Finite Group Databasebeta

← #224506

Subgroups — #224506

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic, complemented
3#22625C211yesmaximal, characteristic, complemented
4#9652C212yesmaximal
5#9651C212yesmaximal
6#9650C211yesmaximal, characteristic, complemented
7#9650C211yesmaximal, characteristic, complemented
8#8562C212yesmaximal, complemented
9#8557C211yesmaximal, characteristic
10#8154C212yesmaximal, complemented
11C4×#323C211yesmaximal, characteristic
12#2494—22no—
13#2494—22no—
14#2494—21no—
15#2494—21no—
16#2494—21no—
17#2494—21no—
18C2×#401C412yes—
19C2×#397C412yes—
20C2×#389C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388C2212yes—
23C2×#387C2212yes—
24C2×#382C411yescharacteristic
25C2×#380C411yescharacteristic
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1917—22no—
32#1916—22no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1797C2211yescharacteristic
36#1797C2212yes—
37#1770—21no—
38C8×C4⋊C4—21no—
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43C2×#324C2212yes—
44C2×#324C2212yes—
45C2×#324C2212yes—
46C2×#324C2212yes—
47C2×#323C2211yescharacteristic
48C2×#323C2211yescharacteristic
49C2×#323C2212yes—
50#1280C2212yes—
51#1280C2211yescharacteristic
52#1280C2211yescharacteristic
53C24×C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8C2311yescharacteristic
72C2×C22⋊C8C2311yescharacteristic
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2311yescharacteristic
76C2×C22⋊C8C2311yescharacteristic
77C2×#97—22no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81#401—22no—
82#401—22no—
83#401—22no—
84#401—22no—
85#397—22no—
86#397—22no—
87#397—22no—
88#397—22no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#382—22no—
105#382—21no—
106#382—21no—
107#380—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2311yescharacteristic
132#336—21no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#324—24no—
141#324—24no—
142#324—42no—
143#324—42no—
144#323—22no—
145#323—22no—
146#323—42no—
147C25C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—82no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—82no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—82no—
228C4⋊C8—82no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8#4112yes—
240C22⋊C8#4112yes—
241#97—82no—
242C4×C8—42no—
243C4×C8—42no—
244C4×C8—82no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24#13112yes—
276C24C23×C411yescharacteristic
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4#13112yes—
337C22×C4#13112yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C42—22no—
389C42—22no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—42no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23C22×C22⋊C411yescharacteristic
426C23C4×#4111yescharacteristic
427C23#52011yescharacteristic
428C23C2×#13111yescharacteristic
429C23C2×#13111yescharacteristic
430C23C2×#13111yescharacteristic
431C23#56811yescharacteristic
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
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—84no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4#40812yes—
498C2×C4#40812yes—
499C2×C4#41012yes—
500C2×C4#41312yes—
501C2×C4#54811yescharacteristic
502C2×C4#54811yescharacteristic
503C2×C4#55011yescharacteristic
504C2×C4#55011yescharacteristic
505C8—42no—
506C8—42no—
507C8—81no—
508C8—81no—
509C8—82no—
510C8—82no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22#234111yescharacteristic
535C22C2×#40811yescharacteristic
536C22#288011yescharacteristic
537C22C2×#41011yescharacteristic
538C22#288311yescharacteristic
539C22C2×#41311yescharacteristic
540C22#288911yescharacteristic
541C22#296611yescharacteristic
542C22#296911yescharacteristic
543C22#299311yescharacteristic
544C22#299411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4#147314yes—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—82no—
567C2C2×#147311yescharacteristic
568C2#1738811yescharacteristic
569C2#1741311yescharacteristic
570C2#1742411yescharacteristic
571C2#2253111yescharacteristic
572C2#2253511yescharacteristic
573C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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. #22625C2semidirect
6. #9650C2semidirect
7. #9650C2semidirect
8. #8562C2semidirect
10. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
#226292
#226252
#96522
#96512
#96502
#85622
#85572
#81542
C4×#3232