Finite Group Databasebeta

← #215731

Subgroups — #215731

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4#9883C212yesmaximal, complemented
5#9882C212yesmaximal, complemented
6#9880C214yesmaximal, complemented
7#8700C214yesmaximal
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#431—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C4×C22⋊Q8—21no—
20C4×C22⋊Q8—21no—
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387C2212yes—
24C2×#385C412yes—
25C2×#380C411yescharacteristic
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C4⋊C4C412yes—
28C2×C4×C22⋊C4C2212yes—
29#1817—21no—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1779C2212yes—
36#1779C2212yes—
37#1749—21no—
38#1749C2212yes—
39C4×C22⋊C8—21no—
40C4×C22⋊C8C2212yes—
41#1280C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—28no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—44no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C42—44no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C412yes—
68C22×C42C2×C412yes—
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#432—44no—
81#431—44no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×#97—22no—
91C2×C4×C8—22no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C25C22×C411yescharacteristic
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8—42no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—24no—
171C22×C8—24no—
172C22⋊Q8—28no—
173C22⋊Q8—28no—
174C22⋊Q8—28no—
175C22⋊Q8—28no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×Q8—44no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4M1618yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—28no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C4⋊C8—44no—
241C4⋊C8—44no—
242C22⋊C8—28no—
243C22⋊C8—28no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248#97—44no—
249C4×C8—44no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260C24—22no—
261C24—22no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268C24#13114yes—
269C24C23×C411yescharacteristic
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—42no—
273C2×Q8—416no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
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—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
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—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—416no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×M1614yes—
359C22×C4C2×M1614yes—
360C22×C4C2×M1614yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C42−1+414yes—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—44no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—44no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—44no—
402C42—44no—
403C42—44no—
404C42—44no—
405C42#9914yes—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—28no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23#40812yes—
430C23#40812yes—
431C23C22×C22⋊C411yescharacteristic
432C23#51912yes—
433C23C22×M1612yes—
434C23C2×#13112yes—
435Q8—48no—
436Q8—48no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4#40712yes—
500C2×C4#40712yes—
501C2×C4C2×#9912yes—
502C2×C4C2×#9912yes—
503C2×C4C2×#9912yes—
504C2×C4#41012yes—
505C2×C4#41312yes—
506C2×C4#41312yes—
507C2×C4#53612yes—
508C2×C4#53612yes—
509C8—84no—
510C8—84no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—416no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22#231011yescharacteristic
533C22C2×#40711yescharacteristic
534C22C2×#40811yescharacteristic
535C22C22×#9911yescharacteristic
536C22C2×#41011yescharacteristic
537C22C2×#41311yescharacteristic
538C22C2×#41311yescharacteristic
539C22#298414yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—44no—
551C4—44no—
552C4—41no—
553C4—41no—
554C4—82no—
555C4—82no—
556C2—24no—
557C2—42no—
558C2—42no—
559C2#1736111yescharacteristic
560C2#1736211yescharacteristic
561C2#1738111yescharacteristic
562C2#1738211yescharacteristic
563C2#2182011yescharacteristic
564C2#2182511yescharacteristic
565C2#2183111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9883C2semidirect
5. #9882C2semidirect
6. #9880C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#207152
#207142
#98832
#98822
#98802
#87002