Finite Group Databasebeta

← #482432

Subgroups — #482432

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#11902C214yesmaximal, complemented
5#11813C212yesmaximal, complemented
6#11812C212yesmaximal, complemented
7#11784C214yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11#2851—21no—
12#2851—21no—
13#2628C411yescharacteristic, complemented
14#2628C411yescharacteristic, complemented
15#2628C2211yescharacteristic
16#2628C2211yescharacteristic
17#2581—21no—
18#2581C412yescomplemented
19#2580—21no—
20#2580C2212yes—
21#2564—21no—
22#2564—21no—
23#2564C412yescomplemented
24#2564C2212yes—
25#2316—21no—
26#2311—21no—
27#2283—21no—
28C2×#382C411yescharacteristic, complemented
29#1820C2212yes—
30#1799—21no—
31#1799C2212yes—
32#1782—21no—
33C4×#99—21no—
34C4×#99C2212yes—
35#1465—21no—
36#1465C2212yes—
37#1464—21no—
38#1464C2212yes—
39#1461—21no—
40#1460C2212yes—
41#1352C2218yes—
42#1351C2218yes—
43C22×#41D411yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8C2×C412yes—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C4×#41—41no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117D411yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75#432—22no—
76#432—41no—
77#431—22no—
78#431—41no—
79#423—41no—
80C2×#99—22no—
81C2×#99—22no—
82#407—22no—
83C4×M16—41no—
84#400—21no—
85#400—21no—
86#400C2×C412yes—
87#400C2×C412yes—
88#399—21no—
89#399—21no—
90#399C2×C412yes—
91#399C2312yes—
92#395—21no—
93#395—21no—
94#395C2×C412yes—
95#395C2×C412yes—
96#393—21no—
97#393—21no—
98#393C2×C412yes—
99#393C2312yes—
100#391—22no—
101#390—22no—
102#389—22no—
103#389—22no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#383—41no—
119#382—21no—
120#382—21no—
121#382—41no—
122#382C2×C412yes—
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#381—22no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—22no—
131#380—22no—
132#380—22no—
133C4×C4⋊C4—41no—
134C4×C22⋊C4—22no—
135#376—41no—
136C2×#95—21no—
137C2×#95C2×C412yes—
138#342—22no—
139#342C2314yes—
140#324—22no—
141#324—22no—
142#323—22no—
143#323C2314yes—
144C2×#41—24no—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C412yes—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C2×D411yescharacteristic
153C22×D4C2×D411yescharacteristic
154C23×C4—22no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—42no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169#125—41no—
170#125—41no—
171#125—41no—
172#125—41no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—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—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—42no—
204#117—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C22⋊C4C22⋊C411yescharacteristic
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—42no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C4⋊C8—42no—
263C4⋊C8—42no—
264#99—44no—
265#99—44no—
266C22⋊C8—24no—
267C22⋊C8—42no—
268#97—42no—
269C4×C8—42no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95C22×C412yes—
287#95C22×C412yes—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C22×D411yescharacteristic
291#41—48no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4#13714yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
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—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—82no—
346C22×C4—82no—
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×C22⋊C411yescharacteristic
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359M16—84no—
360C2×C8—44no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C42—21no—
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—44no—
417C42—84no—
418C42#9911yescharacteristic
419C42#9911yescharacteristic
420C42#9911yescharacteristic
421C42#9911yescharacteristic
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23#41912yes—
436C23C22×C22⋊C411yescharacteristic
437C23C2×#13712yes—
438Q8—44no—
439Q8—84no—
440Q8—84no—
441D4—44no—
442D4—44no—
443D4—44no—
444C2×C4—24no—
445C2×C4—24no—
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—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
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—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—164no—
501C2×C4C2×#9911yescharacteristic
502C2×C4C2×#9911yescharacteristic
503C2×C4C2×#9911yescharacteristic
504C2×C4C2×#9911yescharacteristic
505C2×C4C2×#9911yescharacteristic
506C2×C4C2×#9911yescharacteristic
507C2×C4#41912yes—
508C2×C4#42112yes—
509C8—84no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22C22×#9911yescharacteristic
528C22C2×#41911yescharacteristic
529C22C2×#42111yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—164no—
550C2—21no—
551C2—21no—
552C2—42no—
553C2—42no—
554C2—81no—
555C2#1757811yescharacteristic
556C2#3334711yescharacteristic
557C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #11902C2semidirect
5. #11813C2semidirect
6. #11812C2semidirect
7. #11784C2semidirect
8. C2×#533C4semidirect
13. #2628C4semidirect
14. #2628C4semidirect
18. #2581C4semidirect
23. #2564C4semidirect
28. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#353342
#207232
#173662
#119022
#118132
#118122
#117842