Finite Group Databasebeta

← #193474

Subgroups — #193474

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4C2×#1497C212yesmaximal, complemented
5#8942C214yesmaximal, complemented
6#8938C214yesmaximal, complemented
7#7900C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C42C2211yescharacteristic
10#2920—28no—
11C2×#432—21no—
12C2×#432C2212yes—
13C2×#431—21no—
14C2×#431C2212yes—
15C2×#431C2212yes—
16C2×#423C2212yes—
17C2×#407—21no—
18C2×C4×M16—21no—
19#2857C411yescharacteristic
20#2857C411yescharacteristic
21#2857C411yescharacteristic
22#2857C411yescharacteristic
23#2855—21no—
24#2855C412yes—
25#2851—21no—
26#2851—21no—
27#2848—21no—
28#2848C412yes—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8C2212yes—
32C4×C22⋊Q8C2212yes—
33#2296—21no—
34#2285C2211yescharacteristic
35#2280—21no—
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1779—21no—
40C4×C22⋊C8C2212yes—
41C4×C22⋊C8C2212yes—
42#1497—28no—
43C2×Q8⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C22×M16—22no—
49#533—24no—
50#533—24no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2312yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C42—24no—
65C22×C42—24no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#432—28no—
79#432—24no—
80#432D418yes—
81#431—28no—
82#431—28no—
83#431—24no—
84#431—24no—
85#423—28no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8C2314yes—
89#407—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C4×M16—28no—
96C2×#97—22no—
97C2×C4×C8—22no—
98C2×C4×C8C2314yes—
99#391—21no—
100#391—21no—
101#391C2×C412yes—
102#391C2×C412yes—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389C2×C412yes—
110#389C2×C412yes—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#382—21no—
124#381—22no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380C2×C412yes—
133#380C2×C412yes—
134#380C2×C412yes—
135#379—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142#336—22no—
143#336—22no—
144Q8⋊C8—216no—
145C25C22⋊C411yescharacteristic
146C22×Q8—21no—
147C22×Q8C22×C412yes—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—28no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8—24no—
180C22×C8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183#125—44no—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—42no—
193C4×Q8—42no—
194#117—44no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D414yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4114yes—
254C2×C42C2411yescharacteristic
255C4⋊C8—28no—
256C4⋊C8—24no—
257C4⋊C8—44no—
258C4⋊C8C2×D418yes—
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—44no—
262C22⋊C8—44no—
263C22⋊C8—44no—
264#97—28no—
265C4×C8—28no—
266C4×C8—24no—
267C4×C8—24no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24C2×C22⋊C411yescharacteristic
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—48no—
350C22×C4—416no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C4×D414yes—
363C22×C4#12012yes—
364C22×C4#12114yes—
365C22×C4#12312yes—
366C22×C4#13114yes—
367C22×C4#13714yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371M16—416no—
372C2×C8—28no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—44no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C42—22no—
401C42—22no—
402C42—21no—
403C42—21no—
404C42—22no—
405C42—22no—
406C42—21no—
407C42—21no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—42no—
415C42—42no—
416C42—44no—
417C42C4×D414yes—
418C42#12012yes—
419C42#12114yes—
420C42#12312yes—
421C42C22×D412yes—
422C42C2×#4112yes—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23#38612yes—
449C23#40812yes—
450C23#41912yes—
451C23C22×C22⋊C411yescharacteristic
452Q8—48no—
453Q8—84no—
454C2×C4—28no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4#38612yes—
510C2×C4#38612yes—
511C2×C4#38612yes—
512C2×C4#40812yes—
513C2×C4#41912yes—
514C2×C4#42112yes—
515C2×C4#42112yes—
516C2×C4C2×C4×D412yes—
517C2×C4C2×#12011yescharacteristic
518C2×C4C2×#12112yes—
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#13112yes—
521C2×C4#56812yes—
522C2×C4C2×#13712yes—
523C2×C4#57512yes—
524C8—48no—
525C8—84no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—416no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22C2×#38611yescharacteristic
549C22C2×#40811yescharacteristic
550C22#287911yescharacteristic
551C22C2×#41911yescharacteristic
552C22#289611yescharacteristic
553C22C2×#42111yescharacteristic
554C22C2×#42111yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4#293314yes—
570C4#300712yes—
571C4#309512yes—
572C2—42no—
573C2—42no—
574C2—42no—
575C2#1747711yescharacteristic
576C2#1747911yescharacteristic
577C2#1749111yescharacteristic
578C2#1749911yescharacteristic
579C2#2089711yescharacteristic
580C2#2090211yescharacteristic
581C2#2091111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1497C2semidirect
5. #8942C2semidirect
6. #8938C2semidirect
7. #7900C2semidirect
43. C2×Q8⋊C8C22semidirect
44. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366252
#207142
C2×#14972
#89422
#89382
#79002