Finite Group Databasebeta

← #469238

Subgroups — #469238

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2054C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic, complemented
3#15157C212yesmaximal, complemented
4#11315C212yesmaximal, complemented
5#11231C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11121C212yesmaximal, complemented
8#11022C212yesmaximal
9#10927C212yesmaximal, complemented
10C23×D8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C2×#393—22no—
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22#2054—24no—
23#1985—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1439—22no—
32#1439—22no—
33#1439—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×Q8D411yescharacteristic
38C23×D4C2312yes—
39C22×D8—24no—
40C22×D8—22no—
41C22×D8—42no—
42C22×D8—41no—
43C22×D8—41no—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—41no—
50C2×C4×Q8—41no—
51C2×C4×Q8—41no—
52C2×C4×Q8—41no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#106—42no—
67C2×#106—41no—
68C2×C4⋊C8—41no—
69C2×C4⋊C8—41no—
70C2×C4⋊C8D414yes—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#97—22no—
81C2×#97—41no—
82C2×#97—41no—
83#393—48no—
84#390—24no—
85#390—24no—
86#390—42no—
87#390—42no—
88#390—42no—
89#390—42no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—22no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—41no—
101#336—24no—
102#336—24no—
103C25—22no—
104C22×Q8—22no—
105C22×Q8—24no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—42no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4#4114yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2411yescharacteristic
131C2×D8—48no—
132C2×D8—44no—
133C2×D8—44no—
134C2×D8—48no—
135C2×D8—44no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8#4114yes—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×Q8—44no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C2×D412yes—
210C2×C42#4112yes—
211#106—44no—
212C4⋊C8—44no—
213#102—48no—
214#102—48no—
215#102—48no—
216#97—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—82no—
221#95—82no—
222#95—82no—
223C24—24no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24C22×D411yescharacteristic
239C2×Q8—44no—
240C2×Q8—48no—
241C2×Q8—44no—
242C2×Q8—48no—
243C2×Q8—44no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4AGL(1,ℤ/8)14yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C4H(5,𝔽2)12yes—
326C22×C42−1+412yes—
327D8—416no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C4⋊C4—48no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C22⋊C4—88no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—88no—
363C22⋊C4—88no—
364C22⋊C4—88no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23C2×#12111yescharacteristic
409C23#53411yescharacteristic
410C23#53411yescharacteristic
411C23#54812yes—
412C23#55012yes—
413C23C2×AGL(1,ℤ/8)12yes—
414C23#57512yes—
415C23#57612yes—
416C23#57712yes—
417Q8—416no—
418D4—44no—
419D4—44no—
420D4—44no—
421D4—44no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—82no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#52912yes—
472C2×C4#53411yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53811yescharacteristic
475C2×C4#53811yescharacteristic
476C2×C4#53812yes—
477C2×C4#53812yes—
478C2×C4#53912yes—
479C2×C4#54212yes—
480C2×C4#54212yes—
481C2×C4#54612yes—
482C2×C4#54812yes—
483C2×C4#55111yescharacteristic
484C2×C4#55112yes—
485C2×C4#55511yescharacteristic
486C2×C4H(3,𝔽4)11yescharacteristic
487C2×C4#56212yes—
488C2×C4#56311yescharacteristic
489C8—42no—
490C8—42no—
491C8—44no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—84no—
498C22—84no—
499C22—162no—
500C22—162no—
501C22—162no—
502C22—162no—
503C22—162no—
504C22—162no—
505C22—162no—
506C22—162no—
507C22#242612yes—
508C22#261711yescharacteristic
509C22#272811yescharacteristic
510C22#273012yes—
511C22#273311yescharacteristic
512C22#305512yes—
513C22#305811yescharacteristic
514C22#306011yescharacteristic
515C22#319211yescharacteristic
516C22#319412yes—
517C22#319711yescharacteristic
518C22#319812yes—
519C22#319812yes—
520C22#336612yes—
521C22#337012yes—
522C22#337312yes—
523C22#337612yes—
524C22#339712yes—
525C22#339912yes—
526C22#340112yes—
527C22#340312yes—
528C4—42no—
529C4—42no—
530C4—44no—
531C4—82no—
532C4—162no—
533C4—162no—
534C4—162no—
535C4#257211yescharacteristic
536C4#265312yes—
537C4#268411yescharacteristic
538C4#269612yes—
539C4#273111yescharacteristic
540C4#283211yescharacteristic
541C2—164no—
542C2#1665211yescharacteristic
543C2#2530412yes—
544C2#2530912yes—
545C2#2973611yescharacteristic
546C2#2974211yescharacteristic
547C2#3255412yes—
548C2#3262212yes—
549C2#3262312yes—
550C2#3267512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1985C2semidirect
3. #15157C2semidirect
4. #11315C2semidirect
5. #11231C2semidirect
6. #11189C2semidirect
7. #11121C2semidirect
9. #10927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20542
C2×#19852
#151572
#113152
#112312
#111892
#111212
#110222
#109272