Finite Group Databasebeta

← #226206

Subgroups — #226206

Order 512.

523 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18057C211yesmaximal, characteristic
2#18055C211yesmaximal, characteristic
3#17863C211yesmaximal, characteristic
4#17839C211yesmaximal, characteristic, complemented
5#17834C211yesmaximal, characteristic, complemented
6#9634C211yesmaximal, characteristic, complemented
7#9633C211yesmaximal, characteristic, complemented
8#8900C211yesmaximal, characteristic, complemented
9#8900C211yesmaximal, characteristic, complemented
10#8899C211yesmaximal, characteristic, complemented
11#8898C211yesmaximal, characteristic, complemented
12#8828C211yesmaximal, characteristic, complemented
13#8672C211yesmaximal, characteristic, complemented
14#8671C211yesmaximal, characteristic, complemented
15#8294C211yesmaximal, characteristic, complemented
16#2991C2211yescharacteristic
17#2990C2211yescharacteristic
18#2990C2211yescharacteristic
19#2990C2211yescharacteristic
20#2965C2211yescharacteristic
21C8×#41C2211yescharacteristic
22C4×#135C2211yescharacteristic
23#1848C2211yescharacteristic
24#1837C2211yescharacteristic
25C4×#108C2211yescharacteristic
26#1591—21no—
27#1590—21no—
28#1587—21no—
29#1586—21no—
30#1585—21no—
31#1585—21no—
32#1584C411yescharacteristic
33#1584C411yescharacteristic
34#1583C411yescharacteristic
35#1583C411yescharacteristic
36#1582C411yescharacteristic
37#1582C411yescharacteristic
38#1581C411yescharacteristic
39#1581C411yescharacteristic
40C8×SD16—21no—
41C8×D8—21no—
42#1574C2211yescharacteristic
43#1574C2211yescharacteristic
44#1574C2211yescharacteristic
45#1574C2211yescharacteristic
46#1573—21no—
47#1573—21no—
48#1572—21no—
49#1571—21no—
50#1569—21no—
51#1568—21no—
52#1567C2211yescharacteristic
53#1567C2211yescharacteristic
54#1566C2211yescharacteristic
55#1566C2211yescharacteristic
56#1565—21no—
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#334—21no—
60#1489C2211yescharacteristic
61#1489C2211yescharacteristic
62#1489C2211yescharacteristic
63#1489C2211yescharacteristic
64#1476C2211yescharacteristic
65#1476C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#135Q811yescharacteristic
76C4×#41C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#446—21no—
79#446—21no—
80#446—21no—
81#446C2×C411yescharacteristic
82#446C2×C411yescharacteristic
83C8×Q8—21no—
84C8×Q8C2×C411yescharacteristic
85C8×Q8C2×C411yescharacteristic
86C4×Q16C2×C411yescharacteristic
87C4×Q16D411yescharacteristic
88C4×SD16C2×C411yescharacteristic
89C4×SD16C2×C411yescharacteristic
90C4×SD16D411yescharacteristic
91C4×SD16D411yescharacteristic
92C4×D8C2×C411yescharacteristic
93C4×D8D411yescharacteristic
94#436—21no—
95#436—41no—
96#436C2×C411yescharacteristic
97#436C2×C411yescharacteristic
98#435—21no—
99#435—21no—
100#435—41no—
101#435—41no—
102#435—41no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4—21no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4C2×C411yescharacteristic
109C8×D4C2×C411yescharacteristic
110#432—21no—
111#432—21no—
112#432—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120C2×#108—21no—
121C2×#108—41no—
122#427Q811yescharacteristic
123#423—21no—
124#423C2311yescharacteristic
125#423C2311yescharacteristic
126#423C2311yescharacteristic
127#423C2311yescharacteristic
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131#416Q811yescharacteristic
132#416Q811yescharacteristic
133C4×M16—21no—
134C4×M16C2311yescharacteristic
135C4×M16C2311yescharacteristic
136C2×C4×C8—21no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#341—21no—
141#341—41no—
142#335—21no—
143#335—21no—
144#335—21no—
145#335—21no—
146#335—21no—
147#335—21no—
148#335—21no—
149#335—21no—
150#335C2×C411yescharacteristic
151#335C2×C411yescharacteristic
152#335D411yescharacteristic
153#335D411yescharacteristic
154#334—21no—
155#334—21no—
156#334—21no—
157#334—21no—
158#334—21no—
159#334—21no—
160#334—21no—
161#334—21no—
162#334C2×C411yescharacteristic
163#334C2×C411yescharacteristic
164#334D411yescharacteristic
165#334D411yescharacteristic
166Q8⋊C8—21no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8C2×C411yescharacteristic
171Q8⋊C8C2×C411yescharacteristic
172Q8⋊C8C2×C411yescharacteristic
173Q8⋊C8C2×C411yescharacteristic
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8C2×C411yescharacteristic
179D4⋊C8C2×C411yescharacteristic
180D4⋊C8C2×C411yescharacteristic
181D4⋊C8C2×C411yescharacteristic
182#322—21no—
183#322—21no—
184#322C2×C411yescharacteristic
185#322C2×C411yescharacteristic
186#322C2×C411yescharacteristic
187#322C2×C411yescharacteristic
188C82—21no—
189C82—21no—
190C2×#41C2×Q811yescharacteristic
191C2×#41C2×Q811yescharacteristic
192#135—42no—
193C2×Q16C4⋊C411yescharacteristic
194C2×SD16C4⋊C411yescharacteristic
195C2×SD16C4⋊C411yescharacteristic
196C2×D8C4⋊C411yescharacteristic
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—41no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—41no—
210C22×C8—41no—
211C22×C8—41no—
212C22×C8C2×Q811yescharacteristic
213C22×C8C2×Q811yescharacteristic
214C22×C8#4111yescharacteristic
215C22×C8#4111yescharacteristic
216C4×Q8C22×C411yescharacteristic
217C4×Q8C22×C411yescharacteristic
218C4×Q8C2×D411yescharacteristic
219C4×Q8C2×D411yescharacteristic
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4C22×C411yescharacteristic
225C4×D4C22×C411yescharacteristic
226C4×D4C2×D411yescharacteristic
227C4×D4C2×D411yescharacteristic
228#117—21no—
229#117—21no—
230#117C2×Q811yescharacteristic
231#117C2×Q811yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C2411yescharacteristic
235#108—24no—
236#108—82no—
237#107C4⋊C411yescharacteristic
238#107C4⋊C411yescharacteristic
239#106C4⋊C411yescharacteristic
240#106C4⋊C411yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
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—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8C22×C411yescharacteristic
258C4⋊C8C22×C411yescharacteristic
259C4⋊C8C22×C411yescharacteristic
260C4⋊C8C22×C411yescharacteristic
261C4⋊C8C2×D411yescharacteristic
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D411yescharacteristic
264C4⋊C8C2×D411yescharacteristic
265#103C4⋊C411yescharacteristic
266#103C4⋊C411yescharacteristic
267#103C4⋊C411yescharacteristic
268#103C4⋊C411yescharacteristic
269#102C4⋊C411yescharacteristic
270#102C4⋊C411yescharacteristic
271#102C4⋊C411yescharacteristic
272#102C4⋊C411yescharacteristic
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—41no—
282C22⋊C8—41no—
283C22⋊C8—41no—
284C22⋊C8—41no—
285#97—21no—
286#97—21no—
287#97—21no—
288#97—21no—
289#97—41no—
290#97—41no—
291#97—41no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—41no—
302C4×C8C22×C411yescharacteristic
303C4×C8C22×C411yescharacteristic
304C4×C8C22×C411yescharacteristic
305C4×C8C22×C411yescharacteristic
306C4×C8C22×C411yescharacteristic
307C4×C8C22×C411yescharacteristic
308C4×C8C2×D411yescharacteristic
309C4×C8C2×D411yescharacteristic
310C4×C8C2×D411yescharacteristic
311C4×C8C2×D411yescharacteristic
312#41—22no—
313#41—41no—
314#41—41no—
315#41—41no—
316C2×Q8C2×C4⋊C411yescharacteristic
317C2×Q8C2×C4⋊C411yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4C2×C4⋊C411yescharacteristic
321C2×D4C2×C4⋊C411yescharacteristic
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331Q16—41no—
332SD16—22no—
333SD16—41no—
334D8—22no—
335M16—22no—
336M16—22no—
337M16—41no—
338M16—41no—
339M16—42no—
340M16—81no—
341M16—81no—
342M16—81no—
343M16—81no—
344M16—81no—
345M16—161no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—81no—
370C2×C8—161no—
371C2×C8—161no—
372C2×C8—161no—
373C2×C8C2×C4⋊C411yescharacteristic
374C2×C8C2×C4⋊C411yescharacteristic
375C2×C8C2×C4⋊C411yescharacteristic
376C2×C8C2×C4⋊C411yescharacteristic
377C2×C8C4×D411yescharacteristic
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C2×C8C4×D411yescharacteristic
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4C2×C4⋊C411yescharacteristic
386C4⋊C4C2×C4⋊C411yescharacteristic
387C4⋊C4C2×C4⋊C411yescharacteristic
388C4⋊C4C2×C4⋊C411yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42C23×C411yescharacteristic
402C42C22×D411yescharacteristic
403C42C22×D411yescharacteristic
404C422−1+411yescharacteristic
405C23—21no—
406C23—21no—
407C23#55411yescharacteristic
408Q8—21no—
409Q8—21no—
410Q8—21no—
411Q8—21no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—42no—
416D4—81no—
417D4#43012yes—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—42no—
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—81no—
440C2×C4C22×C4⋊C411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4#52011yescharacteristic
443C2×C4D4211yescharacteristic
444C2×C4#54711yescharacteristic
445C2×C4D4×Q811yescharacteristic
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—21no—
452C8—21no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—42no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8#44312yes—
470C22—21no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22D4×C4⋊C411yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4C2×#43011yescharacteristic
490C4#291611yescharacteristic
491C4C2×#44311yescharacteristic
492C4#295711yescharacteristic
493C4#328811yescharacteristic
494C4#328911yescharacteristic
495C2—41no—
496C2—42no—
497C2—81no—
498C2#2266811yescharacteristic
499C2#2268811yescharacteristic
500C2#2269311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC2×#135Q8
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
4. #17839C2semidirect
5. #17834C2semidirect
6. #9634C2semidirect
7. #9633C2semidirect
8. #8900C2semidirect
9. #8900C2semidirect
10. #8899C2semidirect
11. #8898C2semidirect
12. #8828C2semidirect
13. #8672C2semidirect
14. #8671C2semidirect
15. #8294C2semidirect

Maximal subgroups

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

subgroupindex
#180572
#180552
#178632
#178392
#178342
#96342
#96332
#89002
#88992
#88982
#88282
#86722
#86712
#82942