Finite Group Databasebeta

← #287192

Subgroups — #287192

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29354C211yesmaximal, characteristic
2#29290C211yesmaximal, characteristic, complemented
3#28655C211yesmaximal, characteristic
4#23151C211yesmaximal, characteristic, complemented
5#10589C211yesmaximal, characteristic, complemented
6#10587C211yesmaximal, characteristic
7#10543C211yesmaximal, characteristic, complemented
8#10539C211yesmaximal, characteristic
9#9739C211yesmaximal, characteristic, complemented
10#9735C211yesmaximal, characteristic
11#9707C211yesmaximal, characteristic
12#9605C211yesmaximal, characteristic, complemented
13#9571C211yesmaximal, characteristic
14#9568C211yesmaximal, characteristic, complemented
15#8232C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#486C2211yescharacteristic
18C2×#446—21no—
19#2858C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22#2087—21no—
23#2087—21no—
24#2071—21no—
25#2067—21no—
26#2067—21no—
27#2023—21no—
28#1990C2211yescharacteristic
29#1987—21no—
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1987C2211yescharacteristic
34#1985—21no—
35#1982C2211yescharacteristic
36#1982C2211yescharacteristic
37#1957C2211yescharacteristic
38#1947—21no—
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1936—21no—
44#1936—21no—
45#1925—21no—
46#1925—21no—
47#1917—21no—
48#1916—21no—
49#1842C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#124—21no—
72C2×#124—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75#500—22no—
76#500—41no—
77#486—22no—
78#486—41no—
79#446—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389D411yescharacteristic
120#389D411yescharacteristic
121#384D411yescharacteristic
122#384C2311yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C43C2311yescharacteristic
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#340—21no—
139#340—21no—
140#340—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336D411yescharacteristic
148#336D411yescharacteristic
149#334—41no—
150#334—41no—
151#333—22no—
152#333—22no—
153#333—22no—
154#333—41no—
155#331—22no—
156#331—41no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C4⋊Q8—22no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—41no—
182C4⋊Q8#4112yes—
183#127—21no—
184#127—21no—
185#127—21no—
186#127—21no—
187#124—82no—
188#124—82no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—82no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#107—42no—
225#107—81no—
226#107—81no—
227C4⋊C8—22no—
228C4⋊C8—21no—
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—41no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—81no—
244C4⋊C8#4112yes—
245#102—42no—
246#102—42no—
247#102—42no—
248#102—42no—
249#102—42no—
250#102—42no—
251#102—81no—
252#102—81no—
253#97—42no—
254#97—42no—
255#97—81no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—42no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267C24—41no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—81no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
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×C8AGL(1,ℤ/8)12yes—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
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—81no—
364C4⋊C4—82no—
365C4⋊C4#13712yes—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—22no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—82no—
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C422−1+411yescharacteristic
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23C2×#12111yescharacteristic
397Q8—82no—
398Q8—82no—
399Q8—84no—
400Q8—161no—
401Q8—161no—
402D4—81no—
403D4—81no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4—161no—
411D4—161no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
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—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4#47212yes—
447C2×C4#52911yescharacteristic
448C2×C4#54211yescharacteristic
449C2×C4#54611yescharacteristic
450C2×C4#54811yescharacteristic
451C2×C4#54811yescharacteristic
452C2×C4#55011yescharacteristic
453C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
454C2×C4C2×#13711yescharacteristic
455C2×C4#57511yescharacteristic
456C2×C4#57511yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57711yescharacteristic
459C2×C4#57811yescharacteristic
460C2×C4#57811yescharacteristic
461C8—22no—
462C8—42no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22#242611yescharacteristic
479C22#303611yescharacteristic
480C22#304411yescharacteristic
481C22#305611yescharacteristic
482C22#306011yescharacteristic
483C22C2×#47211yescharacteristic
484C22#306811yescharacteristic
485C4—22no—
486C4—21no—
487C4—21no—
488C4—41no—
489C4—41no—
490C4—42no—
491C4—41no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—82no—
496C4—161no—
497C4—161no—
498C4#330811yescharacteristic
499C4#331311yescharacteristic
500C4#336711yescharacteristic
501C4#336911yescharacteristic
502C4#337111yescharacteristic
503C4#337711yescharacteristic
504C4#340011yescharacteristic
505C4#340111yescharacteristic
506C2—321no—
507C2#1930511yescharacteristic
508C2#1933711yescharacteristic
509C2#1947711yescharacteristic
510C2#1953211yescharacteristic
511C2#2528611yescharacteristic
512C2#2531111yescharacteristic
513C2#2531211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29290C2semidirect
4. #23151C2semidirect
5. #10589C2semidirect
7. #10543C2semidirect
9. #9739C2semidirect
12. #9605C2semidirect
14. #9568C2semidirect
15. #8232C2semidirect

Maximal subgroups

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

subgroupindex
#293542
#292902
#286552
#231512
#105892
#105872
#105432
#105392
#97392
#97352
#97072
#96052
#95712
#95682
#82322