Finite Group Databasebeta

← #194154

Subgroups — #194154

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic, complemented
4#21312C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#8985C214yesmaximal, complemented
7#8973C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4—21no—
14#3101—24no—
15#3101C418yescomplemented
16C2×#483—21no—
17C2×#476C2212yes—
18C2×#461C2212yes—
19C2×#450—21no—
20C2×C4×SD16—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24D4×C4⋊C4—22no—
25D4×C4⋊C4—22no—
26D4×C4⋊C4—22no—
27D4×C4⋊C4C2214yes—
28C2×#389—21no—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yescomplemented
34#1989—22no—
35#1988C2212yescomplemented
36#1987C2212yescomplemented
37#1956C2212yescomplemented
38#1872—22no—
39#1803—22no—
40#1736—21no—
41#1736—21no—
42#1486—24no—
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45#1279C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C22×SD16—22no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533C2×C414yescomplemented
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—48no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4D414yes—
71C2×C4×D4D412yes—
72C2×C4×D4C2312yes—
73C2×#117—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D412yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#483—28no—
90#476—24no—
91#476C2×C418yescomplemented
92#461—24no—
93#461C2×C418yescomplemented
94#450—28no—
95C4×SD16—216no—
96#431—22no—
97#431—22no—
98#431—22no—
99#431C2×C414yescomplemented
100C2×#106—22no—
101#423—22no—
102#423—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×#103—22no—
108C2×#103—24no—
109C2×#102—22no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2312yes—
114#389—24no—
115#389—24no—
116#385—28no—
117#385—28no—
118#384—21no—
119#384—21no—
120#384D412yes—
121#384C2312yes—
122#380—24no—
123#380—24no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C2×#95—22no—
129C2×#95—21no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135D4⋊C8—28no—
136C25—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4C22⋊C414yes—
144C22×D4C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—28no—
154C23×C4—48no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—48no—
160C23×C4—84no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×SD16—28no—
165C2×SD16—44no—
166C2×SD16—44no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yescomplemented
180C4⋊Q8C22×C412yescomplemented
181#125—24no—
182#125—42no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—416no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4C22⋊C418yes—
202C4×D4C22×C414yes—
203C4×D4C2×D414yes—
204#117—24no—
205#117—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—416no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4SD1614yescomplemented
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—41no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—44no—
249C2×C42—41no—
250C2×C42C22×C412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258#106—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8C22×C414yescomplemented
265#103—28no—
266#103—84no—
267#102—28no—
268C22⋊C8—24no—
269C22⋊C8—42no—
270C4×C8—24no—
271C4×C8—22no—
272C4×C8—22no—
273#95—28no—
274#95—22no—
275#95—22no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24#12011yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—216no—
296C2×D4—216no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4C2×C22⋊C414yes—
302C2×D4AGL(1,ℤ/8)14yes—
303C2×D4#13714yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—416no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—416no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—88no—
342C22×C4—832no—
343C22×C4C2×C22⋊C412yes—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12312yes—
352C22×C4C2×SD1612yes—
353C22×C4#13512yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4#13712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359SD16—416no—
360M16—84no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—416no—
382C4⋊C4—44no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—82no—
395C4⋊C4C2×C22⋊C414yes—
396C4⋊C4C2×SD1614yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—416no—
401C22⋊C4—44no—
402C42—44no—
403C42—44no—
404C42—44no—
405C42C2×C22⋊C412yes—
406C42C4×D412yes—
407C42C4×D412yes—
408C42#12112yes—
409C42#12112yes—
410C42#12312yes—
411C42C23×C411yescharacteristic
412C42C22×D411yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—216no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—48no—
424C23—48no—
425C23—48no—
426C23—48no—
427C23—48no—
428C23—48no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23#38612yes—
434C23#45012yes—
435C23#45212yes—
436C23C2×#12011yescharacteristic
437Q8—44no—
438Q8—44no—
439Q8—82no—
440Q8—82no—
441D4—216no—
442D4—24no—
443D4—24no—
444D4—24no—
445D4#41718yes—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—416no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—832no—
478C2×C4—84no—
479C2×C4—88no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4C4×SD1614yes—
488C2×C4#44014yes—
489C2×C4#45012yes—
490C2×C4#45212yes—
491C2×C4#46114yes—
492C2×C4#46314yes—
493C2×C4C22×C22⋊C411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12311yescharacteristic
499C2×C4C22×SD1611yescharacteristic
500C2×C4C2×#13511yescharacteristic
501C2×C4C2×AGL(1,ℤ/8)12yes—
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×#13712yes—
504C2×C4C2×#13711yescharacteristic
505C8—44no—
506C8—82no—
507C8—82no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—28no—
515C22—432no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22#187714yes—
523C22C2×#38611yescharacteristic
524C22C2×#45011yescharacteristic
525C22C2×#45211yescharacteristic
526C22#300311yescharacteristic
527C22#300411yescharacteristic
528C22#301911yescharacteristic
529C22#302011yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—44no—
535C4—44no—
536C4—84no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4#187714yes—
541C4C2×#41712yes—
542C4C2×C4×SD1611yescharacteristic
543C4C2×#44011yescharacteristic
544C4C2×#46111yescharacteristic
545C4C2×#46311yescharacteristic
546C4#309212yes—
547C2—24no—
548C2—48no—
549C2#1874711yescharacteristic
550C2#1875311yescharacteristic
551C2#1877711yescharacteristic
552C2#1878311yescharacteristic
553C2#2090911yescharacteristic
554C2C2×#187711yescharacteristic
555C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21860C2semidirect
4. #21312C2semidirect
6. #8985C2semidirect
7. #8973C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
15. #3101C4semidirect
33. #1990C22semidirect
35. #1988C22semidirect
36. #1987C22semidirect
37. #1956C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
54. #533C2×C4semidirect
91. #476C2×C4semidirect
93. #461C2×C4semidirect
99. #431C2×C4semidirect
179. C4⋊Q8C22×C4semidirect
180. C4⋊Q8C22×C4semidirect
234. C2×C4⋊C4SD16semidirect
264. C4⋊C8C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×D4×C4⋊C42
#218602
#213122
C2×#14862
#89852
#89732
#80432
#78922