Finite Group Databasebeta

← #216569

Subgroups — #216569

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3#13949C211yesmaximal, characteristic, complemented
4#9912C211yesmaximal, characteristic, complemented
5#9909C211yesmaximal, characteristic, complemented
6#9715C211yesmaximal, characteristic
7#9712C211yesmaximal, characteristic
8#9114C212yesmaximal, complemented
9#9114C212yesmaximal, complemented
10#8577C212yesmaximal, complemented
11#8571C212yesmaximal, complemented
12C2×#539—21no—
13C2×#431C2212yes—
14#2857C2211yescharacteristic
15#2856C2211yescharacteristic
16#2854C2212yes—
17#2639—21no—
18#2618—21no—
19#2560—22no—
20#2558—22no—
21#2512C412yescomplemented
22#2506C412yescomplemented
23#2452C412yescomplemented
24#2445C412yescomplemented
25#2325—21no—
26C4×#121C2212yes—
27C2×#387—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—22no—
30#1939—22no—
31#1797C2212yes—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34C2×#327C2212yes—
35C2×#327C2212yes—
36C2×#327C2212yes—
37C2×#327C2212yes—
38#1442—21no—
39#1441—21no—
40#1436—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434C2211yescharacteristic
44#1434C2211yescharacteristic
45#1434C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#539—44no—
52#539—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#126—22no—
56C2×#125—22no—
57C2×#123—22no—
58C2×#123—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2312yes—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#431—42no—
75#431—42no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C4×C8—22no—
83#400—22no—
84#398—22no—
85#397—22no—
86#396—22no—
87#396—22no—
88#394—22no—
89#394—22no—
90#393—22no—
91#391—21no—
92#390—21no—
93#390—21no—
94#390C2×C412yes—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2×C412yes—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—44no—
109#386—22no—
110#386—22no—
111#386—22no—
112#385—22no—
113#384C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115#380—21no—
116#380C2312yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C22⋊C4—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138#336—22no—
139#336—21no—
140#336—21no—
141#327—42no—
142#327—42no—
143#327—42no—
144#327—42no—
145C22×Q8—21no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—81no—
165#127—81no—
166#126—48no—
167#125—48no—
168#123—44no—
169#123—44no—
170#123—48no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C22⋊Q8—48no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—42no—
179#121—42no—
180#121—42no—
181#121—42no—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—82no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4D814yes—
207C2×C4⋊C4SD1614yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42—81no—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C4⋊C8—44no—
241C4⋊C8—82no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C4×C8—44no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—81no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276C24—22no—
277C24C22×D411yescharacteristic
278C2×Q8—42no—
279C2×Q8—81no—
280C2×Q8—81no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4—81no—
296C2×D4—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—44no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×D812yes—
362C22×C4C2×D812yes—
363C22×C4C2×D812yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×SD1612yes—
366C22×C4C2×SD1612yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4#10218yes—
397C4⋊C4#10218yes—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—42no—
416C42—41no—
417C42—41no—
418C42—42no—
419C42—41no—
420C42—41no—
421C42—44no—
422C42—84no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23C22×C22⋊C411yescharacteristic
435C23#53711yescharacteristic
436C23#53711yescharacteristic
437C23C22×D811yescharacteristic
438C23C22×SD1611yescharacteristic
439C23#57511yescharacteristic
440C23#57511yescharacteristic
441Q8—162no—
442D4—82no—
443D4—82no—
444D4—82no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—162no—
449D4—162no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
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—42no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—164no—
530C2×C4C2×#10214yes—
531C2×C4C2×#10214yes—
532C2×C4C2×#10212yes—
533C2×C4C2×#10212yes—
534C2×C4#52011yescharacteristic
535C2×C4#52011yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53611yescharacteristic
538C8—82no—
539C8—82no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22#231111yescharacteristic
560C22#288311yescharacteristic
561C22#288411yescharacteristic
562C22C22×#10211yescharacteristic
563C22#289611yescharacteristic
564C22#290111yescharacteristic
565C22#290111yescharacteristic
566C22#324511yescharacteristic
567C22#324511yescharacteristic
568C22#324611yescharacteristic
569C22#324611yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—24no—
575C4—24no—
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—161no—
588C4—161no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—162no—
594C2#1761311yescharacteristic
595C2#1761811yescharacteristic
596C2#1762311yescharacteristic
597C2#1762411yescharacteristic
598C2#2185811yescharacteristic
599C2#2187711yescharacteristic
600C2#2188411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
3. #13949C2semidirect
4. #9912C2semidirect
5. #9909C2semidirect
8. #9114C2semidirect
9. #9114C2semidirect
10. #8577C2semidirect
11. #8571C2semidirect
21. #2512C4semidirect
22. #2506C4semidirect
23. #2452C4semidirect
24. #2445C4semidirect

Maximal subgroups

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

subgroupindex
#350062
#219512
#139492
#99122
#99092
#97152
#97122
#91142
#85772
#85712