Finite Group Databasebeta

← #375728

Subgroups — #375728

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2C2×#1746C211yesmaximal, characteristic, complemented
3#13797C211yesmaximal, characteristic, complemented
4#11384C212yesmaximal, complemented
5#11009C212yesmaximal
6#10932C212yesmaximal, complemented
7#10931C212yesmaximal
8#10928C211yesmaximal, characteristic, complemented
9#10927C211yesmaximal, characteristic
10#10926C211yesmaximal, characteristic, complemented
11#10925C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2640—24no—
21#2640C418yescomplemented
22C2×#397—22no—
23C2×#393—22no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30#1746—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—22no—
41#1434—22no—
42C23×Q8—21no—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—42no—
49C2×C4⋊Q8—42no—
50C2×C4⋊Q8C2×C414yes—
51C2×#124—22no—
52C2×#124—42no—
53C2×#124—42no—
54C2×#124C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#97—24no—
81C2×#97—22no—
82C2×C4×C8—24no—
83C2×C4×C8—22no—
84#397—216no—
85#393—216no—
86#391—22no—
87#391—22no—
88#391C2×C414yes—
89#391C2×C414yes—
90#390—22no—
91#390—22no—
92#390C2×C414yes—
93#390C2×C414yes—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96#376C2×C414yes—
97#376D414yes—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106C2×#95D412yes—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—24no—
113C25—21no—
114C22×Q8—24no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C22×Q8C22×C412yes—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158C22×C8#4114yes—
159C4⋊Q8—48no—
160C4⋊Q8—48no—
161#124—48no—
162#124—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4#4114yes—
185C2×C4⋊C4#4114yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—28no—
205C2×C42—82no—
206C2×C42—82no—
207C2×C42C22×C412yes—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213#103—48no—
214#103—48no—
215#102—48no—
216#102—48no—
217#97—28no—
218C4×C8—28no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—82no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95C22⋊C418yes—
227#95C22⋊C418yes—
228C24—22no—
229C24—42no—
230C24—42no—
231C24—41no—
232C24—41no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24C22×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—48no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—88no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—88no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
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—24no—
299C22×C4—24no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C2×C22⋊C414yes—
325C22×C4C2×C22⋊C414yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)12yes—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—88no—
366C22⋊C4—88no—
367C42—22no—
368C42—22no—
369C42—24no—
370C42—24no—
371C42—22no—
372C42—22no—
373C42—24no—
374C42—24no—
375C42—88no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—82no—
396C23—82no—
397C23C22×C22⋊C411yescharacteristic
398C23#52512yes—
399C23#53212yes—
400C23#53511yescharacteristic
401C23#53511yescharacteristic
402C23#57612yes—
403C23#57712yes—
404C23#57712yes—
405C23#57812yes—
406Q8—44no—
407Q8—44no—
408Q8—816no—
409Q8—82no—
410Q8—82no—
411D4—44no—
412D4—44no—
413D4—82no—
414D4—82no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4C4×#4111yescharacteristic
469C2×C4C4×#4111yescharacteristic
470C2×C4C4×#4112yes—
471C2×C4#51812yes—
472C2×C4#52511yescharacteristic
473C2×C4#52511yescharacteristic
474C2×C4#53212yes—
475C2×C4#53512yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55112yes—
479C2×C4#55112yes—
480C2×C4#55512yes—
481C2×C4#55512yes—
482C2×C4#55512yes—
483C2×C4#55512yes—
484C8—44no—
485C8—44no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—84no—
490C22—84no—
491C22—162no—
492C22—162no—
493C22—162no—
494C22—162no—
495C22#229711yescharacteristic
496C22#229711yescharacteristic
497C22#230911yescharacteristic
498C22#261012yes—
499C22#261012yes—
500C22#289712yes—
501C22#289812yes—
502C22#311812yes—
503C22#312112yes—
504C22#312412yes—
505C22#312512yes—
506C22#317312yes—
507C22#317512yes—
508C22#317812yes—
509C22#317912yes—
510C22#322011yescharacteristic
511C22#322111yescharacteristic
512C22#322211yescharacteristic
513C22#322311yescharacteristic
514C22#322412yes—
515C22#322512yes—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—44no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—84no—
524C4—84no—
525C4#237212yes—
526C4#237212yes—
527C4#261012yes—
528C4#261012yes—
529C2—162no—
530C2#1416911yescharacteristic
531C2#2139712yes—
532C2#2141212yes—
533C2#2181311yescharacteristic
534C2#2181411yescharacteristic
535C2#2956612yes—
536C2#2956712yes—
537C2#2957012yes—
538C2#2957112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1746C2semidirect
3. #13797C2semidirect
4. #11384C2semidirect
6. #10932C2semidirect
8. #10928C2semidirect
10. #10926C2semidirect
21. #2640C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
C2×#17462
#137972
#113842
#110092
#109322
#109312
#109282
#109272
#109262
#109252