Finite Group Databasebeta

← #443533

Subgroups — #443533

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic
2C2×#2550C211yesmaximal, characteristic
3C2×#2059C212yesmaximal
4C2×#2057C211yesmaximal, characteristic
5#15128C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11325C212yesmaximal
9#10908C212yesmaximal
10#10880C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C2×#498—28no—
16C2×#498—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2550—24no—
22#2550—24no—
23C2×#398—21no—
24C2×#395—21no—
25C2×#395—21no—
26C2×#391—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2059—28no—
35#2057—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8D414yes—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—24no—
64C2×#125—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#498—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106D418yes—
94C2×C4×C8—24no—
95C2×C4×C8—22no—
96#398—28no—
97#395—28no—
98#395—44no—
99#391—28no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389—42no—
106#389D414yes—
107#382—22no—
108#382—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340Q818yescomplemented
130#336—24no—
131#336—22no—
132#336Q818yescomplemented
133C22×Q8—24no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8—82no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C22×C8C2×Q814yes—
168C22×C8C2×Q814yes—
169C22×C8C2×Q814yes—
170C4⋊Q8—28no—
171C4⋊Q8—28no—
172C4⋊Q8—44no—
173C4⋊Q8—48no—
174C4⋊Q8—48no—
175C4⋊Q8—44no—
176#125—48no—
177#125—44no—
178#125—44no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
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—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×Q814yes—
228C2×C4⋊C4C2×Q814yes—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—82no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×Q814yes—
243#107—48no—
244#106—216no—
245#106—216no—
246C4×C8—28no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—44no—
255#95—44no—
256#95—42no—
257#95—42no—
258#95—82no—
259C24C22×D411yescharacteristic
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—84no—
264C2×Q8—88no—
265C2×Q8—88no—
266C2×Q8—88no—
267C2×Q8—88no—
268C2×Q8—84no—
269C2×Q8—84no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4#12012yes—
329C22×C4#12012yes—
330C22×C4C4⋊Q812yes—
331C22×C4C4⋊Q812yes—
332C22×C4#13512yes—
333C22×C4#13512yes—
334C22×C4AGL(1,ℤ/8)12yes—
335C22×C4#13712yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q812yes—
340C22×C4C22×Q812yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C42−1+412yes—
343C22×C42−1+412yes—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8C4⋊Q818yes—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C4⋊C4#12014yes—
386C4⋊C4C4⋊Q814yes—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—84no—
396C23C2×#12011yescharacteristic
397C23C2×C4⋊Q811yescharacteristic
398C23#53611yescharacteristic
399C23D4212yes—
400C23D4×Q812yes—
401C23C2×#13511yescharacteristic
402C23C2×#13511yescharacteristic
403C23C2×AGL(1,ℤ/8)11yescharacteristic
404C23C2×#13711yescharacteristic
405C23#57712yes—
406C23#57712yes—
407Q8—816no—
408Q8—816no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#44914yes—
474C2×C4#45214yes—
475C2×C4C2×#12011yescharacteristic
476C2×C4C2×C4⋊Q812yes—
477C2×C4C2×C4⋊Q811yescharacteristic
478C2×C4#53411yescharacteristic
479C2×C4#53611yescharacteristic
480C2×C4#53612yes—
481C2×C4#54412yes—
482C2×C4#54412yes—
483C2×C4D4212yes—
484C2×C4D4×Q812yes—
485C2×C4D4×Q812yes—
486C2×C4D4×Q812yes—
487C2×C4D4×Q812yes—
488C2×C4D4×Q812yes—
489C2×C4Q8212yes—
490C2×C4#56312yes—
491C8—28no—
492C8—44no—
493C22#240311yescharacteristic
494C22#243411yescharacteristic
495C22#243411yescharacteristic
496C22#255112yes—
497C22#267712yes—
498C22C2×#44911yescharacteristic
499C22C2×#45211yescharacteristic
500C22#300911yescharacteristic
501C22#301011yescharacteristic
502C22#316011yescharacteristic
503C22#316312yes—
504C22#316511yescharacteristic
505C22#322811yescharacteristic
506C22#323111yescharacteristic
507C22#324412yes—
508C22#326512yes—
509C22#326612yes—
510C22#328212yes—
511C22#328212yes—
512C22#338112yes—
513C22#338112yes—
514C22#338412yes—
515C22#338712yes—
516C4—24no—
517C4—24no—
518C4—44no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—81no—
523C4—82no—
524C4—81no—
525C4—82no—
526C4—84no—
527C4—84no—
528C4#243412yes—
529C4#255112yes—
530C4#267712yes—
531C4#273412yes—
532C2#1576211yescharacteristic
533C2#2385211yescharacteristic
534C2#2385811yescharacteristic
535C2#2581211yescharacteristic
536C2#2582011yescharacteristic
537C2#2583511yescharacteristic
538C2#2583611yescharacteristic
539C2#2881012yes—
540C2#2881912yes—
541C2#3094712yes—
542C2#3095612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
129. #340Q8semidirect
132. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#25502
C2×#20592
C2×#20572
#151282
#113842
#113272
#113252
#109082
#108802
#108742