
#91382
#88342
#86572
#80402