Finite Group Databasebeta

← #480581

Subgroups — #480581

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2#17363C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11575C212yesmaximal, complemented
5#11558C212yesmaximal, complemented
6#11544C214yesmaximal, complemented
7#11472C214yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic
10C2×#432—21no—
11C2×#431—21no—
12#2599—21no—
13#2599—21no—
14#2581—21no—
15#2536—21no—
16#2486C411yescharacteristic
17#2486C2211yescharacteristic
18#2484C2212yes—
19#2409C411yescharacteristic
20#2409C2211yescharacteristic
21#2406C412yes—
22#2399—21no—
23#2398—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×#123C412yes—
27C4×#123C2212yes—
28#2283C411yescharacteristic
29#1467—21no—
30#1466C2212yes—
31#1464—21no—
32#1464C2212yes—
33#1461—21no—
34#1460C2212yes—
35#1459—21no—
36#1459—21no—
37#1459C2212yes—
38#1459C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1326C2218yes—
42#1317C2218yes—
43C22×#41C2×C411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C2×#123C2×C412yes—
46C2×#123C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C4×#41—41no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C412yes—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78#432—22no—
79#432—22no—
80#432—41no—
81#432—41no—
82#431—22no—
83#431—22no—
84#431—41no—
85#431—41no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—41no—
91C2×C4×C8—41no—
92#400—22no—
93#398—21no—
94#398—21no—
95#398—22no—
96#397—21no—
97#397—21no—
98#395—22no—
99#395—21no—
100#395—21no—
101#394—22no—
102#393—21no—
103#393—21no—
104#391—21no—
105#391—21no—
106#389—21no—
107#389—21no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386C2×C412yes—
115#386C2×C412yes—
116#385—22no—
117#385—22no—
118#385C2×C412yes—
119#385C2312yes—
120#383—21no—
121#382—21no—
122#381C2×C412yes—
123#381C2×C412yes—
124#381C2312yes—
125#380—21no—
126#380—21no—
127#380C2×C411yescharacteristic
128#380C2×C411yescharacteristic
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2×C412yes—
135#376—21no—
136#324—22no—
137#324—22no—
138#324—22no—
139#324—22no—
140#323—22no—
141#323—22no—
142#323C2314yes—
143#323C2314yes—
144C2×#41—24no—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—44no—
153C22×Q8C22⋊C411yescharacteristic
154C22×D4C22⋊C411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C22×C8—42no—
169C22×C8—42no—
170#123—28no—
171#123—44no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
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—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22⋊C411yescharacteristic
232C2×C22⋊C4C22×C412yes—
233C2×C22⋊C4C22×C412yes—
234C2×C22⋊C4C22×C411yescharacteristic
235C2×C22⋊C4C2×D411yescharacteristic
236C2×C22⋊C4C2×D411yescharacteristic
237C2×C22⋊C4C2411yescharacteristic
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C22⋊C8—24no—
270C22⋊C8—24no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273#97—42no—
274#97—42no—
275C4×C8—42no—
276C4×C8—42no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24C23×C411yescharacteristic
293#41—48no—
294#41—48no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—48no—
304C2×Q8—44no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4#13114yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
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—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—84no—
370C2×C8—84no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C22⋊C4—24no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
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—44no—
420C42—44no—
421C42—42no—
422C42—42no—
423C42#9911yescharacteristic
424C42#9911yescharacteristic
425C42#9911yescharacteristic
426C42#9911yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23#40812yes—
440C23#41312yes—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#13112yes—
443Q8—44no—
444Q8—44no—
445Q8—48no—
446Q8—44no—
447D4—44no—
448D4—44no—
449D4—44no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—84no—
507C2×C4#40812yes—
508C2×C4C2×#9911yescharacteristic
509C2×C4C2×#9911yescharacteristic
510C2×C4C2×#9911yescharacteristic
511C2×C4C2×#9911yescharacteristic
512C2×C4C2×#9911yescharacteristic
513C2×C4C2×#9911yescharacteristic
514C8—84no—
515C8—84no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—81no—
532C22—81no—
533C22C2×#40811yescharacteristic
534C22C22×#9911yescharacteristic
535C22C2×#41311yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—22no—
541C4—41no—
542C4—44no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4#212012yes—
554C4#212312yes—
555C2—21no—
556C2—21no—
557C2—42no—
558C2—42no—
559C2—42no—
560C2#1738111yescharacteristic
561C2C2×#212011yescharacteristic
562C2C2×#212311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11575C2semidirect
5. #11558C2semidirect
6. #11544C2semidirect
7. #11472C2semidirect

Maximal subgroups

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

subgroupindex
#349712
#173632
#173492
#115752
#115582
#115442
#114722