Finite Group Databasebeta

← #463649

Subgroups — #463649

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15125C211yesmaximal, characteristic
2#14130C211yesmaximal, characteristic
3#11384C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11293C212yesmaximal, complemented
6#11228C212yesmaximal, complemented
7#11200C212yesmaximal
8#11042C211yesmaximal, characteristic
9#10730C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#393—21no—
15C2×#389C2212yes—
16C2×#385—21no—
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30#1446—21no—
31#1444—21no—
32#1442—21no—
33#1442—21no—
34#1439—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C23×Q8—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—41no—
66C2×C4⋊C8—42no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—42no—
69C2×C22⋊C8—42no—
70#393—44no—
71#389—24no—
72#389—42no—
73#385—44no—
74#380—22no—
75#380—24no—
76#380—42no—
77#380Q814yescomplemented
78C4×C22⋊C4—41no—
79C4×C22⋊C4—41no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—42no—
100#336—41no—
101C25C2×D411yescharacteristic
102C22×Q8—44no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—82no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—44no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×Q812yes—
130C23×C4#4112yes—
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—44no—
147C4⋊Q8—48no—
148C4⋊Q8—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
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—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×Q814yes—
181C2×C4⋊C4#4114yes—
182C2×C4⋊C4#4114yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4Q1614yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C4⋊C8—44no—
204C22⋊C8—48no—
205#95—24no—
206#95—41no—
207#95—41no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—82no—
215#95—82no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—21no—
221C24—21no—
222C24—22no—
223C24—22no—
224C24C22⋊Q812yes—
225C24C2×Q1612yes—
226C24AGL(1,ℤ/8)12yes—
227C24C22×D411yescharacteristic
228C2×Q8—48no—
229C2×Q8—88no—
230C2×Q8—84no—
231C2×Q8—88no—
232C2×Q8—84no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
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—88no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4C22⋊Q812yes—
306C22×C4C2×Q1612yes—
307C22×C4C2×Q1612yes—
308C22×C4AGL(1,ℤ/8)12yes—
309C22×C4C22×Q811yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4112yes—
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)12yes—
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C42−1+411yescharacteristic
318C22×C42−1+411yescharacteristic
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—88no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—88no—
356C42—42no—
357C42—42no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23#47714yes—
385C23C2×C22⋊Q811yescharacteristic
386C23#53411yescharacteristic
387C23#53611yescharacteristic
388C23#54612yes—
389C23#54612yes—
390C23C22×Q1611yescharacteristic
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23C2×AGL(1,ℤ/8)11yescharacteristic
393C23#57511yescharacteristic
394C23#57712yes—
395C23#57712yes—
396Q8—816no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—816no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4#52912yes—
458C2×C4#53812yes—
459C2×C4#53812yes—
460C2×C4#53812yes—
461C2×C4#53911yescharacteristic
462C2×C4#53911yescharacteristic
463C2×C4#54212yes—
464C2×C4#54212yes—
465C2×C4#54411yescharacteristic
466C2×C4#54411yescharacteristic
467C2×C4#55111yescharacteristic
468C2×C4#55111yescharacteristic
469C2×C4#55412yes—
470C2×C4#55412yes—
471C2×C4#55612yes—
472C2×C4#56011yescharacteristic
473C2×C4H(3,𝔽4)11yescharacteristic
474C2×C4#56212yes—
475C8—82no—
476C8—82no—
477C22—24no—
478C22—28no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22#245012yes—
489C22#262711yescharacteristic
490C22#268411yescharacteristic
491C22#269711yescharacteristic
492C22#270012yes—
493C22C2×#47711yescharacteristic
494C22#307711yescharacteristic
495C22#308412yes—
496C22#318911yescharacteristic
497C22#319211yescharacteristic
498C22#320011yescharacteristic
499C22#320011yescharacteristic
500C22#323011yescharacteristic
501C22#323411yescharacteristic
502C22#324412yes—
503C22#329912yes—
504C22#329912yes—
505C22#330612yes—
506C22#330612yes—
507C22#331012yes—
508C22#332112yes—
509C22#332412yes—
510C22#332612yes—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—84no—
517C4—84no—
518C4—84no—
519C4—162no—
520C4#252311yescharacteristic
521C4#265312yes—
522C4#274211yescharacteristic
523C4#276311yescharacteristic
524C4#278812yes—
525C4#282411yescharacteristic
526C2—44no—
527C2#1648111yescharacteristic
528C2#2641412yes—
529C2#2642412yes—
530C2#2996911yescharacteristic
531C2#2997711yescharacteristic
532C2#3126311yescharacteristic
533C2#3127111yescharacteristic
534C2#3166911yescharacteristic
535C2#3168111yescharacteristic
536C2#3174412yes—
537C2#3176612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11384C2semidirect
4. #11316C2semidirect
5. #11293C2semidirect
6. #11228C2semidirect
77. #380Q8semidirect

Maximal subgroups

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

subgroupindex
#151252
#141302
#113842
#113162
#112932
#112282
#112002
#110422
#107302