Finite Group Databasebeta

← #224136

Subgroups — #224136

Order 512.

1179 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#36545C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic
3Q8×#95C211yesmaximal, characteristic
4#9910C212yesmaximal
5#9909C211yesmaximal, characteristic
6#9887C212yesmaximal
7#9885C211yesmaximal, characteristic
8#8578C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8219C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12C2×#557C411yescharacteristic, complemented
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#446—21no—
16#2857C2211yescharacteristic
17#2851C2212yes—
18#2584C412yescomplemented
19#2582C411yescharacteristic, complemented
20#2581C412yescomplemented
21#2546C411yescharacteristic, complemented
22#2473—22no—
23#2462—22no—
24#2446—21no—
25#2434—21no—
26#2365—21no—
27#2360—21no—
28#2355C411yescharacteristic, complemented
29C4×C22⋊Q8C2211yescharacteristic
30C4×C22⋊Q8C2212yes—
31#1985—22no—
32#1984—21no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2211yescharacteristic
40C4⋊#95—21no—
41C4⋊#95—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4×#95C2211yescharacteristic
45C4×#95C2212yes—
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#557—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125—22no—
59C2×#125C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2311yescharacteristic
62C2×C4×Q8—28no—
63C2×C4×Q8—21no—
64C2×C4×Q8—44no—
65C2×C4×Q8—44no—
66C2×C4×Q8—44no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D412yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D412yes—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#446—44no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2311yescharacteristic
89C2×#97—22no—
90C2×C4×C8—21no—
91#400—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#395—22no—
96#395—22no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391C2×C411yescharacteristic
102#391C2×C412yes—
103#389—21no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C411yescharacteristic
111#389C2×C412yes—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—21no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—21no—
123#384C2×C411yescharacteristic
124#382C2×C412yes—
125#380—21no—
126#380C2311yescharacteristic
127#380C2312yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146#328—22no—
147#328—24no—
148#328Q814yescomplemented
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C4⋊Q8—42no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172#125—44no—
173#125—48no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8C2×Q812yes—
179C22⋊Q8#4112yes—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—44no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—44no—
186C4×Q8—44no—
187C4×Q8—42no—
188C4×Q8—44no—
189C4×Q8—432no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C42—28no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C4⋊C8—42no—
247C4⋊C8—44no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—22no—
251C22⋊C8—21no—
252C22⋊C8—22no—
253C22⋊C8C2×Q812yes—
254#97—44no—
255C4×C8—42no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—21no—
275#95—22no—
276#95—44no—
277#95—42no—
278#95—82no—
279#95—81no—
280#95C2×Q812yes—
281C24C22×D411yescharacteristic
282C2×Q8—216no—
283C2×Q8—216no—
284C2×Q8—21no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8#9914yes—
298C2×Q8#10414yes—
299C2×Q8C4×Q812yescomplemented
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C42−1+411yescharacteristic
358C22×C42−1+411yescharacteristic
359C2×C8—22no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—42no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—81no—
394C4⋊C4—81no—
395C4⋊C4—82no—
396C4⋊C4C4×Q812yescomplemented
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—22no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42—41no—
411C42—416no—
412C42—48no—
413C42—44no—
414C42—42no—
415C42—42no—
416C42—82no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23C22×C22⋊C411yescharacteristic
422C23#54811yescharacteristic
423C23D4×Q811yescharacteristic
424C23C2×AGL(1,ℤ/8)11yescharacteristic
425C23C2×#13711yescharacteristic
426C23#57711yescharacteristic
427C23#57811yescharacteristic
428Q8—216no—
429Q8—28no—
430Q8—28no—
431Q8—42no—
432Q8—42no—
433Q8—84no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4C2×#9914yes—
508C2×C4C2×#9912yes—
509C2×C4#41712yes—
510C2×C4#41712yes—
511C2×C4C2×#10414yes—
512C2×C4C2×#10412yes—
513C2×C4C2×C4×Q811yescharacteristic
514C2×C4#51911yescharacteristic
515C2×C4#54811yescharacteristic
516C2×C4D4×Q811yescharacteristic
517C8—42no—
518C8—81no—
519C8—82no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22Q8×C22⋊C411yescharacteristic
533C22C22×#9911yescharacteristic
534C22#288411yescharacteristic
535C22C2×#41711yescharacteristic
536C22#289811yescharacteristic
537C22C22×#10411yescharacteristic
538C22#290111yescharacteristic
539C22#336811yescharacteristic
540C22#337411yescharacteristic
541C22#338611yescharacteristic
542C22#338711yescharacteristic
543C4—24no—
544C4—22no—
545C4—24no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—84no—
558C4—82no—
559C4—82no—
560C4—81no—
561C4—81no—
562C4—82no—
563C4#150414yes—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2C2×#150411yescharacteristic
569C2#1754911yescharacteristic
570C2#1760611yescharacteristic
571C2#1762811yescharacteristic
572C2Q8×#9911yescharacteristic
573C2#2252411yescharacteristic
574C2Q8×#10411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. C2×#557C4semidirect
18. #2584C4semidirect
19. #2582C4semidirect
20. #2581C4semidirect
21. #2546C4semidirect
28. #2355C4semidirect
148. #328Q8semidirect
299. C2×Q8C4×Q8semidirect
396. C4⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#365452
#225102
Q8×#952
#99102
#99092
#98872
#98852
#85782
#85722
#82192
C4×#3282