Finite Group Databasebeta

← #164780

Subgroups — #164780

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36259C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#20630C211yesmaximal, characteristic, complemented
4#7714C212yesmaximal, complemented
5#7685C211yesmaximal, characteristic, complemented
6#7668C211yesmaximal, characteristic, complemented
7#7658C212yesmaximal, complemented
8#7641C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7573C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C2×#547C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×C8×Q8—21no—
16C2×#424—21no—
17C2×#408—21no—
18#2857C2211yescharacteristic
19#2854C2212yes—
20#2546C411yescharacteristic
21#2538C411yescharacteristic
22#2486C412yes—
23#2429—21no—
24#2425—22no—
25#2416C412yes—
26#2415—21no—
27#2413—22no—
28D4×C4⋊C4C411yescharacteristic
29#2339—21no—
30C4×#121C2211yescharacteristic
31C2×#389C2212yes—
32#1954—21no—
33#1954—22no—
34#1798—22no—
35#1797C2212yes—
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41—21no—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57#547—44no—
58#547—44no—
59#547—44no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#123—22no—
63C2×#123—22no—
64C2×#123—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×#121—22no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C412yes—
71C4×#41—44no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C412yes—
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D412yes—
81C2×#117D411yescharacteristic
82C2×#117D412yes—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4C2×C412yes—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C8×Q8—44no—
89#424—44no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94#408—44no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—22no—
103#398—22no—
104#398—22no—
105#397—22no—
106#396—22no—
107#396—22no—
108#394—22no—
109#391—21no—
110#391—21no—
111#390C2×C412yes—
112#390C2311yescharacteristic
113#389—21no—
114#389—21no—
115#389—22no—
116#389—22no—
117#389C2×C412yes—
118#389C2×C412yes—
119#389C2×C412yes—
120#389C2312yes—
121#389C2311yescharacteristic
122#388—22no—
123#388—22no—
124#388—21no—
125#386—22no—
126#386—21no—
127#386—22no—
128#385—22no—
129#384C2×C411yescharacteristic
130#381—22no—
131#380—21no—
132#380—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135#380C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C22⋊C4—21no—
138C2×#95—22no—
139#336—21no—
140#336—21no—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145#336C2312yes—
146#336C2311yescharacteristic
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—44no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41D814yes—
153C2×#41SD1614yes—
154C22×Q8—21no—
155C22×Q8C22⋊C411yescharacteristic
156C22×D4—22no—
157C22×D4—21no—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C412yes—
160C22×D4C22×C411yescharacteristic
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—44no—
184#123—48no—
185#123—48no—
186#123—44no—
187#123—44no—
188C22⋊Q8—44no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—44no—
192#121—22no—
193#121—41no—
194#121—48no—
195#121—44no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×D4—24no—
199C4×D4—44no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—42no—
205C4×D4—42no—
206#117—24no—
207#117—44no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C22⋊C412yes—
245C2×C42—21no—
246C2×C42—44no—
247C2×C42—42no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2411yescharacteristic
255C4⋊C8—44no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258C4⋊C8—42no—
259C22⋊C8—44no—
260C22⋊C8—42no—
261C22⋊C8—44no—
262C22⋊C8—44no—
263C22⋊C8—82no—
264C22⋊C8—81no—
265C4×C8—44no—
266C4×C8—42no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—22no—
276C24—21no—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C412yes—
279#41—48no—
280#41—48no—
281#41—816no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8#10214yes—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—81no—
302C2×D4#10214yes—
303C2×D4#10218yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—82no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×D812yes—
345C22×C4C2×D814yes—
346C22×C4C2×SD1612yes—
347C22×C4C2×SD1614yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—84no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—84no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—81no—
392C4⋊C4#9914yes—
393C4⋊C4#13112yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C42—24no—
410C42—22no—
411C42—24no—
412C42—22no—
413C42—41no—
414C42—42no—
415C42—41no—
416C42—42no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C2×#10212yes—
437C23C2×#10214yes—
438C23C22×C22⋊C411yescharacteristic
439Q8—28no—
440Q8—84no—
441D4—28no—
442D4—48no—
443D4—81no—
444D4—81no—
445D4—84no—
446D4—88no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—88no—
494C2×C4—88no—
495C2×C4—84no—
496C2×C4C2×#9914yes—
497C2×C4C2×#9912yes—
498C2×C4C2×#10214yes—
499C2×C4C2×#10212yes—
500C2×C4#42112yes—
501C2×C4C2×#13111yescharacteristic
502C2×C4#56811yescharacteristic
503C2×C4C22×D811yescharacteristic
504C2×C4C22×SD1611yescharacteristic
505C2×C4#57711yescharacteristic
506C2×C4#57811yescharacteristic
507C8—42no—
508C8—82no—
509C8—81no—
510C8—81no—
511C8—81no—
512C8—82no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—42no—
520C22—41no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22#288011yescharacteristic
535C22C22×#9911yescharacteristic
536C22#288911yescharacteristic
537C22C22×#10211yescharacteristic
538C22#289811yescharacteristic
539C22C2×#42111yescharacteristic
540C22#290111yescharacteristic
541C4—21no—
542C4—21no—
543C4—24no—
544C4—21no—
545C4—21no—
546C4—24no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—81no—
554C4—82no—
555C4—81no—
556C4—82no—
557C4#151914yes—
558C2—44no—
559C2—42no—
560C2—161no—
561C2#1741611yescharacteristic
562C2#1750711yescharacteristic
563C2#1752511yescharacteristic
564C2C2×#151911yescharacteristic
565C2#1760611yescharacteristic
566C2#1768211yescharacteristic
567C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36259C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22530C2semidirect
3. #20630C2semidirect
4. #7714C2semidirect
5. #7685C2semidirect
6. #7668C2semidirect
7. #7658C2semidirect
8. #7641C2semidirect
9. #7616C2semidirect
10. #7573C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#362592
#225302
#206302
#77142
#76852
#76682
#76582
#76412
#76162
#75732
#75652