Finite Group Databasebeta

← #230051

Subgroups — #230051

Order 512.

573 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#473C211yesmaximal, characteristic
2#17894C211yesmaximal, characteristic
3#17880C211yesmaximal, characteristic, complemented
4#17874C211yesmaximal, characteristic
5#17799C211yesmaximal, characteristic
6#9662C211yesmaximal, characteristic, complemented
7#9610C211yesmaximal, characteristic, complemented
8#9551C211yesmaximal, characteristic, complemented
9#9549C211yesmaximal, characteristic, complemented
10#8955C211yesmaximal, characteristic
11#8955C211yesmaximal, characteristic
12#8838C211yesmaximal, characteristic, complemented
13#8836C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8186C211yesmaximal, characteristic
16C2×#473C411yescharacteristic
17#2989C2211yescharacteristic
18#2989C2211yescharacteristic
19#2983C2211yescharacteristic
20#2982C2211yescharacteristic
21C4×#137C2211yescharacteristic
22C4×#137C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1979C411yescharacteristic
28#1952C411yescharacteristic
29#1930C411yescharacteristic
30#1930C411yescharacteristic
31#1908C411yescharacteristic
32#1908C411yescharacteristic
33#1816C411yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#101C2211yescharacteristic
36C4×#101C2211yescharacteristic
37#1593—21no—
38#1592—21no—
39#1591—21no—
40#1590—21no—
41#1588—21no—
42#1587—21no—
43#1587—21no—
44#1587—21no—
45#1584—21no—
46#1584—21no—
47#1584—21no—
48#1583—21no—
49#1569—21no—
50#1569—21no—
51#1568—21no—
52#1567—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58C2×#335C2211yescharacteristic
59#1479C2211yescharacteristic
60#1479C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×Q8⋊C8C2211yescharacteristic
65#1452C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1278C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#137C2×C411yescharacteristic
76C2×#137C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C4×#41C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83#473—24no—
84#446—21no—
85#446—21no—
86#446—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92C8×Q8—21no—
93C8×Q8—21no—
94#441—21no—
95#441—21no—
96#440—21no—
97#440—21no—
98C4×Q16—21no—
99C4×Q16—21no—
100C4×Q16—21no—
101C4×Q16—21no—
102C4×Q16—21no—
103C4×Q16—21no—
104C4×SD16—21no—
105C4×SD16—21no—
106#436—21no—
107#436—21no—
108#435—41no—
109#435—41no—
110#432—21no—
111#432—21no—
112#432—41no—
113#431—41no—
114#431C2311yescharacteristic
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118C2×#108C2×C411yescharacteristic
119C2×#108C2×C411yescharacteristic
120#428C2×C411yescharacteristic
121#428C2×C411yescharacteristic
122C2×#107C2×C411yescharacteristic
123#423—21no—
124#423—21no—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127#419C2×C411yescharacteristic
128#419C2×C411yescharacteristic
129#417C2×C411yescharacteristic
130#417C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#101C2×C411yescharacteristic
134C2×#101C2×C411yescharacteristic
135C2×#101C2×C411yescharacteristic
136C2×#101C2×C411yescharacteristic
137#405—21no—
138#405—21no—
139#405C2×C411yescharacteristic
140#405C2×C411yescharacteristic
141C4×M16—21no—
142C4×M16C2311yescharacteristic
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C2×C4×C8—21no—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#343C2×C411yescharacteristic
150#343C2×C411yescharacteristic
151#343C2×C411yescharacteristic
152#343C2×C411yescharacteristic
153#341C2×C411yescharacteristic
154#341C2×C411yescharacteristic
155#336—41no—
156#336—41no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335D411yescharacteristic
164#335D411yescharacteristic
165#334—21no—
166#334—21no—
167#334—21no—
168#334—21no—
169#334D411yescharacteristic
170#334D411yescharacteristic
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8D411yescharacteristic
181Q8⋊C8D411yescharacteristic
182D4⋊C8—21no—
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322—21no—
192C82—21no—
193C2×#41C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195C22×Q8C22×C411yescharacteristic
196#137—22no—
197#137—22no—
198#137—41no—
199#137—41no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—22no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—41no—
206C2×SD16—21no—
207C2×SD16—21no—
208#131—22no—
209#131—41no—
210#131—41no—
211#131—41no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16C22×C411yescharacteristic
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8C2×D411yescharacteristic
239C4×Q8#4111yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C2×D411yescharacteristic
243C4×D4#4111yescharacteristic
244#117—21no—
245#117C22×C411yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C2411yescharacteristic
254#108—22no—
255#108—41no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107—21no—
260#106—21no—
261#106—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275C4⋊C8C2×D411yescharacteristic
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#102—21no—
290#102—21no—
291#101—22no—
292#101—41no—
293#101—41no—
294#101M1614yescomplemented
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301#97—21no—
302#97—21no—
303#97—21no—
304#97—21no—
305#97—21no—
306#97—41no—
307#97—41no—
308#97—41no—
309#97—41no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—41no—
323C4×C8—41no—
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8#4111yescharacteristic
329C4×C8#4111yescharacteristic
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8C4×D411yescharacteristic
346C2×Q8C2×M1612yes—
347C2×D4—21no—
348C2×D4C4×D411yescharacteristic
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4C23×C411yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C42−1+411yescharacteristic
359Q16—42no—
360Q16—42no—
361Q16—42no—
362Q16—42no—
363SD16—42no—
364SD16—42no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—42no—
376M16—42no—
377M16—161no—
378M16—161no—
379M16C2×M1612yes—
380M16C2×M1612yes—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—81no—
408C2×C8—81no—
409C2×C8—81no—
410C2×C8C4×D411yescharacteristic
411C2×C8C4×D411yescharacteristic
412C2×C8C4×D411yescharacteristic
413C2×C8C4×D411yescharacteristic
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4C4×D411yescharacteristic
425C4⋊C4C4×D411yescharacteristic
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42C22×D411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C23—21no—
441C23#52011yescharacteristic
442Q8—21no—
443Q8—22no—
444Q8—21no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451Q8—41no—
452D4—21no—
453D4—21no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4C2×C4×D411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4D4211yescharacteristic
480C2×C4#54711yescharacteristic
481C2×C4#54811yescharacteristic
482C2×C4#55311yescharacteristic
483C2×C4C22×M1611yescharacteristic
484C2×C4#56811yescharacteristic
485C2×C4#56811yescharacteristic
486C2×C4#56811yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8—21no—
491C8—22no—
492C8—22no—
493C8—42no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—81no—
507C8—81no—
508C8—81no—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22#236111yescharacteristic
515C22#298911yescharacteristic
516C22#299511yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4#293011yescharacteristic
528C4#293211yescharacteristic
529C4D4×M1611yescharacteristic
530C4D4×M1611yescharacteristic
531C4#329011yescharacteristic
532C4#329311yescharacteristic
533C2—21no—
534C2—21no—
535C2—81no—
536C2#2298811yescharacteristic
537C2#2304311yescharacteristic
538C2#2304511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC2×#41C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
3. #17880C2semidirect
6. #9662C2semidirect
7. #9610C2semidirect
8. #9551C2semidirect
9. #9549C2semidirect
12. #8838C2semidirect
13. #8836C2semidirect
14. #8296C2semidirect
294. #101M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4732
#178942
#178802
#178742
#177992
#96622
#96102
#95512
#95492
#89552
#88382
#88362
#82962
#81862