Finite Group Databasebeta

← #226205

Subgroups — #226205

Order 512.

514 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18063C211yesmaximal, characteristic
2#18057C211yesmaximal, characteristic
3#17868C211yesmaximal, characteristic
4#17839C211yesmaximal, characteristic, complemented
5#17837C211yesmaximal, characteristic, complemented
6#9635C211yesmaximal, characteristic, complemented
7#9633C211yesmaximal, characteristic, complemented
8#8901C211yesmaximal, characteristic, complemented
9#8901C211yesmaximal, characteristic, complemented
10#8901C211yesmaximal, characteristic, complemented
11#8900C211yesmaximal, characteristic, complemented
12#8827C211yesmaximal, 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#1837C2211yescharacteristic
24#1837C2211yescharacteristic
25C4×#108C2211yescharacteristic
26#1594—21no—
27#1590—21no—
28#1588C411yescharacteristic
29#1588C411yescharacteristic
30#1587—21no—
31#1587C411yescharacteristic
32#1587C411yescharacteristic
33#1586—21no—
34#1586C411yescharacteristic
35#1586C411yescharacteristic
36#1585C411yescharacteristic
37#1585C411yescharacteristic
38#1583—21no—
39#1583—21no—
40#1579—21no—
41C8×SD16—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#1566C2211yescharacteristic
55#1566C2211yescharacteristic
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—
121#427Q811yescharacteristic
122#423—21no—
123#423—21no—
124#423—21no—
125#423C2311yescharacteristic
126#423C2311yescharacteristic
127#423C2311yescharacteristic
128#423C2311yescharacteristic
129C2×C4⋊C8—21no—
130#416Q811yescharacteristic
131#416Q811yescharacteristic
132C4×M16—21no—
133C4×M16C2311yescharacteristic
134C4×M16C2311yescharacteristic
135C2×C4×C8—21no—
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#341—21no—
140#341—41no—
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#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#107C4⋊C411yescharacteristic
237#107C4⋊C411yescharacteristic
238#106C4⋊C411yescharacteristic
239#106C4⋊C411yescharacteristic
240C4⋊C8—21no—
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—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C22×C411yescharacteristic
257C4⋊C8C22×C411yescharacteristic
258C4⋊C8C22×C411yescharacteristic
259C4⋊C8C22×C411yescharacteristic
260C4⋊C8C2×D411yescharacteristic
261C4⋊C8C2×D411yescharacteristic
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D411yescharacteristic
264#103C4⋊C411yescharacteristic
265#103C4⋊C411yescharacteristic
266#103C4⋊C411yescharacteristic
267#103C4⋊C411yescharacteristic
268#102C4⋊C411yescharacteristic
269#102C4⋊C411yescharacteristic
270#102C4⋊C411yescharacteristic
271#102C4⋊C411yescharacteristic
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—41no—
281C22⋊C8—41no—
282C22⋊C8—41no—
283C22⋊C8—41no—
284#97—21no—
285#97—21no—
286#97—21no—
287#97—21no—
288#97—41no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—41no—
299C4×C8—41no—
300C4×C8—41no—
301C4×C8C22×C411yescharacteristic
302C4×C8C22×C411yescharacteristic
303C4×C8C22×C411yescharacteristic
304C4×C8C22×C411yescharacteristic
305C4×C8C22×C411yescharacteristic
306C4×C8C22×C411yescharacteristic
307C4×C8C2×D411yescharacteristic
308C4×C8C2×D411yescharacteristic
309C4×C8C2×D411yescharacteristic
310C4×C8C2×D411yescharacteristic
311#41—22no—
312#41—41no—
313#41—41no—
314#41—41no—
315C2×Q8C2×C4⋊C411yescharacteristic
316C2×Q8C2×C4⋊C411yescharacteristic
317C2×D4—21no—
318C2×D4—21no—
319C2×D4C2×C4⋊C411yescharacteristic
320C2×D4C2×C4⋊C411yescharacteristic
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330Q16—22no—
331SD16—22no—
332SD16—41no—
333D8—41no—
334M16—22no—
335M16—42no—
336M16—41no—
337M16—41no—
338M16—41no—
339M16—81no—
340M16—81no—
341M16—81no—
342M16—81no—
343M16—81no—
344M16—161no—
345M16—161no—
346M16—161no—
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—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—81no—
371C2×C8—161no—
372C2×C8C2×C4⋊C411yescharacteristic
373C2×C8C2×C4⋊C411yescharacteristic
374C2×C8C2×C4⋊C411yescharacteristic
375C2×C8C2×C4⋊C411yescharacteristic
376C2×C8C4×D411yescharacteristic
377C2×C8C4×D411yescharacteristic
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4C2×C4⋊C411yescharacteristic
385C4⋊C4C2×C4⋊C411yescharacteristic
386C4⋊C4C2×C4⋊C411yescharacteristic
387C4⋊C4C2×C4⋊C411yescharacteristic
388C22⋊C4—21no—
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42C23×C411yescharacteristic
401C42C22×D411yescharacteristic
402C42C22×D411yescharacteristic
403C422−1+411yescharacteristic
404C23—21no—
405C23—21no—
406C23#55411yescharacteristic
407Q8—21no—
408Q8—21no—
409Q8—21no—
410Q8#43012yes—
411D4—21no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—42no—
416D4—81no—
417C2×C4—21no—
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—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4C22×C4⋊C411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4#52011yescharacteristic
442C2×C4D4211yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4D4×Q811yescharacteristic
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—21no—
452C8—21no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—41no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22D4×C4⋊C411yescharacteristic
479C4—21no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4C2×#43011yescharacteristic
489C4#291611yescharacteristic
490C4#295511yescharacteristic
491C4#295611yescharacteristic
492C4#329011yescharacteristic
493C4#329011yescharacteristic
494C2—41no—
495C2—81no—
496C2—81no—
497C2#2266811yescharacteristic
498C2#2268711yescharacteristic
499C2#2269111yescharacteristic
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. #9635C2semidirect
7. #9633C2semidirect
8. #8901C2semidirect
9. #8901C2semidirect
10. #8901C2semidirect
11. #8900C2semidirect
12. #8827C2semidirect
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
#180632
#180572
#178682
#178392
#178372
#96352
#96332
#89012
#89002
#88272
#86732
#86712
#82962