Finite Group Databasebeta

← #228663

Subgroups — #228663

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23051C211yesmaximal, characteristic
2#22603C211yesmaximal, characteristic
3C4×#484C211yesmaximal, characteristic
4#9598C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9540C211yesmaximal, characteristic, complemented
8#9537C211yesmaximal, characteristic, complemented
9#9077C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8834C211yesmaximal, characteristic
12#8287C211yesmaximal, characteristic, complemented
13#8203C211yesmaximal, characteristic, complemented
14#8147C211yesmaximal, characteristic
15#7904C211yesmaximal, characteristic
16C2×#484C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#432—21no—
19C2×#432C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C4×C22⋊Q8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1984C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1928C411yescharacteristic
28#1928C411yescharacteristic
29#1924—21no—
30#1923—21no—
31#1921C411yescharacteristic
32#1921C411yescharacteristic
33#1917—21no—
34#1917—21no—
35#1917—21no—
36#1872C411yescharacteristic
37#1852C411yescharacteristic
38#1824C411yescharacteristic
39#1817—21no—
40#1797C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#484—24no—
75C8×Q8—22no—
76C8×Q8—41no—
77#432—22no—
78#432—44no—
79#432—41no—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2×C411yescharacteristic
99C2×C22⋊C8C2×C411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C4×M16—22no—
103C4×M16—41no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#387—21no—
119#385—21no—
120#380C2×C411yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C2×#95—21no—
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—22no—
154Q8⋊C8—22no—
155Q8⋊C8—41no—
156#322—22no—
157#322—41no—
158C22×Q8C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22⋊Q8—22no—
182C22⋊Q8—41no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#107—22no—
222#107—22no—
223#107—41no—
224#107—41no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8#4112yes—
237#103—22no—
238#103—41no—
239#103—41no—
240#103M1614yescomplemented
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—42no—
248#97—42no—
249#97—42no—
250#97—42no—
251#97—42no—
252C4×C8—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—21no—
263C4×C8—42no—
264C4×C8#4112yes—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24C4×D411yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8C2×M1612yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—81no—
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317M16—48no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8C2×M1612yes—
359C2×C8AGL(1,ℤ/8)12yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—81no—
379C4⋊C4C2×M1612yes—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—42no—
389C42—42no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42H(5,𝔽2)11yescharacteristic
401C422−1+411yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23#44412yes—
409C23C2×C4×D411yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
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—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4C4×#4111yescharacteristic
462C2×C4#52011yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4#55111yescharacteristic
466C2×C4#55211yescharacteristic
467C2×C4C22×M1611yescharacteristic
468C2×C4#56811yescharacteristic
469C2×C4#56811yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4#57511yescharacteristic
473C2×C4#57811yescharacteristic
474C2×C4#57811yescharacteristic
475C8—22no—
476C8—22no—
477C8—42no—
478C8—42no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—42no—
486C8—81no—
487C8—81no—
488C8—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22#235911yescharacteristic
497C22#293011yescharacteristic
498C22D4×M1611yescharacteristic
499C22C4×AGL(1,ℤ/8)11yescharacteristic
500C22#295011yescharacteristic
501C22C2×#44411yescharacteristic
502C22#295611yescharacteristic
503C4—21no—
504C4—21no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—41no—
509C4—41no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4#296711yescharacteristic
517C4#297011yescharacteristic
518C4#298911yescharacteristic
519C4#299511yescharacteristic
520C4#330811yescharacteristic
521C4#331511yescharacteristic
522C4#335211yescharacteristic
523C4#335311yescharacteristic
524C2—42no—
525C2#1793811yescharacteristic
526C2#1795311yescharacteristic
527C2#1804811yescharacteristic
528C2#1806211yescharacteristic
529C2#2291211yescharacteristic
530C2#2292511yescharacteristic
531C2#2295011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9598C2semidirect
5. #9555C2semidirect
6. #9548C2semidirect
7. #9540C2semidirect
8. #9537C2semidirect
10. #8846C2semidirect
12. #8287C2semidirect
13. #8203C2semidirect
240. #103M16semidirect

Maximal subgroups

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

subgroupindex
#230512
#226032
C4×#4842
#95982
#95552
#95482
#95402
#95372
#90772
#88462
#88342
#82872
#82032
#81472
#79042