Finite Group Databasebeta

← #224142

Subgroups — #224142

Order 512.

1087 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊Q8C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic
3#13817C211yesmaximal, characteristic
4#9909C212yesmaximal
5#9908C211yesmaximal, characteristic
6#9889C212yesmaximal
7#9884C211yesmaximal, characteristic
8#8574C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8219C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12C2×Q82C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×#446—21no—
15#2857C2211yescharacteristic
16#2857C2212yes—
17#2582C412yescomplemented
18#2582C411yescharacteristic, complemented
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—
31C2×C4×C4⋊C4—21no—
32#1985—22no—
33#1984—21no—
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1749C2212yes—
37C4×C22⋊C8C2211yescharacteristic
38C2×#328C2211yescharacteristic
39C2×#328C2212yes—
40C2×#328C2211yescharacteristic
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
44C4×#95C2211yescharacteristic
45C4×#95C2212yes—
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×C412yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2311yescharacteristic
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—44no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C411yescharacteristic
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×C42—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#446—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2311yescharacteristic
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2311yescharacteristic
86C2×#97—22no—
87C2×C4×C8—21no—
88#398—22no—
89#398—22no—
90#398—22no—
91#395—22no—
92#395—22no—
93#395—22no—
94#391—28no—
95#391—21no—
96#391—22no—
97#391—21no—
98#391—22no—
99#391—22no—
100#391—21no—
101#391C2×C411yescharacteristic
102#391C2×C412yes—
103#389—21no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389—21no—
108#389—22no—
109#389C2×C411yescharacteristic
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2×C412yes—
113#389C2×C412yes—
114#387—22no—
115#387—21no—
116#385—22no—
117#385—21no—
118#384—21no—
119#384—22no—
120#384C2×C411yescharacteristic
121#384C2×C412yes—
122#380—22no—
123#380—21no—
124#380C2311yescharacteristic
125#380C2312yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—44no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141#336—22no—
142#336—21no—
143#336—22no—
144#336—21no—
145#328—22no—
146#328—24no—
147#328Q814yescomplemented
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—42no—
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22⋊C412yes—
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C4⋊Q8—44no—
170C4⋊Q8—42no—
171C4⋊Q8—48no—
172C4⋊Q8—44no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—82no—
176C22⋊Q8—81no—
177C22⋊Q8—82no—
178C22⋊Q8—81no—
179C22⋊Q8C2×Q812yes—
180C22⋊Q8#4112yes—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—44no—
188C4×Q8—42no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
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×C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—82no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—44no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—22no—
250C22⋊C8—21no—
251C22⋊C8—22no—
252C22⋊C8C2×Q812yes—
253#97—44no—
254C4×C8—42no—
255#95—28no—
256#95—28no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—21no—
274#95—22no—
275#95—42no—
276#95—44no—
277#95—81no—
278#95C2×Q812yes—
279C24C22×D411yescharacteristic
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—44no—
295C2×Q8—82no—
296C2×Q8—81no—
297C2×Q8—82no—
298C2×Q8—81no—
299C2×Q8#10414yes—
300C2×Q8C4×Q812yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
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—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4#13712yes—
353C22×C4#13712yes—
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—41no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—42no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—82no—
390C4⋊C4—81no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—81no—
396C4⋊C4#9914yes—
397C4⋊C4C4×Q812yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—41no—
401C22⋊C4—42no—
402C22⋊C4—41no—
403C22⋊C4—42no—
404C42—22no—
405C42—24no—
406C42—24no—
407C42—22no—
408C42—41no—
409C42—42no—
410C42—42no—
411C42—41no—
412C42—48no—
413C42—44no—
414C42—42no—
415C42—44no—
416C42—42no—
417C42—88no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23C22×C22⋊C411yescharacteristic
423C23#54811yescharacteristic
424C23D4×Q811yescharacteristic
425C23C2×#13711yescharacteristic
426C23C2×#13711yescharacteristic
427C23#57611yescharacteristic
428C23#57711yescharacteristic
429Q8—28no—
430Q8—48no—
431Q8—42no—
432Q8—42no—
433Q8—44no—
434Q8—164no—
435Q8—162no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4C2×#9914yes—
510C2×C4C2×#9912yes—
511C2×C4#41912yes—
512C2×C4#41912yes—
513C2×C4C2×#10414yes—
514C2×C4C2×#10412yes—
515C2×C4C2×C4×Q811yescharacteristic
516C2×C4#51911yescharacteristic
517C2×C4#54811yescharacteristic
518C2×C4D4×Q811yescharacteristic
519C8—42no—
520C8—81no—
521C8—82no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22Q8×C22⋊C411yescharacteristic
535C22C22×#9911yescharacteristic
536C22#288411yescharacteristic
537C22C2×#41911yescharacteristic
538C22#289711yescharacteristic
539C22C22×#10411yescharacteristic
540C22#290111yescharacteristic
541C22#336911yescharacteristic
542C22#337511yescharacteristic
543C22#338511yescharacteristic
544C22#338711yescharacteristic
545C4—22no—
546C4—24no—
547C4—24no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—84no—
560C4—82no—
561C4—81no—
562C4—82no—
563C4—81no—
564C4—82no—
565C4#150814yes—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2C2×#150811yescharacteristic
571C2#1755011yescharacteristic
572C2#1760311yescharacteristic
573C2#1763211yescharacteristic
574C2Q8×#9911yescharacteristic
575C2#2252511yescharacteristic
576C2Q8×#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×Q82C4semidirect
17. #2582C4semidirect
18. #2582C4semidirect
19. #2567C4semidirect
20. #2567C4semidirect
26. Q8×C4⋊C4C4semidirect
147. #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
#138172
#99092
#99082
#98892
#98842
#85742
#85722
#82192
C4×#3282