Finite Group Databasebeta

← #6506781

Subgroups — #6506781

Order 512.

1608 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#37307C211yesmaximal, characteristic
2#35574C211yesmaximal, characteristic
3#29307C212yesmaximal, complemented
4#29299C212yesmaximal, complemented
5#28783C214yesmaximal, complemented
6#28774C214yesmaximal, complemented
7#27957C212yesmaximal, complemented
8#26072C214yesmaximal, complemented
9#26060C214yesmaximal, complemented
10#25811C212yesmaximal, complemented
11#21946C212yesmaximal
12#20848C212yesmaximal
13#20530C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C2×#498C2214yes—
17C2×#428C2212yes—
18C22×#106C2212yes—
19C2×#408C2212yes—
20#2857C2212yes—
21#2806—22no—
22#2748—22no—
23#2732—22no—
24#2683—22no—
25#2629C2212yes—
26#2579—22no—
27#2452—22no—
28#2451—22no—
29#2441C2214yes—
30#2440C2214yes—
31#2440C2214yes—
32#2439C2214yes—
33#2409—22no—
34#2317C2211yescharacteristic
35#2316C2211yescharacteristic
36#2311C2212yes—
37#2311C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic
40#2085C2214yes—
41#2084C2214yes—
42#2084C2212yes—
43#2084C2214yes—
44#2083C2212yes—
45#2059C2214yes—
46#2058C2214yes—
47#2058C2214yes—
48#2058C2212yes—
49#2057C2212yes—
50#1992C2212yes—
51#1991C2212yes—
52#1990C2214yes—
53#1944C2214yes—
54#1942C2214yes—
55#1941C2214yes—
56#1941C2214yes—
57#1854C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1854C2214yes—
61#1853C2214yes—
62#1853C2214yes—
63#1853C2214yes—
64#1853C2214yes—
65#1847C2212yes—
66#1844C2212yes—
67#1809C2214yes—
68#1807C2214yes—
69#1753C2214yes—
70C8×C22⋊C4C2214yes—
71#1750C2212yes—
72C4×C22⋊C8C2212yes—
73#1736C2212yes—
74#1735C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C22×M16C2312yes—
78C23×C8C2312yes—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—44no—
81C2×C4⋊Q8C2314yes—
82C2×C4⋊Q8C2312yes—
83C2×#125C2312yes—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87#520—44no—
88C2×C4×Q8—21no—
89C2×C4×D4—22no—
90C2×C4×D4—21no—
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C2×#117C2312yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C42C2311yescharacteristic
102#498—48no—
103#428—44no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106—44no—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113#408Q8116yescomplemented
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2312yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2314yes—
119#399—24no—
120#398—24no—
121#398—24no—
122#398—24no—
123#396—24no—
124#396—24no—
125#396—24no—
126#395—24no—
127#395—24no—
128#393—24no—
129#391—21no—
130#391—22no—
131#391—22no—
132#391—24no—
133#391D412yes—
134#390—22no—
135#390—21no—
136#390D412yes—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—24no—
141#389C2312yes—
142#389C2314yes—
143#389C2312yes—
144#389C2314yes—
145#389C2314yes—
146#387—24no—
147#387—24no—
148#386—22no—
149#386—22no—
150#386—22no—
151#386D412yes—
152#386D412yes—
153#385—21no—
154#385—21no—
155#385—22no—
156#385—24no—
157#385—24no—
158#384C2312yes—
159#384C2312yes—
160#384C2312yes—
161#382C2312yes—
162#380—22no—
163#380—22no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C22⋊C4C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#336C2312yes—
178#336C2312yes—
179#336C2314yes—
180C2×#41—42no—
181C2×#41—41no—
182C2×#41—41no—
183C2×#41C2×Q818yes—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—44no—
187C22×Q8—84no—
188C22×Q8C2×D411yescharacteristic
189C22×D4C2×D412yes—
190C22×D4C2×D411yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C2×M16—42no—
200C2×M16—42no—
201C2×M16C2×Q818yescomplemented
202C22×C8—42no—
203C22×C8—42no—
204C22×C8C2×Q818yescomplemented
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—28no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4⋊Q8—48no—
211#125—42no—
212#125—42no—
213C22⋊Q8—88no—
214C22⋊Q8—88no—
215C22⋊Q8—84no—
216C4×Q8—44no—
217C4×D4—44no—
218C4×D4—48no—
219#117—24no—
220#117—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—42no—
225#117—42no—
226#117—42no—
227#117—42no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2411yescharacteristic
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2411yescharacteristic
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C22⋊C4C2×D412yes—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—42no—
274C2×C42C2×D412yes—
275C2×C42C2412yes—
276C2×C42C2412yes—
277#107—44no—
278#107—44no—
279#106—28no—
280#106—28no—
281#106—44no—
282#106—44no—
283#106—44no—
284#106—44no—
285C22⋊C8C2×Q8116yes—
286C22⋊C8C2×Q8116yescomplemented
287#97—42no—
288C4×C8—42no—
289C4×C8—44no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95C2×D414yes—
300#95C2×D414yes—
301C24C22×D412yes—
302C24C22×D411yescharacteristic
303#41—48no—
304#41—48no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—44no—
310C2×Q8—84no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—816no—
314C2×Q8C4⋊Q814yes—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4C4⋊Q818yes—
320C2×D4C4⋊Q814yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4C22×D414yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D412yes—
352C22×C4C22×Q814yes—
353C22×C4C22×Q814yes—
354C22×C4C22×Q814yes—
355C22×C4C2511yescharacteristic
356M16—48no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—48no—
362C2×C8C22×Q818yes—
363C2×C8C22×Q818yes—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—82no—
391C4⋊C4H(5,𝔽2)14yes—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—82no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—24no—
403C42—22no—
404C42—22no—
405C42—42no—
406C42—44no—
407C42—42no—
408C42—42no—
409C42—82no—
410C23—21no—
411C23—22no—
412C23—21no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23C2×C4⋊Q812yes—
419C23C2×C4⋊Q814yes—
420C23C23×D411yescharacteristic
421Q8—44no—
422Q8—816no—
423Q8—164no—
424D4—44no—
425D4—48no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—164no—
465C2×C4—164no—
466C2×C4C2×C4⋊Q814yes—
467C2×C4C2×C4⋊Q812yes—
468C2×C4#57714yes—
469C2×C4#57714yes—
470C2×C4C23×Q812yes—
471C2×C4C2×H(5,𝔽2)12yes—
472C2×C4#58511yescharacteristic
473C2×C4#58511yescharacteristic
474C8—28no—
475C8—28no—
476C22—24no—
477C22—22no—
478C22—28no—
479C22—24no—
480C22—42no—
481C22—44no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—42no—
486C22C22×C4⋊Q811yescharacteristic
487C22#348411yescharacteristic
488C22#348511yescharacteristic
489C22C2×#57712yes—
490C22C2×#57712yes—
491C4—22no—
492C4—24no—
493C4—24no—
494C4—24no—
495C4—44no—
496C4—44no—
497C4—82no—
498C4—84no—
499C4—82no—
500C4—81no—
501C4—81no—
502C4—84no—
503C4#350112yes—
504C4#350112yes—
505C2—42no—
506C2—44no—
507C2#3539111yescharacteristic
508C2#6066512yes—
509C2#6163111yescharacteristic
510C2#6163311yescharacteristic
511C2#6179212yes—
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. #29307C2semidirect
4. #29299C2semidirect
5. #28783C2semidirect
6. #28774C2semidirect
7. #27957C2semidirect
8. #26072C2semidirect
9. #26060C2semidirect
10. #25811C2semidirect
113. #408Q8semidirect
201. C2×M16C2×Q8semidirect
204. C22×C8C2×Q8semidirect
286. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#373072
#355742
#293072
#292992
#287832
#287742
#279572
#260722
#260602
#258112
#219462
#208482
#205302