Finite Group Databasebeta

← #192439

Subgroups — #192439

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34219C211yesmaximal, characteristic
2#21983C211yesmaximal, characteristic, complemented
3#21861C211yesmaximal, characteristic
4C2×#1500C211yesmaximal, characteristic
5#13963C211yesmaximal, characteristic, complemented
6#8871C212yesmaximal, complemented
7#8870C212yesmaximal
8#8867C212yesmaximal, complemented
9#8866C212yesmaximal
10#8013C211yesmaximal, characteristic, complemented
11#7906C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2373—22no—
16#2372—21no—
17#2371—21no—
18#2356—21no—
19#2353—21no—
20#2346—21no—
21#2343—21no—
22#2329C2212yes—
23#2325C2212yes—
24#2318—21no—
25C4×#123—22no—
26#2283C2211yescharacteristic
27C2×#382C2211yescharacteristic
28#2275—21no—
29C2×#376C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1991—21no—
32#1958—21no—
33#1949—22no—
34#1874C2212yes—
35#1872C2212yes—
36#1853C2212yes—
37#1852C2212yes—
38#1846—21no—
39#1835—21no—
40#1805—21no—
41#1804—21no—
42#1500C418yescomplemented
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#327C2211yescharacteristic
46C2×#327C2211yescharacteristic
47#1444C2212yes—
48#1442C2212yes—
49#1436—22no—
50C4⋊#95—21no—
51#1434—21no—
52#1294C2212yes—
53#1294C2212yes—
54#1280C2212yes—
55#1280C2212yes—
56#529—22no—
57#529—22no—
58#529C2×C414yes—
59#529Q814yes—
60C2×#126—21no—
61C2×#125D411yescharacteristic
62C2×#124D411yescharacteristic
63C2×#123—21no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4D411yescharacteristic
70C2×#117—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2×C414yescomplemented
80C2×#107—22no—
81C2×#106—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×#103—22no—
85C2×#102—22no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2311yescharacteristic
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#391—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#386—22no—
104#386—22no—
105#385—22no—
106#385—22no—
107#383—21no—
108#383—21no—
109#382—21no—
110#382—21no—
111#382—21no—
112#382—21no—
113#382—21no—
114#382D411yescharacteristic
115#382D411yescharacteristic
116#381—21no—
117#381—21no—
118#380—21no—
119#380C2312yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131#376—21no—
132#376—21no—
133#376—21no—
134#376—21no—
135#376—21no—
136#376—21no—
137#376—22no—
138#376C2×C414yescomplemented
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#328C2×C414yescomplemented
156#328C2×C414yescomplemented
157#327C2×C414yes—
158#327C2×C414yes—
159C22×Q8C2×D411yescharacteristic
160C22×D4—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175#126—24no—
176#126—24no—
177#125—22no—
178#125C4⋊C414yescomplemented
179#124—22no—
180#124C4⋊C414yescomplemented
181#123—24no—
182#123—24no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8C22×C412yes—
186C22⋊Q8C2×Q812yes—
187#121—21no—
188#121—21no—
189#121C22×C412yes—
190#121C2×Q812yes—
191C4×Q8—22no—
192C4×Q8C4⋊C414yescomplemented
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4C4⋊C414yescomplemented
199#117—24no—
200#117—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42C22×C412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246#107—82no—
247#106—82no—
248C4⋊C8—24no—
249C4⋊C8—42no—
250#103—82no—
251#102—82no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8C22×C412yes—
257C22⋊C8C22×C412yes—
258C4×C8—24no—
259C4×C8—42no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281#95—44no—
282#95—44no—
283#95—41no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—82no—
288#95C22×C412yes—
289#95C22×C412yes—
290C24—21no—
291C24C22×D411yescharacteristic
292C2×Q8—21no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8C2×C4⋊C412yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4C2×C4⋊C412yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4C22⋊Q812yes—
352C22×C4C22⋊Q812yes—
353C22×C4#13514yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—88no—
386C4⋊C4C2×C4⋊C412yes—
387C4⋊C4C2×C4⋊C412yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—24no—
401C42—44no—
402C42—44no—
403C42—48no—
404C42—44no—
405C42—82no—
406C42—82no—
407C42—82no—
408C42C2×C4⋊C412yes—
409C42C2×C4⋊C412yes—
410C42C4×D412yes—
411C42C4×D412yes—
412C42C22⋊Q812yes—
413C42C22⋊Q812yes—
414C42#12312yes—
415C42#12312yes—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23C2×C4×D411yescharacteristic
425C23C2×#12011yescharacteristic
426C23C2×C22⋊Q811yescharacteristic
427C23C2×#13512yes—
428C23#57512yes—
429Q8—42no—
430Q8—81no—
431D4—42no—
432D4—44no—
433D4—42no—
434D4—42no—
435D4—81no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
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—84no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—164no—
495C2×C4#38512yes—
496C2×C4#38512yes—
497C2×C4#38512yes—
498C2×C4#38512yes—
499C2×C4C22×C4⋊C411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C8—42no—
504C8—42no—
505C8—82no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—82no—
518C22—82no—
519C22C2×#38511yescharacteristic
520C22C4×#13511yescharacteristic
521C22#294311yescharacteristic
522C22#300212yes—
523C22#301812yes—
524C22#302711yescharacteristic
525C22#302811yescharacteristic
526C22#307612yes—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—44no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—162no—
545C2—21no—
546C2—21no—
547C2—22no—
548C2—82no—
549C2#1838612yes—
550C2#1840712yes—
551C2#2086311yescharacteristic
552C2#2087211yescharacteristic
553C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21983C2semidirect
5. #13963C2semidirect
6. #8871C2semidirect
8. #8867C2semidirect
10. #8013C2semidirect
11. #7906C2semidirect
42. #1500C4semidirect
79. #431C2×C4semidirect
138. #376C2×C4semidirect
155. #328C2×C4semidirect
156. #328C2×C4semidirect
178. #125C4⋊C4semidirect
180. #124C4⋊C4semidirect
192. C4×Q8C4⋊C4semidirect
198. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342192
#219832
#218612
C2×#15002
#139632
#88712
#88702
#88672
#88662
#80132
#79062