Finite Group Databasebeta

← #232047

Subgroups — #232047

Order 512.

651 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23819C211yesmaximal, characteristic
2#23258C211yesmaximal, characteristic, complemented
3#23173C211yesmaximal, characteristic
4#9866C211yesmaximal, characteristic, complemented
5#9678C211yesmaximal, characteristic
6#9560C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9545C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic, complemented
11#8845C211yesmaximal, characteristic, complemented
12#8279C211yesmaximal, characteristic, complemented
13#8264C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16C2×C8×D4—21no—
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C4×#127C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2100C411yescharacteristic
22#2100C411yescharacteristic
23#2073C411yescharacteristic
24#2073C411yescharacteristic
25#2041C411yescharacteristic
26#2041C411yescharacteristic
27#2025C411yescharacteristic
28#2025C411yescharacteristic
29#1954—21no—
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1940—21no—
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1917—21no—
36#1916—21no—
37#1869—21no—
38#1869—21no—
39#1770—21no—
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×#127C2×C411yescharacteristic
70C2×#125C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C8×D4—22no—
76C8×D4—22no—
77C8×D4—41no—
78C8×D4—41no—
79C8×D4—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390—21no—
114#390—21no—
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#387—21no—
120#387—21no—
121#384C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C43C2311yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—22no—
150#334—22no—
151D4⋊C8—22no—
152D4⋊C8—22no—
153D4⋊C8—41no—
154D4⋊C8—41no—
155#322—41no—
156C82—41no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—81no—
174C22×C8—81no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183#127—22no—
184#127—22no—
185#125—22no—
186#125—41no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—81no—
198C4×D4—81no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—22no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—41no—
235C4⋊C8—42no—
236C4⋊C8—81no—
237C4⋊C8#4112yes—
238C4⋊C8#4112yes—
239#102—22no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—22no—
244#102—22no—
245#102—41no—
246#102—41no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—81no—
252C22⋊C8—81no—
253#97—81no—
254#97—81no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—42no—
268C4×C8—42no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275C24—21no—
276C24—21no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4#13112yes—
294C2×D4#13112yes—
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—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—84no—
362C2×C8—84no—
363C2×C8AGL(1,ℤ/8)12yes—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4#13112yes—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—81no—
389C22⋊C4—81no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—22no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C42H(5,𝔽2)11yescharacteristic
407C422−1+411yescharacteristic
408C23—22no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×C4×D411yescharacteristic
423D4—22no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4—48no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—84no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
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—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4#44312yes—
473C2×C4C4×#4111yescharacteristic
474C2×C4#52011yescharacteristic
475C2×C4#54611yescharacteristic
476C2×C4#54811yescharacteristic
477C2×C4#55611yescharacteristic
478C2×C4#55911yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4C2×#13111yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4#56811yescharacteristic
483C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57711yescharacteristic
487C8—22no—
488C8—42no—
489C8—42no—
490C8—42no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—42no—
499C8—81no—
500C8—81no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22#236711yescharacteristic
516C22#292911yescharacteristic
517C22#293311yescharacteristic
518C22C4×AGL(1,ℤ/8)11yescharacteristic
519C22#294611yescharacteristic
520C22C2×#44311yescharacteristic
521C22#295711yescharacteristic
522C4—21no—
523C4—21no—
524C4—22no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4#296611yescharacteristic
535C4#296811yescharacteristic
536C4#299211yescharacteristic
537C4#299411yescharacteristic
538C4#330911yescharacteristic
539C4#331011yescharacteristic
540C4#336611yescharacteristic
541C4#337111yescharacteristic
542C2—82no—
543C2—82no—
544C2#1791011yescharacteristic
545C2#1791811yescharacteristic
546C2#1800711yescharacteristic
547C2#1801811yescharacteristic
548C2#2321111yescharacteristic
549C2#2323111yescharacteristic
550C2#2323411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23258C2semidirect
4. #9866C2semidirect
6. #9560C2semidirect
7. #9554C2semidirect
8. #9545C2semidirect
9. #9541C2semidirect
10. #8845C2semidirect
11. #8845C2semidirect
12. #8279C2semidirect
13. #8264C2semidirect
14. #8204C2semidirect
15. C8×#102C2semidirect

Maximal subgroups

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

subgroupindex
#238192
#232582
#231732
#98662
#96782
#95602
#95542
#95452
#95412
#88452
#82792
#82642
#82042
C8×#1022