Finite Group Databasebeta

← #188175

Subgroups — #188175

Order 512.

2178 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic
2#21829C212yesmaximal, complemented
3#17398C212yesmaximal, complemented
4#17398C212yesmaximal, complemented
5#8556C212yesmaximal, complemented
6#8555C212yesmaximal, complemented
7#8554C212yesmaximal, complemented
8#8553C212yesmaximal
9C2×#563—28no—
10C2×#563—28no—
11C2×#563—28no—
12C22×C4⋊Q8C411yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#126C411yescharacteristic
15C22×#126C2211yescharacteristic
16C22×#126C2211yescharacteristic
17C22×#126C2211yescharacteristic
18C22×#125C411yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C411yescharacteristic
21C22×#123C411yescharacteristic
22C22×#123C2211yescharacteristic
23C22×C22⋊Q8C411yescharacteristic
24C22×C22⋊Q8C411yescharacteristic
25C22×C22⋊Q8C411yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27#2920C2214yes—
28#1997—28no—
29#1821C2214yes—
30#1821C2214yes—
31#1473—28no—
32#1473—28no—
33#1463C2214yes—
34#1463C2214yes—
35#1462C2214yes—
36#1462C2214yes—
37C23×Q8C2×C411yescharacteristic
38C23×Q8C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43#563—232no—
44#563—232no—
45#563—232no—
46#563—232no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8D418yes—
53C2×#126—24no—
54C2×#126—24no—
55C2×#126—24no—
56C2×#126—24no—
57C2×#126—24no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#126—24no—
61C2×#126—24no—
62C2×#126—24no—
63C2×#126D418yes—
64C2×#126D418yes—
65C2×#125—24no—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125—24no—
70C2×#125—24no—
71C2×#124—24no—
72C2×#124—24no—
73C2×#124—24no—
74C2×#123—24no—
75C2×#123—24no—
76C2×#123—24no—
77C2×#123—24no—
78C2×#123—24no—
79C2×#123—24no—
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×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C4⋊C4C2×C411yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
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⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2×C411yescharacteristic
109C22×C22⋊C4C2×C411yescharacteristic
110C22×C22⋊C4C2×C411yescharacteristic
111C22×C22⋊C4C2×C411yescharacteristic
112C22×C22⋊C4C2×C411yescharacteristic
113C22×C22⋊C4C2×C411yescharacteristic
114C22×C22⋊C4C2×C411yescharacteristic
115C22×C22⋊C4C2×C411yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2×C411yescharacteristic
119C22×C42C2×C411yescharacteristic
120C22×C42C2311yescharacteristic
121#432—24no—
122#432—24no—
123#431—24no—
124#431—24no—
125#423—24no—
126#407C2318yes—
127C4×M16—24no—
128#343—28no—
129#343—28no—
130#324—28no—
131#324—28no—
132#323—28no—
133#323—28no—
134C25C22×C411yescharacteristic
135C25C22×C411yescharacteristic
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—82no—
144C22×Q8—82no—
145C22×Q8—82no—
146C22×Q8C22⋊C414yes—
147C22×Q8C22⋊C414yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—82no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—82no—
159C23×C4—82no—
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×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2411yescharacteristic
173C2×M16—28no—
174C4⋊Q8—232no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179#126—232no—
180#126—232no—
181#126—28no—
182#126—28no—
183#126—28no—
184#126—28no—
185#126—28no—
186#126—28no—
187#126—28no—
188#126—28no—
189#125—232no—
190#125—232no—
191#124—232no—
192#123—232no—
193#123—232no—
194C22⋊Q8—232no—
195C22⋊Q8—232no—
196C22⋊Q8—232no—
197C22⋊Q8—232no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
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—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
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⋊C4C22⋊C414yes—
237C2×C4⋊C4C22⋊C414yes—
238C2×C4⋊C4C22⋊C414yes—
239C2×C4⋊C4C22⋊C414yes—
240C2×C4⋊C4C22⋊C414yes—
241C2×C4⋊C4C22⋊C414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
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—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C22⋊C414yes—
271C2×C22⋊C4C22⋊C414yes—
272C2×C22⋊C4C22⋊C414yes—
273C2×C22⋊C4C22⋊C414yes—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C2×D414yes—
283C2×C42C2×D414yes—
284C2×C42C2×D414yes—
285C4⋊C8—44no—
286C4⋊C8—44no—
287C22⋊C8—28no—
288C22⋊C8—28no—
289#97—44no—
290C4×C8—44no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24C2×C22⋊C412yes—
302C24C2×C22⋊C412yes—
303C24C23×C411yescharacteristic
304C2×Q8—28no—
305C2×Q8—28no—
306C2×Q8—28no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—84no—
316C2×D4—28no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—84no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
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—28no—
341C22×C4—28no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
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×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380M16—416no—
381C2×C8—48no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—28no—
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—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C42—28no—
409C42—28no—
410C42#12014yes—
411C42#12014yes—
412C42#12014yes—
413C42#12014yes—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—88no—
438C23—88no—
439C23C22×C22⋊C411yescharacteristic
440C23C22×C22⋊C411yescharacteristic
441C23C22×C22⋊C411yescharacteristic
442Q8—88no—
443Q8—88no—
444Q8—88no—
445D4—88no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
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—88no—
496C2×C4—88no—
497C2×C4#37914yes—
498C2×C4#37914yes—
499C2×C4C2×#12014yes—
500C8—88no—
501C22—28no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—84no—
520C22C2×#37911yescharacteristic
521C22#288412yes—
522C22#288812yes—
523C22#288912yes—
524C4—24no—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C2—22no—
538C2—22no—
539C2—22no—
540C2—82no—
541C2—82no—
542C2#2058711yescharacteristic
543C2#2059211yescharacteristic
544C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×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. #21829C2semidirect
3. #17398C2semidirect
4. #17398C2semidirect
5. #8556C2semidirect
6. #8555C2semidirect
7. #8554C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
#218292
#173982
#85562
#85552
#85542
#85532