Finite Group Databasebeta

← #229281

Subgroups — #229281

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23215C211yesmaximal, characteristic, complemented
2#23093C211yesmaximal, characteristic
3#23061C211yesmaximal, characteristic, complemented
4#22707C211yesmaximal, characteristic, complemented
5C2×#1677C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#9568C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic, complemented
10#8980C211yesmaximal, characteristic
11#8976C211yesmaximal, characteristic, complemented
12#8929C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic, complemented
14C8×#106C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20#2369C2211yescharacteristic
21#2366C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1982C2211yescharacteristic
27#1970—21no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1956C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1934—21no—
36#1934—21no—
37#1926—21no—
38#1926—21no—
39#1925—21no—
40#1920—21no—
41#1879—21no—
42#1864—21no—
43#1854—21no—
44#1854—21no—
45#1834C2211yescharacteristic
46#1808—21no—
47C4×#106—21no—
48C4×#106C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×#102—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55#1735—21no—
56#1677C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×D8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77#486—22no—
78#486C2×C414yes—
79#476—22no—
80#476C2×C414yescomplemented
81#459—22no—
82#459C2×C414yes—
83C8×D4—22no—
84C8×D4C2×C414yescomplemented
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106D411yescharacteristic
91C2×#106D411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—41no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—41no—
106C2×#102D411yescharacteristic
107C2×#102D411yescharacteristic
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—41no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#388—41no—
123#388—41no—
124#386—41no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
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#340—21no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#334C2×C414yescomplemented
154#333C2×C414yescomplemented
155#331C2×C414yes—
156D4⋊C8—22no—
157C82—22no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C2×D411yescharacteristic
163C23×C4—21no—
164C23×C4—41no—
165C2×D8—24no—
166C2×D8—81no—
167C2×D8—81no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—81no—
175C22×C8—81no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8C22×C412yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C22×C412yes—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—81no—
193C4×D4—81no—
194C4×D4C22×C412yes—
195C4×D4#4112yes—
196#117—42no—
197#117—42no—
198#117—81no—
199#117—81no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42C2411yescharacteristic
231#107—81no—
232#107—81no—
233#107—81no—
234#106—24no—
235#106—22no—
236#106—22no—
237#106—42no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246#103—82no—
247#102—24no—
248#102—24no—
249#102—24no—
250#102—22no—
251#102—24no—
252#102—22no—
253#102—42no—
254#102—42no—
255#102—81no—
256#102—81no—
257#102—82no—
258C22⋊C8—24no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264C4×C8—81no—
265C4×C8—81no—
266C4×C8C22×C412yes—
267C4×C8#4112yes—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—21no—
276C24—41no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—81no—
280C2×Q8—81no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4C4×D412yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—161no—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325D8—84no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—84no—
351C2×C8C4×D412yes—
352C2×C8C4×D412yes—
353C2×C8#13512yes—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—161no—
379C4⋊C4C4×D412yes—
380C4⋊C4#13512yes—
381C4⋊C4#13512yes—
382C4⋊C4#13512yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—161no—
390C22⋊C4—161no—
391C22⋊C4—161no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—82no—
399C42—82no—
400C42—161no—
401C42C23×C411yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C422−1+411yescharacteristic
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23D4211yescharacteristic
420Q8—82no—
421Q8—161no—
422D4—22no—
423D4—22no—
424D4—41no—
425D4—41no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—82no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—161no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
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—164no—
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4#52011yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4#54811yescharacteristic
478C2×C4#55311yescharacteristic
479C2×C4C2×#13511yescharacteristic
480C2×C4C2×#13511yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4#57611yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C8—22no—
488C8—22no—
489C8—22no—
490C8—42no—
491C8—41no—
492C8—42no—
493C8—42no—
494C8—42no—
495C8—81no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22#236111yescharacteristic
511C22#328011yescharacteristic
512C22#328211yescharacteristic
513C22#328211yescharacteristic
514C22#328411yescharacteristic
515C22#329011yescharacteristic
516C22#329211yescharacteristic
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—161no—
528C4—161no—
529C4C4×#13511yescharacteristic
530C4C4×#13511yescharacteristic
531C4#294611yescharacteristic
532C4#294911yescharacteristic
533C4#333711yescharacteristic
534C4#334011yescharacteristic
535C4#336011yescharacteristic
536C4#336111yescharacteristic
537C2—82no—
538C2—162no—
539C2#1998811yescharacteristic
540C2#1999411yescharacteristic
541C2#1999911yescharacteristic
542C2#2000311yescharacteristic
543C2#2299911yescharacteristic
544C2#2302311yescharacteristic
545C2#2304611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23215C2semidirect
3. #23061C2semidirect
4. #22707C2semidirect
6. #9822C2semidirect
7. #9613C2semidirect
8. #9568C2semidirect
9. #9563C2semidirect
11. #8976C2semidirect
12. #8929C2semidirect
13. #8921C2semidirect
14. C8×#106C2semidirect
15. C8×#102C2semidirect
56. #1677C4semidirect
80. #476C2×C4semidirect
84. C8×D4C2×C4semidirect
153. #334C2×C4semidirect
154. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232152
#230932
#230612
#227072
C2×#16772
#98222
#96132
#95682
#95632
#89802
#89762
#89292
#89212
C8×#1062
C8×#1022