Finite Group Databasebeta

← #370181

Subgroups — #370181

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#13880C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic
6#11267C212yesmaximal, complemented
7#11232C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10849C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#385—22no—
20C2×#384C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1854—24no—
26#1851C418yescomplemented
27#1736—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1444—21no—
38#1444—21no—
39#1437—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—21no—
43C4×#95—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—42no—
49C22×M16—42no—
50C22×M16C2×C414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—42no—
57C2×#117D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42—22no—
77C22×C42C2312yes—
78C2×#107—22no—
79C2×#107—42no—
80C2×#106—22no—
81C2×#106—24no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8C2×C418yescomplemented
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2×C414yes—
88#385—48no—
89#384—24no—
90#384—42no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380Q814yes—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—42no—
97C4×C22⋊C4C2×C414yes—
98C4×C22⋊C4Q814yes—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336C2×C418yescomplemented
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—44no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4C22×C412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×Q812yes—
139C23×C4C2×Q812yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C2×M16—28no—
144C2×M16—48no—
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—48no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C22×C414yes—
163C22×C8C22×C412yes—
164C22×C8C22×C412yes—
165#117—28no—
166#117—44no—
167#117—48no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×Q814yes—
205C2×C4⋊C4#4114yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—81no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4C4⋊C418yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42—82no—
226C2×C42C22×C414yes—
227C2×C42C2×Q814yes—
228#107—44no—
229#106—44no—
230C4⋊C8—48no—
231C22⋊C8—28no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—44no—
237#95—44no—
238#95—82no—
239#95C4⋊C414yes—
240#95C4⋊C414yes—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24C2×C4⋊C412yes—
250C24#12112yes—
251C24C22⋊Q812yes—
252C24C22×D411yescharacteristic
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—48no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—88no—
315C22×C4—88no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4C2×C4⋊C414yes—
319C22×C4C2×C4⋊C412yes—
320C22×C4C2×C4⋊C414yes—
321C22×C4C4×D414yes—
322C22×C4C4×Q814yes—
323C22×C4#12112yes—
324C22×C4C22⋊Q812yes—
325C22×C4#12514yes—
326C22×C4C4⋊Q814yes—
327C22×C4AGL(1,ℤ/8)14yes—
328C22×C4#13714yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C4H(5,𝔽2)12yes—
339M16—416no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—48no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C4⋊C4#12314yes—
380C22⋊C4—216no—
381C22⋊C4—48no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—48no—
389C42—88no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23#38914yes—
413C23C22×C4⋊C411yescharacteristic
414C23C2×#12111yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23#54612yes—
417C23#54612yes—
418C23C2×AGL(1,ℤ/8)12yes—
419C23C2×#13712yes—
420C23#57611yescharacteristic
421C23#57711yescharacteristic
422C23#57711yescharacteristic
423C23#57811yescharacteristic
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—816no—
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—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#38914yes—
483C2×C4#42818yes—
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×Q811yescharacteristic
486C2×C4C4×#4112yes—
487C2×C4#51812yes—
488C2×C4C2×#12311yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4C2×#12511yescharacteristic
491C2×C4#52912yes—
492C2×C4C2×C4⋊Q811yescharacteristic
493C2×C4#53411yescharacteristic
494C2×C4#53511yescharacteristic
495C2×C4#53812yes—
496C2×C4#55112yes—
497C2×C4#55112yes—
498C2×C4#55412yes—
499C2×C4#55412yes—
500C2×C4#55412yes—
501C2×C4#55412yes—
502C8—42no—
503C8—44no—
504C8—42no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—44no—
509C22—44no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22C2×#38911yescharacteristic
517C22#234912yes—
518C22#241111yescharacteristic
519C22#241211yescharacteristic
520C22#245012yes—
521C22C2×#42812yes—
522C22#291211yescharacteristic
523C22#291311yescharacteristic
524C22#303811yescharacteristic
525C22#303911yescharacteristic
526C22#304511yescharacteristic
527C22#304611yescharacteristic
528C22#307912yes—
529C22#308111yescharacteristic
530C22#308211yescharacteristic
531C22#330812yes—
532C22#331212yes—
533C22#331412yes—
534C22#331812yes—
535C22#332412yes—
536C22#332912yes—
537C22#333112yes—
538C22#333212yes—
539C4—28no—
540C4—24no—
541C4—44no—
542C4—44no—
543C4—84no—
544C4—84no—
545C4—82no—
546C4—161no—
547C4—161no—
548C4C4×#12311yescharacteristic
549C4#232111yescharacteristic
550C4#236812yes—
551C4#250211yescharacteristic
552C4#250512yes—
553C4#258311yescharacteristic
554C2—44no—
555C2#1404411yescharacteristic
556C2#2117111yescharacteristic
557C2#2117511yescharacteristic
558C2#2270912yes—
559C2#2271512yes—
560C2#2449211yescharacteristic
561C2#2449811yescharacteristic
562C2#2454211yescharacteristic
563C2#2454811yescharacteristic
564C2#2642512yes—
565C2#2643112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11267C2semidirect
7. #11232C2semidirect
8. #10908C2semidirect
9. #10849C2semidirect
26. #1851C4semidirect
84. C2×C4⋊C8C2×C4semidirect
116. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18512
C2×#17362
#138802
#138012
#112672
#112322
#109082
#108492
#107302
#107222