Finite Group Databasebeta

← #226867

Subgroups — #226867

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22829C211yesmaximal, characteristic, complemented
2#22796C211yesmaximal, characteristic
3#21279C211yesmaximal, characteristic
4#9666C211yesmaximal, characteristic
5#9661C211yesmaximal, characteristic
6#9547C211yesmaximal, characteristic, complemented
7#9546C211yesmaximal, characteristic
8#9544C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic
10#8808C211yesmaximal, characteristic, complemented
11#8569C211yesmaximal, characteristic
12#8568C211yesmaximal, characteristic, complemented
13#8291C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8263C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17C2×#433—21no—
18#2295C2211yescharacteristic
19#2293C2211yescharacteristic
20#1984C2211yescharacteristic
21#1984C2211yescharacteristic
22#1973C411yescharacteristic
23#1973C411yescharacteristic
24#1954C2211yescharacteristic
25#1954C2211yescharacteristic
26#1944C411yescharacteristic
27#1944C411yescharacteristic
28#1936C411yescharacteristic
29#1936C411yescharacteristic
30#1935C411yescharacteristic
31#1935C411yescharacteristic
32#1924—21no—
33#1923—21no—
34#1917—21no—
35#1916—21no—
36#1842C2211yescharacteristic
37#1840—21no—
38#1831C2211yescharacteristic
39#1798—21no—
40C4×#107C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1746—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C4×SD16—22no—
72C4×SD16—22no—
73#433—44no—
74C2×#107C2×C411yescharacteristic
75C2×#107C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#103C2×C411yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97C2×C411yescharacteristic
99C2×#97C2×C411yescharacteristic
100C2×#97C2×C411yescharacteristic
101C2×#97C2×C411yescharacteristic
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#388—21no—
119#388—21no—
120#387—21no—
121#387—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126#376—21no—
127C43C2311yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#335—41no—
149#334—22no—
150#334—41no—
151Q8⋊C8—41no—
152Q8⋊C8—41no—
153D4⋊C8—41no—
154D4⋊C8—41no—
155#322—22no—
156#322—22no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C2×SD16—22no—
161C2×SD16—41no—
162C2×SD16—41no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—41no—
214#107—41no—
215#106—22no—
216C4⋊C8—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—41no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229#103—22no—
230#103—22no—
231#103—41no—
232#102—22no—
233#102—22no—
234#102—41no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237C22⋊C8—81no—
238C22⋊C8—81no—
239#97—22no—
240#97—41no—
241#97—41no—
242#97—41no—
243#97—42no—
244#97—42no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8#4112yes—
258C4×C8#4112yes—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302SD16—44no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
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—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×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8#11712yes—
342C2×C8#11712yes—
343C2×C8#13112yes—
344C2×C8AGL(1,ℤ/8)12yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—81no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—81no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C422−1+411yescharacteristic
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×C4×D411yescharacteristic
397Q8—41no—
398Q8—41no—
399Q8—41no—
400Q8—41no—
401D4—41no—
402D4—41no—
403D4—41no—
404D4—41no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
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—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4C2×#11711yescharacteristic
445C2×C4#52011yescharacteristic
446C2×C4#54711yescharacteristic
447C2×C4#55011yescharacteristic
448C2×C4#55511yescharacteristic
449C2×C4#55611yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4#56811yescharacteristic
452C2×C4#56811yescharacteristic
453C2×C4#56811yescharacteristic
454C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
455C2×C4#57511yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57811yescharacteristic
458C8—22no—
459C8—22no—
460C8—22no—
461C8—22no—
462C8—42no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22#235411yescharacteristic
480C22#293211yescharacteristic
481C22#293511yescharacteristic
482C22C4×AGL(1,ℤ/8)11yescharacteristic
483C22#294711yescharacteristic
484C22#295511yescharacteristic
485C22#295611yescharacteristic
486C4—21no—
487C4—21no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4#292111yescharacteristic
500C4#292511yescharacteristic
501C4#299211yescharacteristic
502C4#299511yescharacteristic
503C4#334311yescharacteristic
504C4#334811yescharacteristic
505C4#339711yescharacteristic
506C4#340211yescharacteristic
507C2—161no—
508C2#1798511yescharacteristic
509C2#1798811yescharacteristic
510C2#1807611yescharacteristic
511C2#1808311yescharacteristic
512C2#2279811yescharacteristic
513C2#2280411yescharacteristic
514C2#2282011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×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
1. #22829C2semidirect
6. #9547C2semidirect
8. #9544C2semidirect
10. #8808C2semidirect
12. #8568C2semidirect
13. #8291C2semidirect
14. #8288C2semidirect
15. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#228292
#227962
#212792
#96662
#96612
#95472
#95462
#95442
#95412
#88082
#85692
#85682
#82912
#82882
#82632