Finite Group Databasebeta

← #192332

Subgroups — #192332

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34401C211yesmaximal, characteristic
2#21391C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic, complemented
4C2×#1500C211yesmaximal, characteristic
5#13952C211yesmaximal, characteristic, complemented
6#8869C212yesmaximal
7#8868C212yesmaximal, complemented
8#8867C212yesmaximal, complemented
9#8866C212yesmaximal
10#8022C211yesmaximal, characteristic, complemented
11#7916C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2383—22no—
16#2359—22no—
17#2353—21no—
18#2349—21no—
19#2347—21no—
20#2343—21no—
21#2337—21no—
22#2336—21no—
23#2329C2212yes—
24#2325C2212yes—
25#2320—21no—
26#2318—21no—
27#2285C2211yescharacteristic
28C2×#383C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1989—21no—
32#1983—21no—
33#1947—22no—
34#1874C2212yes—
35#1872C2212yes—
36#1852C2212yes—
37#1852C2212yes—
38C4×#107—21no—
39C4×#107—21no—
40#1763—21no—
41#1762—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—
71C2×#117—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#431—22no—
80#431C2×C414yescomplemented
81C2×#107—22no—
82C2×#107—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×#103—22no—
86C2×#102—22no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2311yescharacteristic
90C2×#97—21no—
91C2×C4×C8—21no—
92#391—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#384D412yes—
113#383—21no—
114#383—21no—
115#383—21no—
116#383—21no—
117#383—21no—
118#382—21no—
119#382—21no—
120#381—21no—
121#380—21no—
122#380—21no—
123#380C2312yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
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—22no—
145C2×#95D411yescharacteristic
146C2×#95D411yescharacteristic
147C2×#95D412yes—
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—22no—
165C23×C4—21no—
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—82no—
197C4×D4—82no—
198C4×D4C4⋊C414yescomplemented
199#117—24no—
200#117—24no—
201#117—81no—
202#117—81no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—21no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—44no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—81no—
242C2×C42—81no—
243C2×C42C22×C412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#107—44no—
252#107—44no—
253C4⋊C8—24no—
254C4⋊C8—42no—
255#103—44no—
256#102—44no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8C22×C412yes—
262C22⋊C8C22×C412yes—
263#97—42no—
264C4×C8—24no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—21no—
277#95—22no—
278#95—21no—
279#95—21no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—44no—
286#95—44no—
287#95—41no—
288#95—41no—
289#95—82no—
290#95C22×C412yes—
291#95C22×C412yes—
292C24—21no—
293C24C22×D411yescharacteristic
294C2×Q8—21no—
295C2×Q8—42no—
296C2×Q8C2×C4⋊C412yes—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4C2×C4⋊C412yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×Q811yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—88no—
389C4⋊C4C2×C4⋊C412yes—
390C4⋊C4C2×C4⋊C412yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C42—24no—
404C42—48no—
405C42—81no—
406C42—81no—
407C42—82no—
408C42—82no—
409C42—82no—
410C42—82no—
411C42—82no—
412C42C2×C4⋊C412yes—
413C42C2×C4⋊C412yes—
414C42C4×D412yes—
415C42C4×D412yes—
416C42C22⋊Q812yes—
417C42C22⋊Q812yes—
418C42#12312yes—
419C42#12312yes—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23C2×C4×D411yescharacteristic
428C23C2×#12011yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#57611yescharacteristic
431C23#57611yescharacteristic
432C23#57811yescharacteristic
433C23#57811yescharacteristic
434Q8—42no—
435Q8—81no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—81no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—164no—
502C2×C4#38512yes—
503C2×C4#38512yes—
504C2×C4#38512yes—
505C2×C4#38512yes—
506C2×C4C22×C4⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C22⋊Q811yescharacteristic
509C2×C4C2×#12311yescharacteristic
510C8—42no—
511C8—42no—
512C8—82no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22C2×#38511yescharacteristic
527C22#295111yescharacteristic
528C22#295111yescharacteristic
529C22#300811yescharacteristic
530C22#301111yescharacteristic
531C22#301812yes—
532C22#302512yes—
533C22#308311yescharacteristic
534C22#308311yescharacteristic
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—44no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—82no—
548C4—82no—
549C4—84no—
550C4—82no—
551C4—82no—
552C4—82no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2—82no—
558C2#1828012yes—
559C2#1828312yes—
560C2#2086311yescharacteristic
561C2#2088111yescharacteristic
562C2#2088811yescharacteristic
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
3. #21358C2semidirect
5. #13952C2semidirect
7. #8868C2semidirect
8. #8867C2semidirect
10. #8022C2semidirect
11. #7916C2semidirect
42. #1500C4semidirect
80. #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
#344012
#213912
#213582
C2×#15002
#139522
#88692
#88682
#88672
#88662
#80222
#79162