Finite Group Databasebeta

← #6471019

Subgroups — #6471019

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic
2#34266C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#32954C212yesmaximal, complemented
5#29970C212yesmaximal, complemented
6#29963C212yesmaximal, complemented
7#26434C214yesmaximal, complemented
8#26417C214yesmaximal, complemented
9#26353C214yesmaximal, complemented
10#26340C214yesmaximal, complemented
11#21937C211yesmaximal, characteristic
12#21771C212yesmaximal
13#20848C212yesmaximal
14C2×#536C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#475C2214yes—
18C2×#428C2212yes—
19C22×#106C2212yes—
20C2×#408C2212yes—
21#2857C2212yes—
22#2806—22no—
23#2748—22no—
24#2732—22no—
25#2683—22no—
26#2629C2212yes—
27#2441C2214yes—
28#2440C2214yes—
29#2367—22no—
30#2365—22no—
31#2362C2214yes—
32#2360C2214yes—
33#2322—22no—
34#2311C2212yes—
35#2310C2211yescharacteristic
36C4×#123—22no—
37#2297C2211yescharacteristic
38#2296C2211yescharacteristic
39#2274C2211yescharacteristic
40#2100C2212yes—
41#2100C2212yes—
42#2096C2212yes—
43#2095C2212yes—
44#2088C2214yes—
45#2086C2214yes—
46#2068C2214yes—
47#2066C2214yes—
48#2058C2212yes—
49#2057C2212yes—
50#2026C2212yes—
51#2026C2212yes—
52#1989C2214yes—
53#1948C2214yes—
54#1920C2214yes—
55#1919C2214yes—
56#1894C2214yes—
57#1892C2214yes—
58#1873C2214yes—
59#1872C2214yes—
60#1865C2214yes—
61#1865C2214yes—
62#1854C2214yes—
63#1854C2214yes—
64#1853C2214yes—
65#1853C2214yes—
66#1851C2214yes—
67#1850C2214yes—
68#1825C2214yes—
69#1803C2214yes—
70#1798C2212yes—
71#1797C2212yes—
72#1787C2214yes—
73#1736C2212yes—
74#1735C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C22×M16C2312yes—
78C23×C8C2312yes—
79#536—24no—
80#536Q818yescomplemented
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2312yes—
83C2×#125C2312yes—
84C2×#124—21no—
85C2×#123—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2314yes—
89#520—44no—
90C2×C4×Q8—21no—
91C2×C4×Q8—22no—
92C2×C4×Q8—44no—
93C2×C4×Q8C2314yes—
94C2×C4×D4—22no—
95C2×C4×D4—21no—
96C2×#117C2312yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107#475—48no—
108#428—44no—
109C2×#107C2314yes—
110C2×#106—44no—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120#408—28no—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123#399—24no—
124#398—24no—
125#398—24no—
126#396—24no—
127#396—24no—
128#395—24no—
129#391—22no—
130#391—22no—
131#391—22no—
132#391D412yes—
133#391C2312yes—
134#391C2312yes—
135#390—22no—
136#390D412yes—
137#389—22no—
138#389—24no—
139#389—24no—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2314yes—
144#388—24no—
145#388—24no—
146#387—24no—
147#387—24no—
148#387—24no—
149#386—22no—
150#386—22no—
151#386D412yes—
152#386D412yes—
153#385—21no—
154#385—21no—
155#385—24no—
156#385—24no—
157#384C2312yes—
158#384C2312yes—
159#381—22no—
160#380—22no—
161#380C2314yes—
162#380C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#336C2312yes—
179#336C2312yes—
180#336C2312yes—
181#336C2312yes—
182C2×#41—42no—
183C2×#41—42no—
184C2×#41C2×Q814yes—
185C2×#41#4114yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2×Q814yes—
193C22×Q8C2412yes—
194C22×D4C2×D412yes—
195C22×D4C2×D411yescharacteristic
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C2×M16—24no—
205C2×M16—42no—
206C2×M16—42no—
207C22×C8—24no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8C2414yes—
211C22×C8C2414yes—
212C4⋊Q8—28no—
213C4⋊Q8—44no—
214#125—44no—
215#124—28no—
216#123—216no—
217C22⋊Q8—28no—
218C22⋊Q8—88no—
219C22⋊Q8—84no—
220C22⋊Q8C2×Q8116yes—
221C4×Q8—28no—
222C4×Q8—44no—
223C4×Q8—48no—
224C4×Q8—44no—
225C4×D4—44no—
226C4×D4—48no—
227#117—22no—
228#117—24no—
229#117—42no—
230#117—42no—
231#117—42no—
232#117—42no—
233#117—42no—
234#117C2×Q814yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2411yescharacteristic
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2411yescharacteristic
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4C2×D414yes—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D414yes—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42C2×D412yes—
281C2×C42C2412yes—
282C2×C42C2412yes—
283#107—44no—
284#106—28no—
285#106—28no—
286#106—44no—
287C4⋊C8—44no—
288C4⋊C8—44no—
289#103—44no—
290#103—44no—
291#103—44no—
292#103—44no—
293C22⋊C8—28no—
294C22⋊C8—28no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C2×D414yes—
305#95C2×D414yes—
306C24C22×D412yes—
307C24C22×D411yescharacteristic
308#41—48no—
309#41—48no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—48no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—48no—
320C2×Q8—44no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8C22⋊Q814yes—
324C2×Q8C22×Q814yes—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4C22⋊Q818yes—
330C2×D4C22⋊Q814yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4C22×D414yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×Q812yes—
363C22×C4C22×Q814yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4114yes—
366C22×C4C2511yescharacteristic
367M16—48no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—48no—
375C4⋊C4—24no—
376C4⋊C4—216no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—82no—
398C4⋊C4C22×Q814yes—
399C4⋊C4H(5,𝔽2)14yes—
400C22⋊C4—24no—
401C22⋊C4—216no—
402C22⋊C4—28no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—82no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—42no—
414C42—48no—
415C42—44no—
416C42—42no—
417C42—84no—
418C42—82no—
419C23—24no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—21no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23C2×C22⋊Q814yes—
428C23C2×C22⋊Q812yes—
429C23C23×D411yescharacteristic
430Q8—28no—
431Q8—44no—
432Q8—48no—
433Q8—44no—
434Q8—164no—
435D4—48no—
436D4—44no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4—164no—
478C2×C4—164no—
479C2×C4C2×C22⋊Q812yes—
480C2×C4C2×C22⋊Q814yes—
481C2×C4#57714yes—
482C2×C4C23×Q811yescharacteristic
483C2×C4C22×#4111yescharacteristic
484C2×C4C2×H(5,𝔽2)12yes—
485C2×C4#58512yes—
486C8—44no—
487C8—44no—
488C22—28no—
489C22—24no—
490C22—22no—
491C22—24no—
492C22—44no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—41no—
497C22—42no—
498C22C22×C22⋊Q811yescharacteristic
499C22#348411yescharacteristic
500C22#348511yescharacteristic
501C22C2×#57712yes—
502C22#358612yes—
503C4—24no—
504C4—22no—
505C4—24no—
506C4—44no—
507C4—44no—
508C4—44no—
509C4—42no—
510C4—44no—
511C4—84no—
512C4—81no—
513C4—84no—
514C4—81no—
515C4#345912yes—
516C4#350112yes—
517C2—42no—
518C2—44no—
519C2#3498511yescharacteristic
520C2#6049712yes—
521C2#6163311yescharacteristic
522C2#6164911yescharacteristic
523C2#6180912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33157C2semidirect
4. #32954C2semidirect
5. #29970C2semidirect
6. #29963C2semidirect
7. #26434C2semidirect
8. #26417C2semidirect
9. #26353C2semidirect
10. #26340C2semidirect
80. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#373072
#342662
#331572
#329542
#299702
#299632
#264342
#264172
#263532
#263402
#219372
#217712
#208482