Finite Group Databasebeta

← #472207

Subgroups — #472207

Order 512.

1366 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
1C2×#2096C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1968C211yesmaximal, characteristic
5#15191C211yesmaximal, characteristic, complemented
6#15155C211yesmaximal, characteristic
7#11323C212yesmaximal, complemented
8#11121C212yesmaximal
9#11050C212yesmaximal, complemented
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×#399—22no—
22C2×#399—22no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#384C2211yescharacteristic
28C2×#384C2212yes—
29#2096—24no—
30#2023—24no—
31#1990—24no—
32#1968—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1446—22no—
40#1442—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—24no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—24no—
72C2×#106—22no—
73C2×#106—42no—
74C2×#106—42no—
75C2×#106—42no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×C4×C8—42no—
87C2×C4×C8D414yes—
88#399—48no—
89#399—48no—
90#391—24no—
91#391—22no—
92#391—22no—
93#391—42no—
94#390—22no—
95#390—22no—
96#390—42no—
97#390—42no—
98#384—22no—
99#384—22no—
100#384—42no—
101#384—42no—
102C2×#95—21no—
103C2×#95—24no—
104C2×#95—21no—
105C2×#95—24no—
106C2×#95D412yes—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—42no—
113#336D414yes—
114C25—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—44no—
127C22×Q8—82no—
128C22×Q8C2×Q812yes—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C2×Q812yes—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4C2×D414yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C2×SD16—28no—
149C2×SD16—48no—
150C2×SD16—44no—
151C2×SD16—44no—
152C2×SD16—48no—
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—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8C2×D414yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×Q812yes—
169C22×C8#4112yes—
170C4⋊Q8—28no—
171C4⋊Q8—48no—
172C4⋊Q8—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
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—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219#106—44no—
220#106—48no—
221#103—44no—
222#103—48no—
223#102—48no—
224#102—44no—
225C4×C8—44no—
226#95—24no—
227#95—48no—
228#95—42no—
229#95—42no—
230#95—84no—
231#95SD1618yescomplemented
232C24—22no—
233C24—41no—
234C24—42no—
235C24—42no—
236C24—41no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—48no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—816no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
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—42no—
315C22×C4#12712yes—
316C22×C4#12712yes—
317C22×C4C2×SD1618yes—
318C22×C4C2×SD1614yes—
319C22×C4AGL(1,ℤ/8)14yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×Q811yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4112yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)12yes—
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+411yescharacteristic
329C22×C42−1+411yescharacteristic
330SD16—416no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8C22⋊Q814yes—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—816no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C22⋊C4—816no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—22no—
370C42—22no—
371C42—24no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—84no—
388C23—82no—
389C23—84no—
390C23—84no—
391C23—84no—
392C23—81no—
393C23C2×#12711yescharacteristic
394C23#53512yes—
395C23#54612yes—
396C23#54711yescharacteristic
397C23D4×Q811yescharacteristic
398C23C22×SD1612yes—
399C23C2×AGL(1,ℤ/8)12yes—
400C23#57712yes—
401C23#57812yes—
402Q8—42no—
403Q8—42no—
404Q8—44no—
405Q8—816no—
406Q8—81no—
407Q8—81no—
408Q8—82no—
409D4—42no—
410D4—42no—
411D4—44no—
412D4—81no—
413D4—81no—
414D4—82no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—84no—
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—81no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4#49214yes—
452C2×C4#49614yes—
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4#52511yescharacteristic
455C2×C4#53411yescharacteristic
456C2×C4#53511yescharacteristic
457C2×C4#53511yescharacteristic
458C2×C4#53512yes—
459C2×C4#53611yescharacteristic
460C2×C4#53812yes—
461C2×C4#54212yes—
462C2×C4#54212yes—
463C2×C4#54312yes—
464C2×C4#54612yes—
465C2×C4#54711yescharacteristic
466C2×C4D4×Q811yescharacteristic
467C2×C4#55412yes—
468C2×C4#55912yes—
469C2×C4#56212yes—
470C2×C4#56212yes—
471C8—24no—
472C8—42no—
473C8—44no—
474C22—88no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—164no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22#253911yescharacteristic
484C22#254211yescharacteristic
485C22#265711yescharacteristic
486C22#269212yes—
487C22#271012yes—
488C22C2×#49211yescharacteristic
489C22C2×#49611yescharacteristic
490C22#315411yescharacteristic
491C22#315611yescharacteristic
492C22#320814yes—
493C22#322514yes—
494C22#329814yes—
495C22#330614yes—
496C22#333512yes—
497C22#335312yes—
498C22Q8×SD1612yes—
499C22#338612yes—
500C4—24no—
501C4—24no—
502C4—48no—
503C4—82no—
504C4—84no—
505C4—161no—
506C4—162no—
507C4—161no—
508C4#258011yescharacteristic
509C4#262611yescharacteristic
510C4#263211yescharacteristic
511C4#269212yes—
512C4#271012yes—
513C4#281611yescharacteristic
514C2—162no—
515C2#1681411yescharacteristic
516C2#2833611yescharacteristic
517C2#2834311yescharacteristic
518C2#2853011yescharacteristic
519C2#2854011yescharacteristic
520C2#3025911yescharacteristic
521C2#3027511yescharacteristic
522C2#3152014yes—
523C2#3205014yes—
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
1. C2×#2096C2semidirect
3. C2×#1990C2semidirect
5. #15191C2semidirect
7. #11323C2semidirect
9. #11050C2semidirect
11. #10797C2semidirect
50. C22×SD16Q8semidirect
231. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#20232
C2×#19902
C2×#19682
#151912
#151552
#113232
#111212
#110502
#110232
#107972