Finite Group Databasebeta

← #482667

Subgroups — #482667

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#11908C212yesmaximal, complemented
5#11883C212yesmaximal, complemented
6#11831C214yesmaximal, complemented
7#11768C214yesmaximal
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C2×#423—21no—
12C22×#101—21no—
13#2584—21no—
14#2584—21no—
15#2583C412yes—
16#2583C2212yes—
17#2567C411yescharacteristic
18#2567C2211yescharacteristic
19#2546C411yescharacteristic
20#2546C2211yescharacteristic
21#2502—21no—
22#2434—21no—
23#2399C412yes—
24#2399C2212yes—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27#2296—21no—
28#2296—21no—
29C4×#117C411yescharacteristic
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34C2×#343C2212yes—
35#1467—21no—
36#1467C2212yes—
37#1463—21no—
38#1463C2212yes—
39#1461—21no—
40#1461C2212yes—
41#1354C2218yes—
42#1346C2218yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45#533—41no—
46#533—41no—
47#533—41no—
48#533—41no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78#432—22no—
79#432—41no—
80#431—22no—
81#431—41no—
82#423—22no—
83#423—22no—
84#423—41no—
85#423—41no—
86#423—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×#101—22no—
90C2×#101—22no—
91C2×#101—44no—
92#407—22no—
93C4×M16—41no—
94#398—22no—
95#398—22no—
96#398C2×C412yes—
97#398C2×C412yes—
98#398C2×C412yes—
99#398C2312yes—
100#395—22no—
101#395—22no—
102#395C2×C412yes—
103#395C2×C412yes—
104#395C2×C412yes—
105#395C2312yes—
106#391—21no—
107#391—21no—
108#391—21no—
109#391—21no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126#384—21no—
127#384—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4C2×C412yes—
137#376C2×C412yes—
138C43—21no—
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2314yes—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—42no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22×C412yes—
154C22×Q8C2×D411yescharacteristic
155C22×Q8C2×D411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—44no—
167C2×M16—44no—
168C22×C8—42no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—44no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176#125—42no—
177#125—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202#117—28no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2411yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C4⋊C8—42no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278#101—44no—
279#101—44no—
280C22⋊C8—24no—
281C22⋊C8—42no—
282#97—42no—
283C4×C8—42no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95C22×C412yes—
294#95C22×C412yes—
295C24C2×C22⋊C411yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—82no—
317C2×Q8AGL(1,ℤ/8)12yes—
318C2×Q8#13712yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4#9911yescharacteristic
347C22×C4#9911yescharacteristic
348C22×C4#9911yescharacteristic
349C22×C4#9911yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364M16—44no—
365M16—44no—
366M16—84no—
367M16—84no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—84no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C22⋊C4—28no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—28no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—82no—
435C23—21no—
436C23—22no—
437C23—21no—
438C23—41no—
439C23C2×#9911yescharacteristic
440C23C2×#9911yescharacteristic
441C23C22×C22⋊C411yescharacteristic
442Q8—42no—
443Q8—42no—
444Q8—44no—
445Q8—44no—
446Q8—42no—
447Q8—42no—
448Q8—84no—
449Q8—84no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#35614yes—
508C2×C4C2×#9911yescharacteristic
509C2×C4C2×#9911yescharacteristic
510C2×C4C2×#9911yescharacteristic
511C2×C4C2×#9911yescharacteristic
512C2×C4#41712yes—
513C2×C4#41712yes—
514C2×C4#42112yes—
515C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C8—84no—
518C8—84no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22C2×#35612yes—
529C22C22×#9911yescharacteristic
530C22C2×#41711yescharacteristic
531C22C2×#42111yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—44no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—82no—
550C4—82no—
551C4C2×#35612yes—
552C4C2×#35612yes—
553C2—21no—
554C2—21no—
555C2—22no—
556C2—41no—
557C2#1759211yescharacteristic
558C2#3334711yescharacteristic
559C2C22×#35611yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11908C2semidirect
5. #11883C2semidirect
6. #11831C2semidirect

Maximal subgroups

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

subgroupindex
#349532
C2×#18012
#173652
#119082
#118832
#118312
#117682