Finite Group Databasebeta

← #482131

Subgroups — #482131

Order 512.

881 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
1#35179C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3C2×#1461C211yesmaximal, characteristic
4#11908C212yesmaximal, complemented
5#11907C212yesmaximal, complemented
6#11905C214yesmaximal, complemented
7#11722C214yesmaximal
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#431—21no—
11C2×#423—21no—
12C22×#101—21no—
13#2855—21no—
14#2855—21no—
15#2627C412yes—
16#2627C2212yes—
17#2582—21no—
18#2582—21no—
19#2581C412yes—
20#2581C2212yes—
21#2568C411yescharacteristic
22#2568C2211yescharacteristic
23#2567C411yescharacteristic
24#2567C2211yescharacteristic
25#2565—21no—
26#2565—21no—
27#2317—21no—
28#2310—21no—
29#2285C411yescharacteristic
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34C2×#343C2212yes—
35#1461—21no—
36#1461—21no—
37#1461C2212yes—
38#1461C2212yes—
39C2×#324—21no—
40C2×#324C2212yes—
41#1354C2218yes—
42#1353C2218yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45#533—41no—
46#533—41no—
47#533—41no—
48#533—41no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76#431—22no—
77#431—22no—
78#431—41no—
79#431—41no—
80#423—22no—
81#423—22no—
82#423—41no—
83#423—41no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86C2×C4⋊C8—41no—
87C2×#101—22no—
88C2×#101—22no—
89C2×#101—44no—
90C2×C22⋊C8—22no—
91C2×C4×C8—41no—
92#400C2×C412yes—
93#400C2×C412yes—
94#400C2×C412yes—
95#400C2312yes—
96#398—22no—
97#398—22no—
98#398—22no—
99#398—22no—
100#395C2×C412yes—
101#395C2×C412yes—
102#395C2×C412yes—
103#395C2312yes—
104#391—21no—
105#391—21no—
106#391—22no—
107#391—21no—
108#391—21no—
109#391—22no—
110#391—21no—
111#391—21no—
112#391—21no—
113#391—21no—
114#389—22no—
115#389—22no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#387—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#385—22no—
125#385—22no—
126#384—21no—
127#384—21no—
128#384—21no—
129#382—21no—
130#382—21no—
131#382C2×C412yes—
132#381—22no—
133#381—22no—
134#380—21no—
135#380C2×C412yes—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2314yes—
145C22×Q8—21no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—42no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8SD1614yes—
152C22×Q8Q1614yes—
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—44no—
166C2×M16—44no—
167C22×C8—42no—
168C22×C8—42no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173#125—42no—
174#125—42no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C4×Q8—41no—
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—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201#117—44no—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
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—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2411yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
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—21no—
252C2×C42—21no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
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—
269C4⋊C8—42no—
270C4⋊C8—42no—
271#101—44no—
272#101—44no—
273C22⋊C8—24no—
274C22⋊C8—42no—
275C4×C8—42no—
276C4×C8—42no—
277#95—22no—
278#95—22no—
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—
290#95C22×C412yes—
291#95C22×C412yes—
292C24C2×C22⋊C411yescharacteristic
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—44no—
306C2×Q8—42no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×Q8#10314yes—
311C2×Q8#10314yes—
312C2×Q8C2×SD1612yes—
313C2×Q8C2×SD1612yes—
314C2×Q8C2×Q1612yes—
315C2×Q8C2×Q1612yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—42no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4#9911yescharacteristic
340C22×C4#9911yescharacteristic
341C22×C4#9911yescharacteristic
342C22×C4#9911yescharacteristic
343C22×C4#10314yes—
344C22×C4#10314yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
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×SD1612yes—
357C22×C4C2×Q1612yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361M16—44no—
362M16—44no—
363M16—84no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—84no—
367C2×C8—84no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C42—21no—
400C42—21no—
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—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—42no—
424C42—42no—
425C42—44no—
426C23—41no—
427C23—41no—
428C23C2×#9911yescharacteristic
429C23C2×#9911yescharacteristic
430C23C2×#10312yes—
431C23#42112yes—
432C23C22×C22⋊C411yescharacteristic
433Q8—44no—
434Q8—44no—
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—82no—
440Q8—84no—
441Q8—82no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4C2×#9911yescharacteristic
498C2×C4C2×#9911yescharacteristic
499C2×C4C2×#9911yescharacteristic
500C2×C4C2×#9911yescharacteristic
501C2×C4C2×#10312yes—
502C2×C4C2×#10312yes—
503C2×C4C2×#10312yes—
504C2×C4C2×#10312yes—
505C2×C4C2×#10312yes—
506C2×C4C22×SD1611yescharacteristic
507C2×C4C22×Q1611yescharacteristic
508C8—84no—
509C8—84no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22#152012yes—
519C22#152012yes—
520C22C22×#9911yescharacteristic
521C22C22×#10311yescharacteristic
522C22C2×#42111yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4#213312yes—
544C2—21no—
545C2—21no—
546C2—22no—
547C2—22no—
548C2C2×#152011yescharacteristic
549C2#3334711yescharacteristic
550C2C2×#213311yescharacteristic
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. #11907C2semidirect
6. #11905C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#18012
C2×#14612
#119082
#119072
#119052
#117222