Finite Group Databasebeta

← #6378564

Subgroups — #6378564

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#22358C214yesmaximal, complemented
3#22330C214yesmaximal, complemented
4#22299C211yesmaximal, characteristic, complemented
5#22260C211yesmaximal, characteristic, complemented
6#22133C212yesmaximal, complemented
7#22013C212yesmaximal, complemented
8#21926C211yesmaximal, characteristic, complemented
9#21681C211yesmaximal, characteristic, complemented
10#21673C211yesmaximal, characteristic, complemented
11#21571C214yesmaximal, complemented
12#21496C214yesmaximal, complemented
13#21487C212yesmaximal, complemented
14#20485C212yesmaximal, complemented
15C8×#117C211yesmaximal, characteristic, complemented
16C2×#436C2212yes—
17C2×#436C2214yes—
18C2×#435C2214yes—
19C2×C8×D4C2212yes—
20C2×#431C2211yescharacteristic
21C2×#405C2214yes—
22C22×C4×C8C2211yescharacteristic
23#2651C414yes—
24#2648C414yes—
25#2647C411yescharacteristic
26#2644C412yes—
27#2641C411yescharacteristic
28#2629C411yescharacteristic
29#2624C412yes—
30#2619C411yescharacteristic
31#2395C2214yes—
32#2392C2214yes—
33C4×#127C2211yescharacteristic
34C4×#125C2211yescharacteristic
35C4×#121C2212yes—
36#2293C2212yes—
37C4×#117C2211yescharacteristic
38#1954C2214yes—
39#1954C2214yes—
40#1954C2212yes—
41#1954C2212yes—
42#1939C2214yes—
43#1939C2212yes—
44#1939C2212yes—
45#1924C2214yes—
46#1924C2214yes—
47#1924C2214yes—
48#1923C2214yes—
49#1923C2214yes—
50#1917C2214yes—
51#1916C2214yes—
52#1916C2214yes—
53#1869C2214yes—
54#1869C2214yes—
55#1869C2214yes—
56#1869C2214yes—
57#1850C2214yes—
58#1842C2211yescharacteristic
59#1831C2211yescharacteristic
60#1818C2214yes—
61#1798C2214yes—
62#1797C2212yes—
63#1771C2214yes—
64#1770C2212yes—
65C8×C4⋊C4C2212yes—
66C8×C4⋊C4C2212yes—
67C8×C4⋊C4C2212yes—
68#1767C2212yes—
69#1753C2214yes—
70#1753C2214yes—
71C8×C22⋊C4C2214yes—
72C8×C22⋊C4C2212yes—
73C8×C22⋊C4C2212yes—
74#1750C2214yes—
75#1744C2211yescharacteristic
76#1744C2211yescharacteristic
77C42×C8C2211yescharacteristic
78C42×C8C2211yescharacteristic
79C22×M16C2314yes—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2314yes—
83C2×#127C2×C411yescharacteristic
84C2×#125C2×C411yescharacteristic
85C2×#121C2×C412yes—
86C2×C4×D4C2312yes—
87C2×C4×D4C2314yes—
88C2×C4×D4C2314yes—
89C2×#117C2×C411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C2×#117C2314yes—
92C22×C42C2311yescharacteristic
93#436—28no—
94#436—48no—
95#435—48no—
96C8×D4—28no—
97#431—24no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2314yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112#405—48no—
113C2×#97C2314yes—
114C2×#97C2314yes—
115C2×C4×C8—24no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2312yes—
122C2×C4×C8C2314yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#400C2×C414yes—
128#399C2×C414yes—
129#398C2×C414yes—
130#398C2×C414yes—
131#397C2×C414yes—
132#397C2×C414yes—
133#397C2×C414yes—
134#397C2×C414yes—
135#396C2×C414yes—
136#396C2×C414yes—
137#394C2×C414yes—
138#394C2×C414yes—
139#394C2×C414yes—
140#394C2×C414yes—
141#394C2×C414yes—
142#392C2×C414yes—
143#390C2×C412yes—
144#390C2×C414yes—
145#390C2×C412yes—
146#390C2×C412yes—
147#390C2×C412yes—
148#389C2×C412yes—
149#389C2×C412yes—
150#389C2×C412yes—
151#389C2×C412yes—
152#388C2×C414yes—
153#388C2×C414yes—
154#388C2×C414yes—
155#387C2×C414yes—
156#386C2×C414yes—
157#384C2×C411yescharacteristic
158#382C2×C411yescharacteristic
159#380C2×C412yes—
160C4×C4⋊C4C2×C412yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2312yes—
163C4×C4⋊C4C2312yes—
164C4×C22⋊C4C2×C412yes—
165C4×C22⋊C4C2314yes—
166C4×C22⋊C4C2312yes—
167C4×C22⋊C4C2314yes—
168#376C2×C411yescharacteristic
169#376C2×C411yescharacteristic
170C43C2311yescharacteristic
171C43C2311yescharacteristic
172#336C2314yes—
173#336C2312yes—
174#336C2314yes—
175#336C2312yes—
176#336C2314yes—
177#336C2314yes—
178#336C2314yes—
179#336C2314yes—
180#336C2312yes—
181#336C2312yes—
182#336C2314yes—
183C22×D4C22×C412yes—
184C22×D4C22×C414yes—
185C22×D4C22×C414yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—28no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—28no—
196C22×C8—24no—
197C22×C8—42no—
198C22×C8—44no—
199C22×C8—44no—
200C22×C8C2411yescharacteristic
201C22×C8C2414yes—
202C22×C8C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2412yes—
205C22×C8C2412yes—
206C22×C8C2411yescharacteristic
207C22×C8C2411yescharacteristic
208#127—24no—
209#125—24no—
210#121—28no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—44no—
217#117—24no—
218#117—24no—
219#117—44no—
220#117—44no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C414yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4C22×C412yes—
233C2×C22⋊C4C22×C412yes—
234C2×C22⋊C4C22×C414yes—
235C2×C22⋊C4C22×C414yes—
236C2×C22⋊C4C22×C414yes—
237C2×C22⋊C4C22×C414yes—
238C2×C22⋊C4C22×C414yes—
239C2×C22⋊C4C22×C414yes—
240C2×C22⋊C4C22×C414yes—
241C2×C22⋊C4C22×C414yes—
242C2×C22⋊C4C22×C414yes—
243C2×C22⋊C4C22×C414yes—
244C2×C22⋊C4C22×C414yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C412yes—
253C2×C42C22×C412yes—
254C2×C42C22×C411yescharacteristic
255C2×C42C2412yes—
256C2×C42C2414yes—
257C2×C42C2412yes—
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260C4⋊C8—24no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—44no—
264C4⋊C8—44no—
265C4⋊C8—44no—
266C22⋊C8—24no—
267C22⋊C8—28no—
268C22⋊C8—28no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274#97—28no—
275#97—44no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—24no—
279C4×C8—24no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—42no—
283C4×C8—42no—
284C4×C8—44no—
285C4×C8#4114yes—
286C4×C8#4114yes—
287#95C22×C412yes—
288#95C22×C412yes—
289#95C22×C412yes—
290#95C22×C412yes—
291#95C22×C414yes—
292#95C22×C414yes—
293#95C22×C414yes—
294#95C22×C414yes—
295#95C22×C414yes—
296#95C22×C414yes—
297#95C22×C414yes—
298C24C23×C414yes—
299C24C23×C411yescharacteristic
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4#13114yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C23×C414yes—
326C22×C4C23×C412yes—
327C22×C4C23×C412yes—
328C22×C4C23×C412yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C23×C414yes—
331C22×C4C23×C414yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C23×C412yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C22×C4C2511yescharacteristic
337M16—416no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—44no—
355C2×C8—416no—
356C2×C8—44no—
357C2×C8C2×#4112yes—
358C2×C8C2×#4114yes—
359C2×C8C2×#4112yes—
360C2×C8H(5,𝔽2)14yes—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C22⋊C4—24no—
373C22⋊C4—28no—
374C22⋊C4—24no—
375C22⋊C4—28no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—21no—
391C42—21no—
392C42—22no—
393C42—22no—
394C42—24no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—42no—
399C42—42no—
400C42#13114yes—
401C42C2×#4112yes—
402C42C2×#4112yes—
403C23—21no—
404C23—22no—
405C23—21no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23C2×#13112yes—
415C23C24×C411yescharacteristic
416D4—48no—
417D4—88no—
418D4—88no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4C4×#4112yes—
456C2×C4C4×#4112yes—
457C2×C4C2×#13114yes—
458C2×C4C2×#13112yes—
459C2×C4C2×#13112yes—
460C2×C4C2×#13112yes—
461C2×C4C22×#4111yescharacteristic
462C2×C4C22×#4111yescharacteristic
463C2×C4C2×H(5,𝔽2)12yes—
464C2×C4#58512yes—
465C8—22no—
466C8—24no—
467C8—41no—
468C8—44no—
469C8—41no—
470C8—42no—
471C8—44no—
472C8#54014yes—
473C22—22no—
474C22—24no—
475C22—41no—
476C22—41no—
477C22—42no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22C2×C4×#4111yescharacteristic
486C22C4×H(5,𝔽2)12yes—
487C22C22×#13111yescharacteristic
488C22C22×#13111yescharacteristic
489C22#357312yes—
490C4—22no—
491C4—22no—
492C4—24no—
493C4—41no—
494C4—44no—
495C4—41no—
496C4—44no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4#296614yes—
502C4C2×#54011yescharacteristic
503C4#345511yescharacteristic
504C4#346012yes—
505C2—42no—
506C2—84no—
507C2C4×#54011yescharacteristic
508C2C2×#296611yescharacteristic
509C2#6020611yescharacteristic
510C2#6032112yes—
511C2#6032112yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radical#2619C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22358C2semidirect
3. #22330C2semidirect
4. #22299C2semidirect
5. #22260C2semidirect
6. #22133C2semidirect
7. #22013C2semidirect
8. #21926C2semidirect
9. #21681C2semidirect
10. #21673C2semidirect
11. #21571C2semidirect
12. #21496C2semidirect
13. #21487C2semidirect
14. #20485C2semidirect
15. C8×#117C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#223582
#223302
#222992
#222602
#221332
#220132
#219262
#216812
#216732
#215712
#214962
#214872
#204852
C8×#1172