Finite Group Databasebeta

← #224829

Subgroups — #224829

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic, complemented
3#13933C211yesmaximal, characteristic, complemented
4#9716C212yesmaximal
5#9711C211yesmaximal, characteristic, complemented
6#9682C212yesmaximal
7#9679C211yesmaximal, characteristic, complemented
8#8578C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10#8217C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2850C2212yes—
14#2596—22no—
15#2571C411yescharacteristic, complemented
16#2547C411yescharacteristic, complemented
17#2514C412yescomplemented
18#2510—22no—
19#2494—21no—
20#2487—21no—
21#2448C412yescomplemented
22#2365C411yescharacteristic, complemented
23#2354C411yescharacteristic, complemented
24#2352—21no—
25#2347—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2211yescharacteristic
28C2×#389—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1939—21no—
31#1917—22no—
32#1797C2211yescharacteristic
33#1797C2212yes—
34#1770—21no—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#327C2211yescharacteristic
42C2×#327C2211yescharacteristic
43C2×#327C2212yes—
44#1442—22no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×C22⋊Q8C2312yes—
56C2×#121C2311yescharacteristic
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2311yescharacteristic
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2311yescharacteristic
73C2×#97—22no—
74C2×C4×C8—21no—
75#401—22no—
76#400—22no—
77#400—22no—
78#399—22no—
79#397—22no—
80#397—22no—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2×C412yes—
84#390C2×C412yes—
85#390C2×C411yescharacteristic
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—44no—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#388—22no—
94#388—22no—
95#388—22no—
96#388—21no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#386—22no—
104#386—21no—
105#385—22no—
106#384C2×C411yescharacteristic
107#383—22no—
108#382—21no—
109#382C2×C412yes—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380C2311yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95D412yes—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137#336—21no—
138#336—22no—
139#336—21no—
140#336—22no—
141#328—24no—
142#328—24no—
143#328—42no—
144#328—42no—
145#327—22no—
146#327—22no—
147#327—42no—
148C22×Q8C22×C412yes—
149C22×D4—21no—
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C4⋊Q8—42no—
164#124—82no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169#121—21no—
170#121—21no—
171#121—21no—
172#121—21no—
173C4×Q8—82no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177#117—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—82no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C4⋊C8—42no—
232C4⋊C8—82no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—21no—
238C22⋊C8#4112yes—
239#97—82no—
240C4×C8—42no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—42no—
260#95—44no—
261#95—82no—
262#95—81no—
263#95—82no—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95#4112yes—
269C24—21no—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
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—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4#13712yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C42−1+411yescharacteristic
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—82no—
374C4⋊C4—81no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4—88no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—42no—
387C22⋊C4—81no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C42—22no—
392C42—22no—
393C42—42no—
394C42—42no—
395C42—41no—
396C42—41no—
397C42—42no—
398C42—42no—
399C42—44no—
400C42—42no—
401C42—82no—
402C42—84no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23C22×C22⋊C411yescharacteristic
415C23#54811yescharacteristic
416C23#55011yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23C2×#13711yescharacteristic
419C23#57711yescharacteristic
420C23#57811yescharacteristic
421Q8—82no—
422Q8—82no—
423D4—81no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—81no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4#41012yes—
503C2×C4#41712yes—
504C2×C4#41712yes—
505C2×C4#42112yes—
506C2×C4C4×#4111yescharacteristic
507C2×C4#52011yescharacteristic
508C2×C4#54811yescharacteristic
509C2×C4#55011yescharacteristic
510C8—42no—
511C8—81no—
512C8—82no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#234111yescharacteristic
533C22C2×#41011yescharacteristic
534C22#288311yescharacteristic
535C22C2×#41711yescharacteristic
536C22#289811yescharacteristic
537C22C2×#42111yescharacteristic
538C22#290111yescharacteristic
539C22#336711yescharacteristic
540C22#336911yescharacteristic
541C22#339711yescharacteristic
542C22#339811yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4—82no—
559C4—81no—
560C4—81no—
561C4—162no—
562C4#151214yes—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—161no—
568C2C2×#151211yescharacteristic
569C2#1760011yescharacteristic
570C2#1761211yescharacteristic
571C2#1763011yescharacteristic
572C2#2253511yescharacteristic
573C2#2254411yescharacteristic
574C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
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. #22629C2semidirect
3. #13933C2semidirect
5. #9711C2semidirect
7. #9679C2semidirect
8. #8578C2semidirect
10. #8217C2semidirect
15. #2571C4semidirect
16. #2547C4semidirect
17. #2514C4semidirect
21. #2448C4semidirect
22. #2365C4semidirect
23. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#365862
#226292
#139332
#97162
#97112
#96822
#96792
#85782
#85712
#82172
C4×#3272