Finite Group Databasebeta

← #229595

Subgroups — #229595

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23113C211yesmaximal, characteristic
2#22982C211yesmaximal, characteristic, complemented
3#19156C211yesmaximal, characteristic
4#18067C211yesmaximal, characteristic, complemented
5#17830C211yesmaximal, characteristic, complemented
6#9851C211yesmaximal, characteristic, complemented
7#9624C211yesmaximal, characteristic, complemented
8#9579C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic, complemented
10#8995C211yesmaximal, characteristic
11#8994C211yesmaximal, characteristic, complemented
12#8933C211yesmaximal, characteristic, complemented
13#8932C211yesmaximal, characteristic, complemented
14#8275C211yesmaximal, characteristic, complemented
15#8196C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20#2962C2211yescharacteristic
21#2935C2211yescharacteristic
22#2925C2211yescharacteristic
23#2906C2211yescharacteristic
24#1994C2211yescharacteristic
25#1974—21no—
26#1962C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1945—21no—
30#1929—21no—
31#1922—21no—
32#1876—21no—
33#1866—21no—
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36#1764—21no—
37#1764—21no—
38C4×#104C2211yescharacteristic
39C4×#97C2211yescharacteristic
40#1700C411yescharacteristic, complemented
41#1699C411yescharacteristic, complemented
42#1698C411yescharacteristic, complemented
43#1697C411yescharacteristic, complemented
44#1692C411yescharacteristic, complemented
45#1691C411yescharacteristic, complemented
46#1686C411yescharacteristic, complemented
47#1685C411yescharacteristic, complemented
48#1590—21no—
49#1583—21no—
50#1582—21no—
51C8×SD16—21no—
52#1574—21no—
53#1573—21no—
54#1571C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1567C2211yescharacteristic
58#1565—21no—
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61#1542C2211yescharacteristic
62#1540C2211yescharacteristic
63#1493C2211yescharacteristic
64#1492C2211yescharacteristic
65#1481C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68C8×M16C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77C2×#131—21no—
78#533C2311yescharacteristic
79#532C2311yescharacteristic
80C4×#41C2311yescharacteristic
81#498C2×C411yescharacteristic
82#492C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#486C2×C411yescharacteristic
85#479—21no—
86#478—21no—
87#477C2×C411yescharacteristic
88#476C2×C411yescharacteristic
89#475C2×C411yescharacteristic
90#474C2×C411yescharacteristic
91#465—21no—
92#465—21no—
93#464—21no—
94#463—21no—
95#462C2×C411yescharacteristic
96#461C2×C411yescharacteristic
97#460C2×C411yescharacteristic
98#459C2×C411yescharacteristic
99#446—21no—
100#446—41no—
101#446C2×C411yescharacteristic
102#446C2×C411yescharacteristic
103C8×Q8—21no—
104C4×SD16—21no—
105C4×SD16—21no—
106#436—21no—
107#436C2×C411yescharacteristic
108#436C2×C411yescharacteristic
109#435—21no—
110#435—21no—
111#435—41no—
112C8×D4—21no—
113#433—41no—
114#432—21no—
115#431—21no—
116#431—21no—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119C2×#108—21no—
120C2×#108—21no—
121#427—21no—
122#424—21no—
123#424—21no—
124#424—41no—
125#423—21no—
126#423C2311yescharacteristic
127#423C2311yescharacteristic
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×#104—21no—
134C2×#104—21no—
135C2×#104D411yescharacteristic
136C2×#104D411yescharacteristic
137#416—21no—
138#416—21no—
139#408—21no—
140#408—21no—
141#405—21no—
142#405—21no—
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C2×#97—21no—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C43C2311yescharacteristic
152#341—21no—
153#341—21no—
154#339—21no—
155#339—21no—
156#339D411yescharacteristic
157#339D411yescharacteristic
158#336—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335C2×C411yescharacteristic
165#335C2×C411yescharacteristic
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
168#334—21no—
169#334—21no—
170#334D411yescharacteristic
171#334D411yescharacteristic
172#334D411yescharacteristic
173#334D411yescharacteristic
174#333C2×C411yescharacteristic
175#333C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#332C2×C411yescharacteristic
180#331C2×C411yescharacteristic
181#331C2×C411yescharacteristic
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185Q8⋊C8—41no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—41no—
190#322—21no—
191C82—21no—
192C2×#41—21no—
193C2×#41C2×D411yescharacteristic
194#135—81no—
195#135—81no—
196C2×Q16—21no—
197C2×SD16—21no—
198C2×SD16—21no—
199C2×SD16—21no—
200C2×SD16—21no—
201C2×D8—21no—
202#131—42no—
203#131—42no—
204#131—81no—
205#131—81no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—41no—
211C2×M16—41no—
212C2×M16C2×D411yescharacteristic
213C2×M16C2×D411yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—41no—
220C22×C8—41no—
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226#127C22×C411yescharacteristic
227#125—21no—
228#124—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8C22×C411yescharacteristic
236C4×Q8#4111yescharacteristic
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C22×C411yescharacteristic
242C4×D4#4111yescharacteristic
243#117—21no—
244#117—21no—
245#117C2×D411yescharacteristic
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2411yescharacteristic
252#108—81no—
253#108—81no—
254#107—21no—
255#107—21no—
256#106—21no—
257#106—21no—
258#106—21no—
259#106—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8C22×C411yescharacteristic
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C2×D411yescharacteristic
282C4⋊C8C2×D411yescharacteristic
283C4⋊C8C2×D411yescharacteristic
284C4⋊C8C2×D411yescharacteristic
285#104—22no—
286#104—41no—
287#104—42no—
288#104—81no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#103—21no—
294#103—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309#97—24no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—41no—
314#97—42no—
315#97—41no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—41no—
322C4×C8—41no—
323C4×C8—81no—
324C4×C8—81no—
325C4×C8C22×C411yescharacteristic
326C4×C8C22×C411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329#41—21no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8C4×D411yescharacteristic
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C2×D4C4×D411yescharacteristic
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C42−1+411yescharacteristic
356Q16—81no—
357SD16—42no—
358SD16—81no—
359SD16—81no—
360SD16—81no—
361D8—81no—
362M16—21no—
363M16—21no—
364M16—21no—
365M16—42no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—81no—
371M16—82no—
372M16—82no—
373M16#13712yes—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—82no—
403C2×C8—82no—
404C2×C8C4×D411yescharacteristic
405C2×C8C4×D411yescharacteristic
406C2×C8C4×D411yescharacteristic
407C2×C8C4×D411yescharacteristic
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4C4×D411yescharacteristic
419C4⋊C4C4×D411yescharacteristic
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—41no—
436C42—81no—
437C42C23×C411yescharacteristic
438C42C22×D411yescharacteristic
439C42C2×#4111yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C23—21no—
442C23—41no—
443C23#54711yescharacteristic
444Q8—21no—
445Q8—21no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—81no—
451D4—21no—
452D4—21no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—81no—
460D4—81no—
461D4—81no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×D411yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4D4211yescharacteristic
488C2×C4#54811yescharacteristic
489C2×C4#55311yescharacteristic
490C2×C4C2×#13711yescharacteristic
491C2×C4#57511yescharacteristic
492C2×C4#57611yescharacteristic
493C2×C4#57711yescharacteristic
494C8—21no—
495C8—21no—
496C8—21no—
497C8—22no—
498C8—42no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—81no—
511C8—81no—
512C8—81no—
513C8#44312yes—
514C22—21no—
515C22—21no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22#236111yescharacteristic
523C22#335111yescharacteristic
524C22#335711yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4C4×#13711yescharacteristic
537C4#294611yescharacteristic
538C4C2×#44311yescharacteristic
539C4#295611yescharacteristic
540C4#326611yescharacteristic
541C4#326811yescharacteristic
542C2—21no—
543C2—21no—
544C2—81no—
545C2—161no—
546C2#2301511yescharacteristic
547C2#2303511yescharacteristic
548C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3035C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22982C2semidirect
4. #18067C2semidirect
5. #17830C2semidirect
6. #9851C2semidirect
7. #9624C2semidirect
8. #9579C2semidirect
9. #9577C2semidirect
11. #8994C2semidirect
12. #8933C2semidirect
13. #8932C2semidirect
14. #8275C2semidirect
15. #8196C2semidirect
40. #1700C4semidirect
41. #1699C4semidirect
42. #1698C4semidirect
43. #1697C4semidirect
44. #1692C4semidirect
45. #1691C4semidirect
46. #1686C4semidirect
47. #1685C4semidirect

Maximal subgroups

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

subgroupindex
#231132
#229822
#191562
#180672
#178302
#98512
#96242
#95792
#95772
#89952
#89942
#89332
#89322
#82752
#81962