Finite Group Databasebeta

← #472203

Subgroups — #472203

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2094C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1968C211yesmaximal, characteristic
5#15199C211yesmaximal, characteristic, complemented
6#15144C211yesmaximal, characteristic
7#11323C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#11049C212yesmaximal
10#11023C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#393—22no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#384C2211yescharacteristic
27C2×#384C2212yes—
28#2094—24no—
29#2024—24no—
30#1990—24no—
31#1968—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1444—22no—
39#1444—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—42no—
49C22×SD16—42no—
50C22×SD16Q814yescomplemented
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—24no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—24no—
72C2×#106—22no—
73C2×#106—42no—
74C2×#106—42no—
75C2×#106—42no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×C4×C8—42no—
87C2×C4×C8D414yes—
88#393—48no—
89#391—22no—
90#391—22no—
91#391—42no—
92#391—42no—
93#390—24no—
94#390—22no—
95#390—22no—
96#390—42no—
97#384—22no—
98#384—22no—
99#384—42no—
100#384—42no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95D412yes—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—24no—
111#336—42no—
112#336D414yes—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—44no—
126C22×Q8—82no—
127C22×Q8C2×Q812yes—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C2×Q812yes—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4C2×D414yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C2×SD16—28no—
148C2×SD16—48no—
149C2×SD16—44no—
150C2×SD16—44no—
151C2×SD16—48no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8C2×D414yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×Q812yes—
168C22×C8#4112yes—
169C4⋊Q8—28no—
170C4⋊Q8—48no—
171C4⋊Q8—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—84no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—84no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218#106—44no—
219#106—48no—
220#103—48no—
221#103—44no—
222#102—44no—
223#102—48no—
224C4×C8—44no—
225#95—24no—
226#95—48no—
227#95—42no—
228#95—42no—
229#95—48no—
230#95SD1618yescomplemented
231C24—22no—
232C24—41no—
233C24—42no—
234C24—42no—
235C24—41no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—48no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—816no—
261C2×Q8—84no—
262C2×Q8—84no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4#12712yes—
315C22×C4#12712yes—
316C22×C4C2×SD1618yes—
317C22×C4C2×SD1614yes—
318C22×C4#13714yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4112yes—
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)12yes—
327C22×C42−1+411yescharacteristic
328C22×C42−1+411yescharacteristic
329SD16—416no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8C22⋊Q814yes—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—816no—
365C22⋊C4—816no—
366C22⋊C4—84no—
367C22⋊C4—84no—
368C42—22no—
369C42—22no—
370C42—24no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—84no—
387C23—82no—
388C23—84no—
389C23—84no—
390C23—81no—
391C23—84no—
392C23C2×#12711yescharacteristic
393C23#53512yes—
394C23#54612yes—
395C23#54711yescharacteristic
396C23D4×Q811yescharacteristic
397C23C22×SD1612yes—
398C23C2×#13712yes—
399C23#57612yes—
400C23#57712yes—
401Q8—42no—
402Q8—42no—
403Q8—44no—
404Q8—816no—
405Q8—81no—
406Q8—81no—
407Q8—82no—
408D4—42no—
409D4—42no—
410D4—44no—
411D4—81no—
412D4—81no—
413D4—82no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4#49214yes—
451C2×C4#49714yes—
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4#52511yescharacteristic
454C2×C4#53411yescharacteristic
455C2×C4#53511yescharacteristic
456C2×C4#53511yescharacteristic
457C2×C4#53512yes—
458C2×C4#53611yescharacteristic
459C2×C4#53812yes—
460C2×C4#54212yes—
461C2×C4#54212yes—
462C2×C4#54312yes—
463C2×C4#54612yes—
464C2×C4#54711yescharacteristic
465C2×C4D4×Q811yescharacteristic
466C2×C4#55412yes—
467C2×C4#55912yes—
468C2×C4#56212yes—
469C2×C4#56212yes—
470C8—24no—
471C8—42no—
472C8—44no—
473C22—88no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—164no—
478C22—161no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22#253911yescharacteristic
483C22#254211yescharacteristic
484C22#265711yescharacteristic
485C22#269212yes—
486C22#271012yes—
487C22C2×#49211yescharacteristic
488C22C2×#49711yescharacteristic
489C22#315411yescharacteristic
490C22#315511yescharacteristic
491C22#320614yes—
492C22#322414yes—
493C22#329514yes—
494C22#331614yes—
495C22#333512yes—
496C22#335712yes—
497C22Q8×SD1612yes—
498C22#338512yes—
499C4—24no—
500C4—24no—
501C4—48no—
502C4—84no—
503C4—82no—
504C4—161no—
505C4—162no—
506C4—161no—
507C4#258011yescharacteristic
508C4#262611yescharacteristic
509C4#263211yescharacteristic
510C4#269212yes—
511C4#271012yes—
512C4#281611yescharacteristic
513C2—162no—
514C2#1681411yescharacteristic
515C2#2833211yescharacteristic
516C2#2834711yescharacteristic
517C2#2853011yescharacteristic
518C2#2853911yescharacteristic
519C2#3026311yescharacteristic
520C2#3027411yescharacteristic
521C2#3152214yes—
522C2#3204814yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2024C2semidirect
3. C2×#1990C2semidirect
5. #15199C2semidirect
7. #11323C2semidirect
8. #11122C2semidirect
11. #10797C2semidirect
50. C22×SD16Q8semidirect
230. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#20942
C2×#20242
C2×#19902
C2×#19682
#151992
#151442
#113232
#111222
#110492
#110232
#107972