Finite Group Databasebeta

← #6854424

Subgroups — #6854424

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36573C211yesmaximal, characteristic
2Q8×#125C211yesmaximal, characteristic, complemented
3#33315C211yesmaximal, characteristic, complemented
4#30959C214yesmaximal, complemented
5#30951C214yesmaximal, complemented
6#30312C212yesmaximal, complemented
7#29214C211yesmaximal, characteristic, complemented
8#29184C212yesmaximal, complemented
9#29149C211yesmaximal, characteristic
10#28516C212yesmaximal
11#26059C214yesmaximal, complemented
12#25068C214yesmaximal, complemented
13#22226C212yesmaximal
14#21656C211yesmaximal, characteristic
15#21346C211yesmaximal, characteristic, complemented
16C2×#499C2212yes—
17C2×#499C2212yes—
18C2×#490C2211yescharacteristic
19C2×#478C2214yes—
20C2×#478C2214yes—
21#2859C2212yes—
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2212yes—
25#2580—22no—
26#2580C2212yes—
27#2552C2212yes—
28#2552C2212yes—
29#2552C2212yes—
30#2507—22no—
31#2507—22no—
32#2507—22no—
33#2454—22no—
34#2454—22no—
35#2454C2214yes—
36Q8×C4⋊C4—22no—
37Q8×C4⋊C4C2212yes—
38#2349—22no—
39#2349C2212yes—
40C4×#127C2212yes—
41C4×#127C2211yescharacteristic
42C4×#125C2212yes—
43C4×#125C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45#2094C2214yes—
46#2093C2214yes—
47#2088C2214yes—
48#2086C2214yes—
49#2084C2214yes—
50#2083C2214yes—
51#2059C2214yes—
52#2059C2214yes—
53#2059C2214yes—
54#2025C2212yes—
55#2025C2212yes—
56#1992C2211yescharacteristic
57#1992C2212yes—
58#1985C2212yes—
59#1961C2211yescharacteristic
60#1961C2212yes—
61#1949C2214yes—
62#1940C2212yes—
63#1934C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1921C2214yes—
67#1920C2214yes—
68#1916C2214yes—
69#1864C2214yes—
70#1862C2214yes—
71#1847C2212yes—
72#1847C2212yes—
73#1847C2212yes—
74#1847C2214yes—
75#1836C2212yes—
76#1836C2211yescharacteristic
77#1802C2214yes—
78#1767C2212yes—
79#1767C2211yescharacteristic
80#1762C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#125—22no—
85C2×#125C2311yescharacteristic
86C2×#125C2312yes—
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×#125C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×C4×Q8—22no—
92C2×C4×Q8D412yes—
93C2×C4×Q8C2311yescharacteristic
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4C2312yes—
98C2×C4×D4C2314yes—
99#499—44no—
100#499—44no—
101#490Q818yescomplemented
102#478—48no—
103#478—48no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2314yes—
111C2×C4⋊C8C2312yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×#97C2311yescharacteristic
119C2×#97C2311yescharacteristic
120C2×C4×C8C2312yes—
121#399—24no—
122#399—22no—
123#398—22no—
124#398—22no—
125#398—22no—
126#398—22no—
127#398—22no—
128#398—22no—
129#396—24no—
130#393—22no—
131#390—22no—
132#390—22no—
133#390—22no—
134#390C2314yes—
135#390C2312yes—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#389C2314yes—
156#389C2314yes—
157#387—22no—
158#387—22no—
159#387—22no—
160#385—24no—
161#385—24no—
162#385—24no—
163#384—22no—
164#384—22no—
165#384—22no—
166#384C2311yescharacteristic
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#384C2312yes—
171#382—22no—
172#382C2312yes—
173#382C2311yescharacteristic
174#382C2312yes—
175#380—24no—
176#380—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2311yescharacteristic
187C4×C4⋊C4C2312yes—
188C2×#95—22no—
189C43C2311yescharacteristic
190C43C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C22×Q8C2×D411yescharacteristic
202C22×D4—22no—
203C22×D4C2412yes—
204C23×C4—24no—
205C23×C4—22no—
206C22×C8C2414yes—
207C22×C8C2412yes—
208C22×C8C2412yes—
209#127—22no—
210#127C2×Q814yes—
211#125—22no—
212#125—44no—
213#125—44no—
214#125—42no—
215#125—42no—
216#125—42no—
217#125—42no—
218#125—84no—
219#125C2×Q814yes—
220C4×Q8—24no—
221C4×Q8—44no—
222C4×Q8—48no—
223C4×D4—28no—
224C4×D4—48no—
225C4×D4—44no—
226C4×D4—44no—
227C4×D4—84no—
228C4×D4—84no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2412yes—
277C2×C42C2412yes—
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281#107—28no—
282#107—44no—
283#107—44no—
284#106—28no—
285#106—44no—
286#106—44no—
287C4⋊C8—28no—
288C4⋊C8—24no—
289#102—28no—
290#102—44no—
291#102—44no—
292#102C2×Q8116yes—
293#97—24no—
294#97—22no—
295#97—24no—
296#97C2×Q814yes—
297C4×C8—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95C2×D411yescharacteristic
312#95C2×D412yes—
313#95C2×D411yescharacteristic
314#95C2×D412yes—
315C24—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×D4—28no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4C22×Q814yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—48no—
332C22×C4—44no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C2511yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8C22×Q814yes—
353C2×C8H(5,𝔽2)14yes—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4C22×Q814yes—
389C22⋊C4—44no—
390C22⋊C4—48no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—24no—
395C42—22no—
396C42—24no—
397C42—24no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—44no—
404C42—44no—
405C42—44no—
406C42C22×Q812yes—
407C422−1+414yes—
408C422−1+412yes—
409C23—24no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23C23×D411yescharacteristic
415Q8—28no—
416D4—28no—
417D4—416no—
418D4—44no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—164no—
455C2×C4—168no—
456C2×C4#53612yes—
457C2×C4#53612yes—
458C2×C4D4×Q814yes—
459C2×C4#57514yes—
460C2×C4C23×Q811yescharacteristic
461C2×C4C2×H(5,𝔽2)12yes—
462C2×C4C2×2−1+412yes—
463C2×C4C2×2−1+411yescharacteristic
464C8—24no—
465C8—24no—
466C8—44no—
467C22—48no—
468C22—44no—
469C22—82no—
470C22—82no—
471C22—84no—
472C22C2×#53611yescharacteristic
473C22C2×D4×Q811yescharacteristic
474C22#349811yescharacteristic
475C22C2×#57512yes—
476C22#358512yes—
477C4—24no—
478C4—22no—
479C4—24no—
480C4—22no—
481C4—44no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4#350312yes—
492C4#354412yes—
493C2—84no—
494C2#3961711yescharacteristic
495C2#6086111yescharacteristic
496C2#6088311yescharacteristic
497C2#6136712yes—
498C2#6245112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#125C2semidirect
3. #33315C2semidirect
4. #30959C2semidirect
5. #30951C2semidirect
6. #30312C2semidirect
7. #29214C2semidirect
8. #29184C2semidirect
11. #26059C2semidirect
12. #25068C2semidirect
15. #21346C2semidirect
101. #490Q8semidirect

Maximal subgroups

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

subgroupindex
#365732
Q8×#1252
#333152
#309592
#309512
#303122
#292142
#291842
#291492
#285162
#260592
#250682
#222262
#216562
#213462