Finite Group Databasebeta

← #6644690

Subgroups — #6644690

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35087C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#33008C212yesmaximal
4#32444C214yesmaximal, complemented
5#32346C214yesmaximal, complemented
6#31640C214yesmaximal, complemented
7#31536C214yesmaximal, complemented
8#29641C212yesmaximal
9#29106C211yesmaximal, characteristic
10#28313C211yesmaximal, characteristic
11#28124C212yesmaximal
12#27966C211yesmaximal, characteristic
13#27905C211yesmaximal, characteristic
14#27647C212yesmaximal
15#21914C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#483C2212yes—
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#451C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2851C2211yescharacteristic
25#2655—22no—
26#2567—22no—
27#2567—22no—
28#2567C2214yes—
29#2567C2214yes—
30#2514C2212yes—
31#2506C2212yes—
32#2493—22no—
33#2473—22no—
34#2453C2212yes—
35#2446C2212yes—
36C2×#395—22no—
37#2336—22no—
38C4×C22⋊Q8C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#389—22no—
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2100C2214yes—
46#2096C2214yes—
47#2096C2212yes—
48#2084C2214yes—
49#2074C2212yes—
50#2072C2214yes—
51#2066C2214yes—
52#2058C2214yes—
53#2058C2214yes—
54#2056C2214yes—
55#2026C2212yes—
56#2024C2214yes—
57#2024C2214yes—
58#2016C2212yes—
59#1992C2212yes—
60#1971C2214yes—
61#1956C2212yes—
62#1948C2214yes—
63#1933C2214yes—
64#1927C2214yes—
65#1917C2214yes—
66#1916C2214yes—
67#1878C2214yes—
68#1878C2214yes—
69#1878C2214yes—
70#1865C2214yes—
71#1865C2214yes—
72#1844C2212yes—
73#1836C2212yes—
74#1823C2212yes—
75#1823C2212yes—
76#1823C2212yes—
77#1817C2212yes—
78#1803C2214yes—
79#1797C2212yes—
80#1787C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8C2311yescharacteristic
85C22×Q16C2314yes—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2314yes—
90C2×#125—22no—
91C2×#125C2311yescharacteristic
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8—22no—
94C2×C4×Q8—44no—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2312yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4D412yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#483—44no—
110#483—44no—
111#483—44no—
112#451—44no—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×#97C2312yes—
126C2×C4×C8C2312yes—
127#401—22no—
128#400—22no—
129#400—22no—
130#400—22no—
131#400—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#395—24no—
136#395—24no—
137#395—24no—
138#395—24no—
139#391—21no—
140#391—21no—
141#391—22no—
142#391C2314yes—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#389—24no—
150#389—24no—
151#389—24no—
152#389—24no—
153#389—22no—
154#389—22no—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#384—21no—
172#384—21no—
173#384—22no—
174#384C2314yes—
175#384C2312yes—
176#384C2312yes—
177#384C2311yescharacteristic
178#383—22no—
179#382C2311yescharacteristic
180#381—22no—
181#380—22no—
182#380C2312yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95D412yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—42no—
209C22×Q8—44no—
210C22×Q8—84no—
211C22×Q8C2412yes—
212C22×Q8C2412yes—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×Q16—28no—
223C2×Q16—44no—
224C2×Q16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—48no—
228C4⋊Q8—48no—
229C4⋊Q8—42no—
230#125—42no—
231#125—48no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C4×Q8—48no—
235C4×Q8—48no—
236C4×Q8—44no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—84no—
262C2×C4⋊C4—84no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2412yes—
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290#106—44no—
291#106—44no—
292#106—44no—
293C4⋊C8—44no—
294#103—28no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302#97—42no—
303C4×C8—42no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326#95C2×D412yes—
327C24C22×D411yescharacteristic
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—416no—
337C2×Q8—42no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—88no—
341C2×Q8—816no—
342C2×Q82−1+414yes—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4H(5,𝔽2)14yes—
380C22×C4C2511yescharacteristic
381Q16—88no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4#13718yes—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C42—24no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—41no—
425C42—42no—
426C42—41no—
427C42—48no—
428C42—44no—
429C42—82no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23#53512yes—
434C23#57714yes—
435C23C23×D411yescharacteristic
436Q8—28no—
437Q8—48no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—816no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—28no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4#53512yes—
489C2×C4C2×#13714yes—
490C2×C4C2×#13714yes—
491C2×C4C2×#13714yes—
492C2×C4C2×H(5,𝔽2)12yes—
493C2×C4C2×2−1+412yes—
494C2×C4#58512yes—
495C8—44no—
496C8—44no—
497C22—24no—
498C22—22no—
499C22—24no—
500C22—22no—
501C22#322214yes—
502C22C2×#53511yescharacteristic
503C22#348511yescharacteristic
504C22#348611yescharacteristic
505C22C22×#13712yes—
506C22C2×#57712yes—
507C4—24no—
508C4—24no—
509C4—24no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—44no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4#353514yes—
523C2—24no—
524C2#3711811yescharacteristic
525C2C2×#322211yescharacteristic
526C2#6080011yescharacteristic
527C2#6173512yes—
528C2#6183812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32444C2semidirect
5. #32346C2semidirect
6. #31640C2semidirect
7. #31536C2semidirect

Maximal subgroups

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

subgroupindex
#350872
C2×#25672
#330082
#324442
#323462
#316402
#315362
#296412
#291062
#283132
#281242
#279662
#279052
#276472
#219142