Finite Group Databasebeta

← #367358

Subgroups — #367358

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1959C211yesmaximal, characteristic
2C2×#1958C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic, complemented
4#14273C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10843C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2211yescharacteristic
28#1959—24no—
29#1958C418yescomplemented
30#1834—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1439—22no—
38#1439—22no—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#124—22no—
47C2×#124—22no—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124C2×C414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
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—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103C2×C418yes—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102C2×C418yes—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8C2×C414yes—
86#391—24no—
87#391—42no—
88#390—24no—
89#390—42no—
90#384—22no—
91#384—22no—
92#384—22no—
93#384D414yes—
94#382—22no—
95#382C2×C414yes—
96C4×C4⋊C4—42no—
97C4×C4⋊C4—42no—
98#376—22no—
99#376—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D412yes—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—24no—
115C25—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8C22×C412yes—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22×C412yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
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—22no—
158C22×C8C22×C414yes—
159C22×C8#4114yes—
160#124—28no—
161#124—28no—
162#124—44no—
163#124—44no—
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—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—44no—
212C2×C42—82no—
213C2×C42—82no—
214C2×C42C22×C412yes—
215C2×C42C2×D412yes—
216C2×C42#4114yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221#103—48no—
222#103—48no—
223#102—48no—
224#102—48no—
225#97—44no—
226C4×C8—28no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—82no—
235#95—82no—
236C24—22no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24C22×D411yescharacteristic
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
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—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
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—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12312yes—
333C22×C4#12312yes—
334C22×C4#12414yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C42—24no—
380C42—24no—
381C42—42no—
382C42—42no—
383C42—88no—
384C42#11714yes—
385C42C4×D414yes—
386C42#12414yes—
387C42#12614yes—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23C2×C4×D411yescharacteristic
412C23C2×#12111yescharacteristic
413C23C2×#12311yescharacteristic
414C23C2×#12411yescharacteristic
415C23#53211yescharacteristic
416C23#53511yescharacteristic
417C23#53511yescharacteristic
418C23#57612yes—
419C23#57712yes—
420C23#57712yes—
421C23#57812yes—
422Q8—44no—
423Q8—44no—
424Q8—82no—
425Q8—82no—
426D4—44no—
427D4—44no—
428D4—82no—
429D4—82no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
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—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4#38814yes—
484C2×C4#38814yes—
485C2×C4C2×#11711yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C4×#4112yes—
488C2×C4#51812yes—
489C2×C4C2×#12411yescharacteristic
490C2×C4C2×#12611yescharacteristic
491C2×C4#52912yes—
492C2×C4#54012yes—
493C2×C4#54612yes—
494C2×C4#55012yes—
495C2×C4#55312yes—
496C2×C4#55312yes—
497C2×C4#55312yes—
498C2×C4#55512yes—
499C2×C4#55912yes—
500C2×C4#55912yes—
501C8—44no—
502C8—44no—
503C22—84no—
504C22—82no—
505C22—82no—
506C22—84no—
507C22—82no—
508C22—162no—
509C22—162no—
510C22—161no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22C2×#38811yescharacteristic
515C22C4×#12411yescharacteristic
516C22#232811yescharacteristic
517C22#250011yescharacteristic
518C22#250411yescharacteristic
519C22#254411yescharacteristic
520C22#254811yescharacteristic
521C22#295112yes—
522C22#295212yes—
523C22#304111yescharacteristic
524C22#304211yescharacteristic
525C22#304311yescharacteristic
526C22#304411yescharacteristic
527C22#309812yes—
528C22#309912yes—
529C22#313612yes—
530C22#313712yes—
531C22#317812yes—
532C22#317912yes—
533C22#322412yes—
534C22#322412yes—
535C22#322512yes—
536C22#322512yes—
537C4—24no—
538C4—24no—
539C4—44no—
540C4—82no—
541C4—84no—
542C4—82no—
543C4—84no—
544C4—161no—
545C4—161no—
546C4#235712yes—
547C4#236312yes—
548C4#249012yes—
549C4#251312yes—
550C2—162no—
551C2#1400211yescharacteristic
552C2#2106711yescharacteristic
553C2#2106811yescharacteristic
554C2#2164312yes—
555C2#2227412yes—
556C2#2790412yes—
557C2#2798412yes—
558C2#2862711yescharacteristic
559C2#2862811yescharacteristic
560C2#2875811yescharacteristic
561C2#2875911yescharacteristic
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
3. C2×#1834C2semidirect
5. #13779C2semidirect
6. #11190C2semidirect
8. #10932C2semidirect
10. #10843C2semidirect
11. #10694C2semidirect
29. #1958C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19592
C2×#19582
C2×#18342
#142732
#137792
#111902
#111892
#109322
#109312
#108432
#106942