Finite Group Databasebeta

← #226201

Subgroups — #226201

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18061C211yesmaximal, characteristic
2#18059C211yesmaximal, characteristic
3#17868C211yesmaximal, characteristic
4#17839C211yesmaximal, characteristic, complemented
5#17837C211yesmaximal, characteristic, complemented
6#9636C211yesmaximal, characteristic, complemented
7#9634C211yesmaximal, characteristic, complemented
8#8899C211yesmaximal, characteristic, complemented
9#8899C211yesmaximal, characteristic, complemented
10#8899C211yesmaximal, characteristic, complemented
11#8898C211yesmaximal, characteristic, complemented
12#8826C211yesmaximal, characteristic, complemented
13#8673C211yesmaximal, characteristic, complemented
14#8671C211yesmaximal, characteristic, complemented
15#8296C211yesmaximal, characteristic, complemented
16#2991C2211yescharacteristic
17#2991C2211yescharacteristic
18#2991C2211yescharacteristic
19#2990C2211yescharacteristic
20#2965C2211yescharacteristic
21C8×#41C2211yescharacteristic
22C4×#135C2211yescharacteristic
23#1848C2211yescharacteristic
24#1848C2211yescharacteristic
25C4×#108C2211yescharacteristic
26#1592—21no—
27#1589—21no—
28#1588—21no—
29#1588—21no—
30#1588C411yescharacteristic
31#1588C411yescharacteristic
32#1587C411yescharacteristic
33#1587C411yescharacteristic
34#1586C411yescharacteristic
35#1586C411yescharacteristic
36#1585C411yescharacteristic
37#1585C411yescharacteristic
38#1584—21no—
39#1581—21no—
40SD16⋊C8—21no—
41C8×Q16—21no—
42#1574—21no—
43#1574—21no—
44#1574C2211yescharacteristic
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1574C2211yescharacteristic
48#1572—21no—
49#1571—21no—
50#1569—21no—
51#1567—21no—
52#1567C2211yescharacteristic
53#1567C2211yescharacteristic
54#1567C2211yescharacteristic
55#1567C2211yescharacteristic
56#1565—21no—
57#1564—21no—
58#1564C2211yescharacteristic
59#1564C2211yescharacteristic
60#1489C2211yescharacteristic
61#1489C2211yescharacteristic
62#1489C2211yescharacteristic
63#1489C2211yescharacteristic
64#1476C2211yescharacteristic
65#1476C2211yescharacteristic
66#1451C2211yescharacteristic
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#446C2×C411yescharacteristic
80#446C2×C411yescharacteristic
81C8×Q8—21no—
82C8×Q8—21no—
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—21no—
96#436—21no—
97#436—41no—
98#436—41no—
99#436—41no—
100#436C2×C411yescharacteristic
101#436C2×C411yescharacteristic
102#435—21no—
103#435—21no—
104#435—41no—
105C8×D4—21no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4C2×C411yescharacteristic
109C8×D4C2×C411yescharacteristic
110#432—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120C2×#108—21no—
121C2×#108—41no—
122C2×#108—41no—
123#427Q811yescharacteristic
124#423—21no—
125#423—21no—
126#423—21no—
127#423C2311yescharacteristic
128#423C2311yescharacteristic
129#423C2311yescharacteristic
130#423C2311yescharacteristic
131C2×C4⋊C8—21no—
132#416Q811yescharacteristic
133#416Q811yescharacteristic
134C4×M16—21no—
135C4×M16C2311yescharacteristic
136C4×M16C2311yescharacteristic
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141#341—21no—
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#335—21no—
151#335—21no—
152#335—21no—
153#335—21no—
154#335C2×C411yescharacteristic
155#335C2×C411yescharacteristic
156#335D411yescharacteristic
157#335D411yescharacteristic
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#322—21no—
185#322C2×C411yescharacteristic
186#322C2×C411yescharacteristic
187#322C2×C411yescharacteristic
188#322C2×C411yescharacteristic
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—
206C2×M16—41no—
207C2×M16—41no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
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—42no—
236#108—82no—
237#108—82no—
238#107C4⋊C411yescharacteristic
239#107C4⋊C411yescharacteristic
240#106C4⋊C411yescharacteristic
241#106C4⋊C411yescharacteristic
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—21no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8C22×C411yescharacteristic
259C4⋊C8C22×C411yescharacteristic
260C4⋊C8C22×C411yescharacteristic
261C4⋊C8C22×C411yescharacteristic
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D411yescharacteristic
264C4⋊C8C2×D411yescharacteristic
265C4⋊C8C2×D411yescharacteristic
266#103C4⋊C411yescharacteristic
267#103C4⋊C411yescharacteristic
268#103C4⋊C411yescharacteristic
269#103C4⋊C411yescharacteristic
270#102C4⋊C411yescharacteristic
271#102C4⋊C411yescharacteristic
272#102C4⋊C411yescharacteristic
273#102C4⋊C411yescharacteristic
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—41no—
283C22⋊C8—41no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286#97—21no—
287#97—21no—
288#97—21no—
289#97—21no—
290#97—41no—
291C4×C8—21no—
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—41no—
301C4×C8—41no—
302C4×C8—41no—
303C4×C8C22×C411yescharacteristic
304C4×C8C22×C411yescharacteristic
305C4×C8C22×C411yescharacteristic
306C4×C8C22×C411yescharacteristic
307C4×C8C22×C411yescharacteristic
308C4×C8C22×C411yescharacteristic
309C4×C8C2×D411yescharacteristic
310C4×C8C2×D411yescharacteristic
311C4×C8C2×D411yescharacteristic
312C4×C8C2×D411yescharacteristic
313#41—22no—
314#41—41no—
315#41—41no—
316#41—41no—
317C2×Q8C2×C4⋊C411yescharacteristic
318C2×Q8C2×C4⋊C411yescharacteristic
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C2×C4⋊C411yescharacteristic
322C2×D4C2×C4⋊C411yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4C22×Q811yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332Q16—22no—
333SD16—22no—
334SD16—41no—
335D8—41no—
336M16—22no—
337M16—42no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—81no—
342M16—81no—
343M16—81no—
344M16—81no—
345M16—81no—
346M16—161no—
347M16—161no—
348M16—161no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—81no—
373C2×C8—161no—
374C2×C8C2×C4⋊C411yescharacteristic
375C2×C8C2×C4⋊C411yescharacteristic
376C2×C8C2×C4⋊C411yescharacteristic
377C2×C8C2×C4⋊C411yescharacteristic
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C2×C8C4×D411yescharacteristic
381C2×C8C4×D411yescharacteristic
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4C2×C4⋊C411yescharacteristic
387C4⋊C4C2×C4⋊C411yescharacteristic
388C4⋊C4C2×C4⋊C411yescharacteristic
389C4⋊C4C2×C4⋊C411yescharacteristic
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C22⋊C4—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42C23×C411yescharacteristic
403C42C22×D411yescharacteristic
404C42C22×D411yescharacteristic
405C422−1+411yescharacteristic
406C23—21no—
407C23—21no—
408C23#55411yescharacteristic
409Q8—21no—
410Q8—21no—
411Q8—21no—
412Q8#43012yes—
413D4—21no—
414D4—21no—
415D4—21no—
416D4—21no—
417D4—42no—
418D4—81no—
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—21no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4C22×C4⋊C411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4#52011yescharacteristic
444C2×C4D4211yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4D4×Q811yescharacteristic
447C8—22no—
448C8—22no—
449C8—21no—
450C8—21no—
451C8—21no—
452C8—21no—
453C8—21no—
454C8—21no—
455C8—21no—
456C8—21no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22D4×C4⋊C411yescharacteristic
481C4—21no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4C2×#43011yescharacteristic
491C4#291611yescharacteristic
492C4#295511yescharacteristic
493C4#295611yescharacteristic
494C4#329211yescharacteristic
495C4#329311yescharacteristic
496C2—41no—
497C2—81no—
498C2—81no—
499C2#2266811yescharacteristic
500C2#2268711yescharacteristic
501C2#2269211yescharacteristic
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. #17837C2semidirect
6. #9636C2semidirect
7. #9634C2semidirect
8. #8899C2semidirect
9. #8899C2semidirect
10. #8899C2semidirect
11. #8898C2semidirect
12. #8826C2semidirect
13. #8673C2semidirect
14. #8671C2semidirect
15. #8296C2semidirect

Maximal subgroups

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

subgroupindex
#180612
#180592
#178682
#178392
#178372
#96362
#96342
#88992
#88982
#88262
#86732
#86712
#82962