Finite Group Databasebeta

← #224129

Subgroups — #224129

Order 512.

1103 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic, complemented
3#13818C211yesmaximal, characteristic, complemented
4#9909C211yesmaximal, characteristic, complemented
5#9909C212yesmaximal, complemented
6#9885C211yesmaximal, characteristic, complemented
7#9884C212yesmaximal, complemented
8#8577C212yesmaximal
9#8571C211yesmaximal, characteristic
10C4×#327C211yesmaximal, characteristic
11C4×#327C212yesmaximal
12C2×#554—21no—
13#2857C2212yes—
14#2857C2211yescharacteristic
15#2564—22no—
16#2556—22no—
17#2551C411yescharacteristic, complemented
18#2542C411yescharacteristic, complemented
19#2506C412yescomplemented
20#2492—21no—
21#2452C412yescomplemented
22#2417—21no—
23#2365C411yescharacteristic, complemented
24#2360—21no—
25D4×C4⋊C4C411yescharacteristic, complemented
26C4×#121C2211yescharacteristic
27C4×#121C2212yes—
28C2×#389—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1916—22no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35C8×C4⋊C4—21no—
36C4×C22⋊C8C2212yes—
37C4×C22⋊C8C2211yescharacteristic
38C2×#327C2212yes—
39C2×#327C2211yescharacteristic
40C2×#327C2211yescharacteristic
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4×#95C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2211yescharacteristic
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#554—44no—
54#554—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C412yes—
58C2×#125—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—21no—
64C2×#121C2311yescharacteristic
65C2×C4×Q8—21no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C412yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42—22no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2311yescharacteristic
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86#400—22no—
87#399—22no—
88#398—22no—
89#396—22no—
90#395—22no—
91#393—22no—
92#391—21no—
93#391—21no—
94#391—22no—
95#390C2×C411yescharacteristic
96#390C2×C412yes—
97#389—28no—
98#389—21no—
99#389—22no—
100#389—21no—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C411yescharacteristic
105#389C2×C412yes—
106#388—22no—
107#388—21no—
108#387—22no—
109#387—22no—
110#386—21no—
111#386—22no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C412yes—
116#380—21no—
117#380—22no—
118#380—21no—
119#380C2312yes—
120#380C2311yescharacteristic
121C4×C4⋊C4—44no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D411yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95D412yes—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140#336—21no—
141#336—22no—
142#336—21no—
143#336—22no—
144#327—24no—
145#327—22no—
146#327Q814yescomplemented
147C22×Q8—21no—
148C22×Q8—22no—
149C22×D4—21no—
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C4⋊Q8—42no—
165C4⋊Q8—44no—
166C4⋊Q8—82no—
167#125—48no—
168C22⋊Q8—44no—
169C22⋊Q8—42no—
170C22⋊Q8—44no—
171C22⋊Q8—42no—
172C22⋊Q8—48no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175#121—21no—
176#121—21no—
177#121C2×Q812yes—
178#121#4112yes—
179C4×Q8—44no—
180C4×D4—42no—
181C4×D4—44no—
182C4×D4—42no—
183C4×D4—44no—
184C4×D4—42no—
185C4×D4—82no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4D814yes—
212C2×C4⋊C4SD1614yes—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C412yes—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C4⋊C8—42no—
246C4⋊C8—82no—
247C22⋊C8—21no—
248C22⋊C8—22no—
249C22⋊C8—21no—
250C22⋊C8—22no—
251C22⋊C8—21no—
252C22⋊C8C2×Q812yes—
253C4×C8—42no—
254C4×C8—82no—
255#95—28no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—42no—
271#95—44no—
272#95—81no—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276#95C22⋊C412yes—
277#95C2×Q812yes—
278C24—21no—
279C24C22×D411yescharacteristic
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—42no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4C4×Q812yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
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—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×D812yes—
351C22×C4C2×D814yes—
352C22×C4C2×SD1612yes—
353C22×C4C2×SD1614yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C42−1+411yescharacteristic
359C22×C42−1+411yescharacteristic
360C2×C8—22no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C4⋊C4—21no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—81no—
387C4⋊C4—82no—
388C4⋊C4—88no—
389C4⋊C4#10214yes—
390C4⋊C4#10218yes—
391C4⋊C4#10214yes—
392C4⋊C4#10414yes—
393C4⋊C4C4×Q812yes—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—42no—
400C22⋊C4—41no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—81no—
405C22⋊C4—82no—
406C42—22no—
407C42—22no—
408C42—42no—
409C42—42no—
410C42—41no—
411C42—41no—
412C42—42no—
413C42—42no—
414C42—48no—
415C42—44no—
416C42—42no—
417C42—88no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C22×C22⋊C411yescharacteristic
430C23#54811yescharacteristic
431C23D4×Q811yescharacteristic
432C23C22×D811yescharacteristic
433C23C22×SD1611yescharacteristic
434C23#57711yescharacteristic
435C23#57811yescharacteristic
436Q8—84no—
437Q8—164no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—42no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—164no—
515C2×C4#41012yes—
516C2×C4C2×#10214yes—
517C2×C4C2×#10212yes—
518C2×C4C2×#10214yes—
519C2×C4C2×#10212yes—
520C2×C4C2×#10414yes—
521C2×C4C2×#10412yes—
522C2×C4C2×C4×Q811yescharacteristic
523C2×C4#51911yescharacteristic
524C2×C4#54811yescharacteristic
525C2×C4D4×Q811yescharacteristic
526C8—42no—
527C8—81no—
528C8—82no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22Q8×C22⋊C411yescharacteristic
549C22C2×#41011yescharacteristic
550C22#288311yescharacteristic
551C22C22×#10211yescharacteristic
552C22#289811yescharacteristic
553C22C22×#10411yescharacteristic
554C22#290111yescharacteristic
555C22#335811yescharacteristic
556C22#335911yescharacteristic
557C22Q8×D811yescharacteristic
558C22Q8×SD1611yescharacteristic
559C4—22no—
560C4—24no—
561C4—24no—
562C4—24no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—44no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—82no—
573C4—84no—
574C4—81no—
575C4—82no—
576C4—82no—
577C4—81no—
578C4#150214yes—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—82no—
584C2C2×#150211yescharacteristic
585C2#1754811yescharacteristic
586C2#1760511yescharacteristic
587C2#1761311yescharacteristic
588C2#2251311yescharacteristic
589C2Q8×#10211yescharacteristic
590C2Q8×#10411yescharacteristic
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. #22625C2semidirect
3. #13818C2semidirect
4. #9909C2semidirect
5. #9909C2semidirect
6. #9885C2semidirect
7. #9884C2semidirect
17. #2551C4semidirect
18. #2542C4semidirect
19. #2506C4semidirect
21. #2452C4semidirect
23. #2365C4semidirect
25. D4×C4⋊C4C4semidirect
146. #327Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
#226252
#138182
#99092
#98852
#98842
#85772
#85712
C4×#3272