Finite Group Databasebeta

← #275398

Subgroups — #275398

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33148C211yesmaximal, characteristic
2#27962C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic
4#21358C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#9622C211yesmaximal, characteristic, complemented
10#9619C211yesmaximal, characteristic, complemented
11#9576C211yesmaximal, characteristic, complemented
12#9576C211yesmaximal, characteristic, complemented
13#9128C211yesmaximal, characteristic
14#8831C211yesmaximal, characteristic
15#8726C211yesmaximal, characteristic
16C2×#432—21no—
17C2×#431C2211yescharacteristic
18#2857C2211yescharacteristic
19#2285C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#2101—21no—
22#2101—21no—
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1987C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1920—21no—
36#1920—21no—
37#1917—21no—
38#1917—21no—
39#1853—21no—
40#1852—21no—
41#1852—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1825—21no—
45#1825—21no—
46#1823—21no—
47#1817—21no—
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50C4×#107—21no—
51C4×#107—21no—
52#1770C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C22⋊Q8—21no—
69C2×#117—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#432—44no—
75#431—22no—
76#431—22no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C22⋊C8D411yescharacteristic
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—21no—
102#389—21no—
103#389—21no—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#382—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C2×#95—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#334—22no—
149#333—22no—
150#333—22no—
151C22×Q8—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C22⋊Q8—82no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—42no—
214#107—42no—
215#107—42no—
216#107—42no—
217#106—42no—
218#106—42no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8#4112yes—
232C4⋊C8#4112yes—
233C22⋊C8—24no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—42no—
237C22⋊C8—42no—
238#97—42no—
239#97—42no—
240C4×C8—24no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24#12111yescharacteristic
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—81no—
257C2×Q8—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—81no—
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#13512yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4#13712yes—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—42no—
367C42—42no—
368C42—82no—
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C422−1+411yescharacteristic
372C422−1+411yescharacteristic
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23#46312yes—
380C23C2×#12111yescharacteristic
381Q8—161no—
382Q8—161no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#46312yes—
430C2×C4#52511yescharacteristic
431C2×C4#53611yescharacteristic
432C2×C4#54711yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#55211yescharacteristic
435C2×C4#55211yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57511yescharacteristic
442C2×C4#57811yescharacteristic
443C2×C4#57811yescharacteristic
444C8—41no—
445C8—41no—
446C8—41no—
447C8—41no—
448C8—42no—
449C8—42no—
450C8—81no—
451C8—81no—
452C8—81no—
453C8—81no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22#241311yescharacteristic
462C22C2×#46311yescharacteristic
463C22#304411yescharacteristic
464C22#305411yescharacteristic
465C22#305611yescharacteristic
466C22#306711yescharacteristic
467C22#306811yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—41no—
472C4—41no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4#311111yescharacteristic
482C4#311611yescharacteristic
483C4#323711yescharacteristic
484C4#324011yescharacteristic
485C4#333611yescharacteristic
486C4#333611yescharacteristic
487C4#334311yescharacteristic
488C4#334311yescharacteristic
489C2—42no—
490C2#1914211yescharacteristic
491C2#1916811yescharacteristic
492C2#1944211yescharacteristic
493C2#1947811yescharacteristic
494C2#2457711yescharacteristic
495C2#2461111yescharacteristic
496C2#2462511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10657C2semidirect
6. #10656C2semidirect
7. #10593C2semidirect
8. #10589C2semidirect
9. #9622C2semidirect
10. #9619C2semidirect
11. #9576C2semidirect
12. #9576C2semidirect

Maximal subgroups

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

subgroupindex
#331482
#279622
#226032
#213582
#106572
#106562
#105932
#105892
#96222
#96192
#95762
#91282
#88312
#87262