Finite Group Databasebeta

← #188145

Subgroups — #188145

Order 512.

2150 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2#21830C212yesmaximal, complemented
3#17398C212yesmaximal
4#17365C212yesmaximal, complemented
5#8556C212yesmaximal
6#8555C212yesmaximal
7#8554C212yesmaximal, complemented
8#8554C212yesmaximal, complemented
9C2×#541—28no—
10C2×#541—28no—
11C2×#541—24no—
12C2×#541—24no—
13C22×C4⋊Q8C411yescharacteristic
14C22×#126C411yescharacteristic
15C22×#126C411yescharacteristic
16C22×#126C2211yescharacteristic
17C22×#126C2211yescharacteristic
18C22×#125C411yescharacteristic
19C22×#125C411yescharacteristic
20C22×#125C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×C22⋊Q8C411yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C411yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×#117C411yescharacteristic
28#2920C2214yes—
29#1998—28no—
30#1821C2214yes—
31#1821C2214yes—
32#1473—28no—
33#1467—24no—
34#1463C2214yes—
35#1463C2214yes—
36#1463C2214yes—
37#1462C2214yes—
38#1461—24no—
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42#541—232no—
43#541—232no—
44#541—232no—
45#541—232no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×#126—24no—
50C2×#126—24no—
51C2×#126—24no—
52C2×#126—24no—
53C2×#126—24no—
54C2×#126—24no—
55C2×#126—24no—
56C2×#126—24no—
57C2×#126—22no—
58C2×#126—22no—
59C2×#126—24no—
60C2×#126—22no—
61C2×#126—22no—
62C2×#126D418yes—
63C2×#125—24no—
64C2×#125—24no—
65C2×#125—24no—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125—24no—
70C2×#125—24no—
71C2×#125—24no—
72C2×#125—24no—
73C2×#125—24no—
74C2×#125—22no—
75C2×#125—22no—
76C2×#125—24no—
77C2×#125—22no—
78C2×#125—22no—
79C2×#125D418yes—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8D414yes—
93C2×C4×Q8D414yes—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—22no—
97C2×#117—22no—
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C411yescharacteristic
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2×C411yescharacteristic
119C22×C22⋊C4C2×C411yescharacteristic
120C22×C22⋊C4C2×C411yescharacteristic
121C22×C22⋊C4C2×C411yescharacteristic
122C22×C22⋊C4C2×C411yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2×C411yescharacteristic
125C22×C42C2×C411yescharacteristic
126C22×C42C2×C411yescharacteristic
127C22×C42C2×C411yescharacteristic
128C22×C42C2×C411yescharacteristic
129C22×C42C2311yescharacteristic
130#432—24no—
131#432—24no—
132#431—24no—
133#431—24no—
134#423—24no—
135#407C2318yes—
136C4×M16—24no—
137#343—28no—
138#343—28no—
139#324—28no—
140#324—28no—
141#324—28no—
142#323—28no—
143C25C22×C411yescharacteristic
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8C22⋊C414yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
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×C4C2411yescharacteristic
170C2×M16—28no—
171C4⋊Q8—232no—
172#126—232no—
173#126—232no—
174#126—216no—
175#126—216no—
176#126—28no—
177#126—28no—
178#126—28no—
179#126—28no—
180#125—232no—
181#125—232no—
182#125—232no—
183#125—216no—
184#125—216no—
185#125—28no—
186#125—28no—
187#125—28no—
188#125—28no—
189C22⋊Q8—232no—
190C22⋊Q8—232no—
191C4×Q8—216no—
192C4×Q8—216no—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×Q8—28no—
196C4×Q8—28no—
197#117—216no—
198#117—216no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
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⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22⋊C414yes—
251C2×C4⋊C4C22⋊C414yes—
252C2×C4⋊C4C22⋊C414yes—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22⋊C414yes—
255C2×C4⋊C4C22⋊C414yes—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C22⋊C414yes—
279C2×C22⋊C4C22⋊C414yes—
280C2×C22⋊C4C2×D414yes—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C22⋊C412yes—
295C2×C42C22⋊C412yes—
296C2×C42C22⋊C412yes—
297C2×C42C22⋊C412yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C4⋊C8—44no—
303C4⋊C8—44no—
304C22⋊C8—28no—
305C22⋊C8—28no—
306#97—44no—
307C4×C8—44no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24C2×C22⋊C412yes—
314C24C23×C411yescharacteristic
315C2×Q8—28no—
316C2×Q8—28no—
317C2×Q8—28no—
318C2×Q8—44no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387M16—416no—
388C2×C8—48no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—28no—
419C42—28no—
420C42—28no—
421C42—28no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C42#12014yes—
427C42#12014yes—
428C42#12014yes—
429C42#12014yes—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—88no—
444C23C22×C22⋊C411yescharacteristic
445C23C22×C22⋊C411yescharacteristic
446C23C22×C22⋊C411yescharacteristic
447Q8—416no—
448Q8—416no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—416no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
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#37912yes—
505C2×C4#37912yes—
506C2×C4#37914yes—
507C2×C4#41314yes—
508C2×C4C2×#12014yes—
509C8—88no—
510C22—28no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22C2×#37911yescharacteristic
524C22#288412yes—
525C22C2×#41312yes—
526C22#288912yes—
527C4—24no—
528C4—24no—
529C4—24no—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C2—22no—
543C2—22no—
544C2—22no—
545C2—82no—
546C2#2058711yescharacteristic
547C2#2059111yescharacteristic
548C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21830C2semidirect
4. #17365C2semidirect
7. #8554C2semidirect
8. #8554C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
#218302
#173982
#173652
#85562
#85552
#85542