Finite Group Databasebeta

← #230083

Subgroups — #230083

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23717C211yesmaximal, characteristic
2#17897C211yesmaximal, characteristic, complemented
3#17894C211yesmaximal, characteristic
4#17876C211yesmaximal, characteristic
5#17812C211yesmaximal, characteristic
6#9662C211yesmaximal, characteristic, complemented
7#9608C211yesmaximal, characteristic, complemented
8#9551C211yesmaximal, characteristic, complemented
9#9549C211yesmaximal, characteristic, complemented
10#8955C211yesmaximal, characteristic
11#8953C211yesmaximal, characteristic
12#8838C211yesmaximal, characteristic, complemented
13#8838C211yesmaximal, characteristic, complemented
14#8296C211yesmaximal, characteristic, complemented
15#8189C211yesmaximal, characteristic
16#2989C2211yescharacteristic
17#2988C2211yescharacteristic
18#2983C2211yescharacteristic
19#2983C2211yescharacteristic
20C4×#137C2211yescharacteristic
21C4×#137C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23#2922C2211yescharacteristic
24#2089C411yescharacteristic
25#2078C411yescharacteristic
26#2064C411yescharacteristic
27#2064C411yescharacteristic
28#2052C411yescharacteristic
29#2052C411yescharacteristic
30#2047C411yescharacteristic
31#2020C411yescharacteristic
32#1984C2211yescharacteristic
33#1984C2211yescharacteristic
34C4×#108C2211yescharacteristic
35C4×#101C2211yescharacteristic
36C4×#101C2211yescharacteristic
37#1593—21no—
38#1593—21no—
39#1593—21no—
40#1592—21no—
41#1591—21no—
42#1591—21no—
43#1591—21no—
44#1590—21no—
45#1587—21no—
46#1585—21no—
47#1584—21no—
48#1582—21no—
49#1574—21no—
50#1573—21no—
51#1572—21no—
52#1571—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#1451C2211yescharacteristic
66#1450C2211yescharacteristic
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
78C4×#41C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81#446—21no—
82#446—21no—
83#446—21no—
84#446—21no—
85#446—21no—
86#446—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90C8×Q8—21no—
91#441—21no—
92#441—21no—
93#440—21no—
94#440—21no—
95C4×Q16—21no—
96C4×Q16—21no—
97C4×Q16—21no—
98C4×Q16—21no—
99C4×Q16—21no—
100C4×Q16—21no—
101C4×SD16—21no—
102C4×SD16—21no—
103#436—21no—
104#435—41no—
105C8×D4—21no—
106C8×D4—41no—
107#433—21no—
108#433—21no—
109#432—21no—
110#432—41no—
111#432—41no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119C2×#108C2×C411yescharacteristic
120C2×#108C2×C411yescharacteristic
121#428C2×C411yescharacteristic
122#428C2×C411yescharacteristic
123C2×#107C2×C411yescharacteristic
124#424C2×C411yescharacteristic
125#424C2×C411yescharacteristic
126#423—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130#419C2×C411yescharacteristic
131#419C2×C411yescharacteristic
132#417C2×C411yescharacteristic
133#417C2×C411yescharacteristic
134C2×#103C2×C411yescharacteristic
135C2×#103C2×C411yescharacteristic
136C2×#101C2×C411yescharacteristic
137C2×#101C2×C411yescharacteristic
138C2×#101C2×C411yescharacteristic
139C2×#101C2×C411yescharacteristic
140#408C2×C411yescharacteristic
141#408C2×C411yescharacteristic
142C4×M16C2311yescharacteristic
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#343C2×C411yescharacteristic
149#343C2×C411yescharacteristic
150#343C2×C411yescharacteristic
151#343C2×C411yescharacteristic
152#341C2×C411yescharacteristic
153#341C2×C411yescharacteristic
154#336—41no—
155#336—41no—
156#335—21no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335D411yescharacteristic
163#335D411yescharacteristic
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334D411yescharacteristic
169#334D411yescharacteristic
170Q8⋊C8—21no—
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⋊C8D411yescharacteristic
180Q8⋊C8D411yescharacteristic
181D4⋊C8—21no—
182D4⋊C8D411yescharacteristic
183D4⋊C8D411yescharacteristic
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C2×#41C22×C411yescharacteristic
193C22×Q8C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195#137—22no—
196#137—41no—
197#137—41no—
198#137—41no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—41no—
204C2×Q16—41no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×M16—41no—
208C2×M16C22×C411yescharacteristic
209C2×M16C22×C411yescharacteristic
210C2×M16C22×C411yescharacteristic
211C2×M16C22×C411yescharacteristic
212C2×M16C22×C411yescharacteristic
213C2×M16C22×C411yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—41no—
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C2×D411yescharacteristic
230C4×Q8#4111yescharacteristic
231C4×D4—21no—
232C4×D4—21no—
233C4×D4C2×D411yescharacteristic
234C4×D4#4111yescharacteristic
235#117—21no—
236#117C22×C411yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42C2411yescharacteristic
245#108—22no—
246#108—41no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#106—21no—
252#106—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C4⋊C8C2×D411yescharacteristic
269C4⋊C8C2×D411yescharacteristic
270C4⋊C8C2×D411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274#103—21no—
275#103—21no—
276#103—21no—
277#103—21no—
278#103—21no—
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#102—21no—
285#102—21no—
286#101—22no—
287#101—41no—
288#101—41no—
289#101—41no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97—41no—
304#97—41no—
305#97—41no—
306#97—41no—
307#97—41no—
308C4×C8—21no—
309C4×C8—21no—
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—41no—
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8#4111yescharacteristic
324C4×C8#4111yescharacteristic
325#41—21no—
326#41—21no—
327#41—21no—
328#41—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8C4×D411yescharacteristic
342C2×D4—21no—
343C2×D4C4×D411yescharacteristic
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4C23×C411yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C42−1+411yescharacteristic
354Q16—42no—
355Q16—42no—
356Q16—42no—
357Q16—42no—
358SD16—42no—
359SD16—42no—
360M16—21no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—21no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—161no—
372M16#13112yes—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
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—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—81no—
398C2×C8—81no—
399C2×C8—81no—
400C2×C8—161no—
401C2×C8C4×D411yescharacteristic
402C2×C8C4×D411yescharacteristic
403C2×C8C4×D411yescharacteristic
404C2×C8C4×D411yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4C4×D411yescharacteristic
416C4⋊C4C4×D411yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42C22×D411yescharacteristic
428C42C22×D411yescharacteristic
429C42C2×#4111yescharacteristic
430C42C2×#4111yescharacteristic
431C23—21no—
432C23#52011yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—41no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443D4—21no—
444D4—21no—
445D4—41no—
446D4—41no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
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—41no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4D4211yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4#54811yescharacteristic
473C2×C4#55311yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4#56811yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4#56811yescharacteristic
478C8—21no—
479C8—21no—
480C8—22no—
481C8—21no—
482C8—21no—
483C8—42no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—21no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22#236111yescharacteristic
506C22#299311yescharacteristic
507C22#299511yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4#293011yescharacteristic
519C4#293011yescharacteristic
520C4#293311yescharacteristic
521C4#293511yescharacteristic
522C4#329011yescharacteristic
523C4#329311yescharacteristic
524C2—21no—
525C2—21no—
526C2—81no—
527C2#2299411yescharacteristic
528C2#2304311yescharacteristic
529C2#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
2. #17897C2semidirect
6. #9662C2semidirect
7. #9608C2semidirect
8. #9551C2semidirect
9. #9549C2semidirect
12. #8838C2semidirect
13. #8838C2semidirect
14. #8296C2semidirect

Maximal subgroups

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

subgroupindex
#237172
#178972
#178942
#178762
#178122
#96622
#96082
#95512
#95492
#89552
#89532
#88382
#82962
#81892