Finite Group Databasebeta

← #197885

Subgroups — #197885

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6#9228C212yesmaximal, complemented
7#9226C212yesmaximal, complemented
8#9220C212yesmaximal, complemented
9#9218C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic
11#7879C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3113—24no—
15#3113C418yescomplemented
16C2×#483—21no—
17C2×#475C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×#451—21no—
22#2984—24no—
23#2984—24no—
24C2×#446—22no—
25C2×#446C2212yes—
26C2×C8×Q8C2212yes—
27C2×#441—21no—
28C2×#441—21no—
29C2×#432—21no—
30C2×#432C2211yescharacteristic
31C2×#432C2211yescharacteristic
32C2×#431—21no—
33C2×#431C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103—21no—
36C2×#384C2211yescharacteristic
37#1990C2212yes—
38#1956C2212yes—
39#1872—21no—
40#1872—21no—
41#1803—22no—
42#1803—21no—
43#1803—21no—
44#1803C2212yes—
45#1803C2212yes—
46C2×#340—21no—
47#1552—24no—
48C2×#332C2212yes—
49C2×#332C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C23×Q8D411yescharacteristic
55C22×Q16—21no—
56C22×Q16—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533C2×C414yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C414yes—
74C2×C4×Q8D414yes—
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#483—28no—
88#475—22no—
89#475—22no—
90#475C2×C414yescomplemented
91#475C2×C414yescomplemented
92#462—22no—
93#462—22no—
94#462C2×C414yescomplemented
95#462C2×C414yescomplemented
96#451—28no—
97#446—216no—
98#446—24no—
99#446—24no—
100C8×Q8—24no—
101C8×Q8—24no—
102#441—28no—
103#441—28no—
104#432—28no—
105#432—22no—
106#432—22no—
107#432C2×C414yes—
108#432D414yes—
109#431—28no—
110#431—22no—
111#431—22no—
112C2×#106—21no—
113C2×#106—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D412yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2312yes—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—22no—
129C2×#103—22no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—44no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×#97—22no—
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—22no—
141C2×C4×C8C2312yes—
142#384—21no—
143#384—21no—
144#384—21no—
145#384C2312yes—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#332—24no—
151#332—24no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8C22⋊C414yes—
161C22×Q8C2×D411yescharacteristic
162C22×Q8C2×D411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—41no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C2×Q16—24no—
170C2×Q16—24no—
171C2×Q16—42no—
172C2×Q16—42no—
173C2×Q16—42no—
174C2×Q16—42no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C412yes—
192#125—24no—
193#125—42no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—24no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8C22⋊C414yes—
213C4×Q8C22⋊C414yes—
214C4×Q8C22×C412yes—
215C4×Q8C22×C412yes—
216C4×Q8C2×D412yes—
217C4×Q8C2×D412yes—
218#117—24no—
219#117—42no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4C22⋊C414yes—
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C22⋊C4—41no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—41no—
255C2×C42C22×C412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42#4111yescharacteristic
261C2×C42#4112yes—
262C2×C42C2411yescharacteristic
263#106—24no—
264#106—24no—
265C4⋊C8—28no—
266C4⋊C8—28no—
267C4⋊C8—24no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—42no—
274C4⋊C8C22×C412yes—
275C4⋊C8C22×C412yes—
276C4⋊C8C2×D414yes—
277#103—24no—
278#103—24no—
279#103—24no—
280#103—24no—
281#103—44no—
282#103—44no—
283#103—82no—
284#103—82no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289#97—28no—
290#97—24no—
291#97—24no—
292C4×C8—28no—
293C4×C8—22no—
294C4×C8—22no—
295C24#12111yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—24no—
301C2×Q8—28no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8C2×C22⋊C412yes—
322C2×Q8C2×C22⋊C412yes—
323C2×Q8AGL(1,ℤ/8)12yes—
324C2×Q8#13712yes—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C22⋊C412yes—
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4#12012yes—
353C22×C4#12112yes—
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12312yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361Q16—48no—
362Q16—48no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—84no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4C2×C22⋊C412yes—
410C4⋊C4C2×C22⋊C412yes—
411C22⋊C4—44no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—24no—
417C42—24no—
418C42—44no—
419C42C2×C22⋊C412yes—
420C42C4×D412yes—
421C42C4×D412yes—
422C42#12012yes—
423C42#12112yes—
424C42#12312yes—
425C42C23×C411yescharacteristic
426C42C22×D411yescharacteristic
427C42C22×D411yescharacteristic
428C42C2×#4111yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23#38612yes—
434C23#47112yes—
435C23#47312yes—
436C23C2×#12111yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—48no—
443Q8—41no—
444Q8—41no—
445Q8—42no—
446Q8—42no—
447Q8—44no—
448Q8—82no—
449Q8—81no—
450Q8—81no—
451Q8#41714yes—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—164no—
492C2×C4#38612yes—
493C2×C4#38612yes—
494C2×C4#38612yes—
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×#12011yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
502C2×C4C2×#13711yescharacteristic
503C2×C4#57511yescharacteristic
504C2×C4#57511yescharacteristic
505C2×C4#57712yes—
506C2×C4#57812yes—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—42no—
511C8—44no—
512C8—82no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22#190814yes—
520C22C2×#38611yescharacteristic
521C22#304311yescharacteristic
522C22#304411yescharacteristic
523C22#305511yescharacteristic
524C22#305611yescharacteristic
525C22C2×#47111yescharacteristic
526C22C2×#47311yescharacteristic
527C4—21no—
528C4—21no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—24no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—82no—
537C4—82no—
538C4—162no—
539C4C2×#41711yescharacteristic
540C4#289611yescharacteristic
541C4#294712yes—
542C4#301312yes—
543C4#305511yescharacteristic
544C4#305611yescharacteristic
545C2—24no—
546C2#1946811yescharacteristic
547C2#1947111yescharacteristic
548C2#1947511yescharacteristic
549C2#1947611yescharacteristic
550C2#2093011yescharacteristic
551C2#2097111yescharacteristic
552C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
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
6. #9228C2semidirect
7. #9226C2semidirect
8. #9220C2semidirect
9. #9218C2semidirect
15. #3113C4semidirect
90. #475C2×C4semidirect
91. #475C2×C4semidirect
94. #462C2×C4semidirect
95. #462C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#29842
#218602
C2×#18032
C2×#15522
#92282
#92262
#92202
#92182
#80402
#78792