Finite Group Databasebeta

← #6480693

Subgroups — #6480693

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic
3#33077C214yesmaximal, complemented
4#30098C212yesmaximal, complemented
5#30079C212yesmaximal, complemented
6#29579C212yesmaximal, complemented
7#29573C212yesmaximal, complemented
8#29326C212yesmaximal, complemented
9#29138C212yesmaximal, complemented
10#27681C212yesmaximal
11#27652C212yesmaximal
12#27620C212yesmaximal
13#24072C212yesmaximal
14#22094C214yesmaximal
15#20785C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×#485C2214yes—
18#2859C2212yes—
19#2859C2212yes—
20#2647C2212yes—
21#2647C2212yes—
22#2571—22no—
23#2552—22no—
24#2442—22no—
25#2442—22no—
26#2373—22no—
27#2373—22no—
28#2371—22no—
29#2370—22no—
30#2330C2211yescharacteristic
31#2327C2211yescharacteristic
32#2323C2212yes—
33#2317C2212yes—
34#2316C2212yes—
35#2314C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#125C2211yescharacteristic
38C4×C22⋊Q8C2212yes—
39#2295C2212yes—
40#2285C2211yescharacteristic
41C2×#384C2211yescharacteristic
42#2276C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2095C2212yes—
48#2095C2212yes—
49#2084C2212yes—
50#2083C2212yes—
51#2083C2214yes—
52#2072C2212yes—
53#2072C2212yes—
54#2059C2214yes—
55#2043C2214yes—
56#1949C2214yes—
57#1947C2214yes—
58#1944C2214yes—
59#1942C2214yes—
60#1935C2214yes—
61#1933C2214yes—
62#1921C2214yes—
63#1919C2214yes—
64#1918C2214yes—
65#1918C2214yes—
66#1872C2214yes—
67#1865C2214yes—
68#1863C2214yes—
69#1863C2214yes—
70#1863C2214yes—
71#1852C2214yes—
72#1839C2212yes—
73#1832C2212yes—
74#1825C2214yes—
75#1818C2212yes—
76#1805C2214yes—
77#1803C2214yes—
78#1797C2212yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83#541—28no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#126—22no—
86C2×#125—22no—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125C2311yescharacteristic
90C2×C22⋊Q8C2312yes—
91C2×C4×Q8C2312yes—
92C2×C4×Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#485—216no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#107C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C2×#97C2312yes—
112C2×C4×C8C2312yes—
113#400—24no—
114#400—24no—
115#398—24no—
116#398—24no—
117#391C2312yes—
118#391C2312yes—
119#389—24no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389C2312yes—
128#389C2314yes—
129#389C2314yes—
130#389C2314yes—
131#389C2314yes—
132#389C2312yes—
133#388—22no—
134#388—22no—
135#388—22no—
136#387—22no—
137#387—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#385—22no—
146#385—22no—
147#385—22no—
148#384—22no—
149#384—22no—
150#384C2311yescharacteristic
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382—22no—
156#382—22no—
157#382C2311yescharacteristic
158#382C2312yes—
159#380—22no—
160#380—24no—
161#380C2312yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—22no—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4D412yes—
178#376—22no—
179#376—22no—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2312yes—
193#336C2312yes—
194#336C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2314yes—
198C22×Q8C2412yes—
199C23×C4—22no—
200C23×C4C2×D412yes—
201C23×C4C2411yescharacteristic
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—24no—
205#126—216no—
206#125—216no—
207#125—24no—
208#125—84no—
209#125—84no—
210C22⋊Q8—24no—
211C22⋊Q8#4118yes—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—28no—
215C4×Q8—44no—
216#117—24no—
217#117—28no—
218#117—42no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—84no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4C2×D412yes—
252C2×C22⋊C4C2×D411yescharacteristic
253C2×C22⋊C4C2×D411yescharacteristic
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273#107—28no—
274#107—44no—
275#107—44no—
276#106—28no—
277C4⋊C8—28no—
278#103—28no—
279#103—28no—
280#103—44no—
281#103—44no—
282C22⋊C8—28no—
283#97—42no—
284C4×C8—42no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95C2×D412yes—
302#95C2×D412yes—
303#95C2×D412yes—
304#95C2×D412yes—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8C2×#4114yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—84no—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C2×#4114yes—
333C22×C4C2511yescharacteristic
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4C2×#4114yes—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—44no—
375C22⋊C4—82no—
376C42—28no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42—44no—
389C42—44no—
390C42—82no—
391C42—82no—
392C42—82no—
393C422−1+414yes—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23#52512yes—
399C23C23×D411yescharacteristic
400Q8—28no—
401Q8—44no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—164no—
443C2×C4#52512yes—
444C2×C4#53614yes—
445C2×C4#57814yes—
446C2×C4C22×#4112yes—
447C2×C4C2×2−1+412yes—
448C2×C4#58512yes—
449C8—44no—
450C8—44no—
451C22—22no—
452C22—24no—
453C22—41no—
454C22—41no—
455C22—42no—
456C22C2×#52511yescharacteristic
457C22C2×#53611yescharacteristic
458C22#344911yescharacteristic
459C22C2×#57812yes—
460C22#358512yes—
461C4—22no—
462C4—24no—
463C4—24no—
464C4—44no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—42no—
470C4—42no—
471C4—44no—
472C4—82no—
473C4—84no—
474C4—84no—
475C4—82no—
476C4#351514yes—
477C2—42no—
478C2#3513811yescharacteristic
479C2#6060012yes—
480C2#6087412yes—
481C2#6101311yescharacteristic
482C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33077C2semidirect
4. #30098C2semidirect
5. #30079C2semidirect
6. #29579C2semidirect
7. #29573C2semidirect
8. #29326C2semidirect
9. #29138C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#342812
#330772
#300982
#300792
#295792
#295732
#293262
#291382
#276812
#276522
#276202
#240722
#220942
#207852