Finite Group Databasebeta

← #224383

Subgroups — #224383

Order 512.

1007 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊Q8C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic
3#13931C211yesmaximal, characteristic
4#9912C212yesmaximal
5#9908C211yesmaximal, characteristic
6#9888C212yesmaximal
7#9884C211yesmaximal, characteristic
8#8576C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8217C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12C2×Q82C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×#446—21no—
15#2857C2211yescharacteristic
16#2857C2212yes—
17#2582C411yescharacteristic, complemented
18#2582C412yescomplemented
19#2567C411yescharacteristic, complemented
20#2567C412yescomplemented
21#2451—21no—
22#2451—22no—
23#2440—21no—
24#2440—22no—
25Q8×C4⋊C4—21no—
26Q8×C4⋊C4C411yescharacteristic, complemented
27Q8×C22⋊C4—21no—
28C4×C22⋊Q8C2211yescharacteristic
29C4×C22⋊Q8C2212yes—
30C2×#391—21no—
31#1985—22no—
32#1984—21no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2211yescharacteristic
40#1445—22no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54Q82—44no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2311yescharacteristic
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#446—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2311yescharacteristic
85C2×#97—22no—
86C2×C4×C8—21no—
87#398—22no—
88#398—22no—
89#398—22no—
90#395—22no—
91#395—22no—
92#395—22no—
93#391—21no—
94#391—22no—
95#391—21no—
96#391—22no—
97#391—21no—
98#391—22no—
99#391—44no—
100#391C2×C411yescharacteristic
101#391C2×C412yes—
102#389—22no—
103#389—21no—
104#389—22no—
105#389—21no—
106#389—21no—
107#389—22no—
108#389C2×C411yescharacteristic
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2×C412yes—
113#387—21no—
114#387—22no—
115#385—21no—
116#385—22no—
117#384—22no—
118#384—21no—
119#384C2×C411yescharacteristic
120#384C2×C412yes—
121#380—22no—
122#380—21no—
123#380C2311yescharacteristic
124#380C2312yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—21no—
142#336—22no—
143#336—21no—
144#328—22no—
145#328—24no—
146#328Q814yescomplemented
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—42no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C4⋊Q8—44no—
169C4⋊Q8—42no—
170C4⋊Q8—48no—
171C4⋊Q8—44no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—81no—
175C22⋊Q8—82no—
176C22⋊Q8—81no—
177C22⋊Q8—82no—
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—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C4⋊C8—42no—
242C4⋊C8—44no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—22no—
246C22⋊C8—21no—
247C22⋊C8—22no—
248C22⋊C8C2×Q812yes—
249#97—44no—
250C4×C8—42no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—21no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—21no—
268#95—22no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—82no—
274#95—81no—
275#95C2×Q812yes—
276C24C22×D411yescharacteristic
277C2×Q8—21no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—42no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—44no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—81no—
293C2×Q8—82no—
294C2×Q8—81no—
295C2×Q8—82no—
296C2×Q8#9914yes—
297C2×Q8C4×Q812yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
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—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4#13712yes—
350C22×C4#13712yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—81no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—81no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4C4×Q812yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—42no—
400C22⋊C4—41no—
401C42—22no—
402C42—24no—
403C42—24no—
404C42—22no—
405C42—41no—
406C42—42no—
407C42—41no—
408C42—42no—
409C42—44no—
410C42—44no—
411C42—42no—
412C42—42no—
413C42—84no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23C22×C22⋊C411yescharacteristic
419C23#54811yescharacteristic
420C23D4×Q811yescharacteristic
421C23C2×#13711yescharacteristic
422C23C2×#13711yescharacteristic
423C23#57611yescharacteristic
424C23#57711yescharacteristic
425Q8—28no—
426Q8—48no—
427Q8—42no—
428Q8—42no—
429Q8—44no—
430Q8—164no—
431Q8—162no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4C2×#9914yes—
507C2×C4C2×#9912yes—
508C2×C4#41912yes—
509C2×C4#41912yes—
510C2×C4#42112yes—
511C2×C4C2×C4×Q811yescharacteristic
512C2×C4#51911yescharacteristic
513C2×C4#54811yescharacteristic
514C2×C4D4×Q811yescharacteristic
515C8—42no—
516C8—81no—
517C8—82no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22Q8×C22⋊C411yescharacteristic
531C22C22×#9911yescharacteristic
532C22#288411yescharacteristic
533C22C2×#41911yescharacteristic
534C22#289711yescharacteristic
535C22C2×#42111yescharacteristic
536C22#290111yescharacteristic
537C22#336911yescharacteristic
538C22#337511yescharacteristic
539C22#338511yescharacteristic
540C22#338711yescharacteristic
541C4—22no—
542C4—24no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4—82no—
559C4—81no—
560C4—162no—
561C4#151114yes—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2C2×#151111yescharacteristic
567C2#1760211yescharacteristic
568C2#1761111yescharacteristic
569C2#1763211yescharacteristic
570C2Q8×#9911yescharacteristic
571C2#2252511yescharacteristic
572C2#2252911yescharacteristic
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×Q82C4semidirect
17. #2582C4semidirect
18. #2582C4semidirect
19. #2567C4semidirect
20. #2567C4semidirect
26. Q8×C4⋊C4C4semidirect
146. #328Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊Q82
#225102
#139312
#99122
#99082
#98882
#98842
#85762
#85722
#82172
C4×#3282