Finite Group Databasebeta

← #275785

Subgroups — #275785

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29925C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#22640C211yesmaximal, characteristic, complemented
4#21987C211yesmaximal, characteristic, complemented
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10599C211yesmaximal, characteristic, complemented
8#10595C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#9571C211yesmaximal, characteristic, complemented
12#9563C211yesmaximal, characteristic, complemented
13#9142C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic, complemented
15#8723C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17#2856C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2283C2211yescharacteristic
20#2092—21no—
21#2073—21no—
22#2071—21no—
23#2071—21no—
24#2034—21no—
25#2033—21no—
26#1982C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1923—21no—
40#1874—21no—
41#1874—21no—
42#1853—21no—
43#1852—21no—
44#1847—21no—
45#1840—21no—
46#1836—21no—
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1768—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#390—21no—
110#390—21no—
111#390—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#383—21no—
119#382D411yescharacteristic
120#382D411yescharacteristic
121#381—21no—
122#380D411yescharacteristic
123#380D411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127#376—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#334—22no—
148#331—22no—
149#331—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168#127—21no—
169#127—21no—
170#127—21no—
171#127—21no—
172#121—42no—
173#121—42no—
174#121—81no—
175#121—81no—
176#121—82no—
177C2×C4⋊C4—21no—
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—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4—81no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—81no—
214#107—81no—
215#107—81no—
216#106—81no—
217#106—81no—
218#106—81no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8#4112yes—
232C4⋊C8#4112yes—
233#102—81no—
234#102—81no—
235#102—81no—
236#102—81no—
237#102—81no—
238#102—81no—
239#102—81no—
240#102—81no—
241C22⋊C8—24no—
242C22⋊C8—41no—
243C22⋊C8—41no—
244C22⋊C8—81no—
245C22⋊C8—81no—
246#97—24no—
247#97—81no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—81no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—41no—
262C24#12111yescharacteristic
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—82no—
282C22×C4—22no—
283C22×C4—21no—
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—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
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—41no—
329C2×C8—41no—
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—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
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—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C22⋊C4—42no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—81no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C422−1+411yescharacteristic
388C422−1+411yescharacteristic
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23#47212yes—
403C23C2×#12111yescharacteristic
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4—81no—
411D4—81no—
412D4—161no—
413D4—161no—
414D4—164no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—42no—
427C2×C4—42no—
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—41no—
446C2×C4—41no—
447C2×C4—41no—
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#52511yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#54711yescharacteristic
461C2×C4#54711yescharacteristic
462C2×C4#55211yescharacteristic
463C2×C4#55211yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57611yescharacteristic
467C2×C4#57611yescharacteristic
468C2×C4#57711yescharacteristic
469C2×C4#57711yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—42no—
477C8—42no—
478C8—81no—
479C8—81no—
480C8—81no—
481C8—81no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—161no—
490C22—161no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22#241311yescharacteristic
497C22#304011yescharacteristic
498C22#304511yescharacteristic
499C22#305711yescharacteristic
500C22#306211yescharacteristic
501C22C2×#47211yescharacteristic
502C22#306911yescharacteristic
503C4—21no—
504C4—21no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—161no—
513C4—161no—
514C4—161no—
515C4#311911yescharacteristic
516C4#312511yescharacteristic
517C4#324111yescharacteristic
518C4#324411yescharacteristic
519C4#335011yescharacteristic
520C4#335011yescharacteristic
521C4#335511yescharacteristic
522C4#335511yescharacteristic
523C2—42no—
524C2—321no—
525C2#1936211yescharacteristic
526C2#1938311yescharacteristic
527C2#1953611yescharacteristic
528C2#1958211yescharacteristic
529C2#2459711yescharacteristic
530C2#2461811yescharacteristic
531C2#2462011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
3. #22640C2semidirect
4. #21987C2semidirect
5. #10658C2semidirect
6. #10657C2semidirect
7. #10599C2semidirect
8. #10595C2semidirect
9. #9619C2semidirect
10. #9613C2semidirect
11. #9571C2semidirect
12. #9563C2semidirect
14. #8834C2semidirect
15. #8723C2semidirect

Maximal subgroups

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

subgroupindex
#299252
#279732
#226402
#219872
#106582
#106572
#105992
#105952
#96192
#96132
#95712
#95632
#91422
#88342
#87232