Finite Group Databasebeta

← #197966

Subgroups — #197966

Order 512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1559C211yesmaximal, characteristic
6#9226C212yesmaximal, complemented
7#9224C212yesmaximal, complemented
8#9218C212yesmaximal, complemented
9#9216C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic
11#7880C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3104—24no—
15#3104C418yescomplemented
16C2×#484—21no—
17C2×#477C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×#451—21no—
22#2984—24no—
23#2984—24no—
24C2×#446—22no—
25C2×#446C2212yes—
26C2×C8×Q8C2212yes—
27C2×C4×Q16—21no—
28C2×C4×Q16—21no—
29C2×#432—21no—
30C2×#432C2211yescharacteristic
31C2×#431—21no—
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#423C2211yescharacteristic
35C2×#419—21no—
36#2285C2211yescharacteristic
37#1990C2212yes—
38#1990C2212yes—
39#1989—22no—
40#1825—21no—
41#1825—21no—
42#1805—21no—
43#1805—21no—
44#1803C2212yes—
45#1803C2212yes—
46#1736—21no—
47#1559—24no—
48C2×#333C2212yes—
49C2×#332C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C23×Q8D411yescharacteristic
55C22×Q16—21no—
56C22×Q16—21no—
57C22×M16—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—
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#484—28no—
86#477—22no—
87#477—22no—
88#477C2×C414yes—
89#477C2×C414yes—
90#462—22no—
91#462—22no—
92#462C2×C414yes—
93#462C2×C414yes—
94#451—28no—
95#446—216no—
96#446—24no—
97#446—24no—
98C8×Q8—24no—
99C8×Q8—24no—
100C4×Q16—28no—
101C4×Q16—28no—
102#432—28no—
103#432—22no—
104#432D414yes—
105#431—28no—
106#431—22no—
107#431—22no—
108#431—22no—
109#431C2×C414yes—
110C2×#107—21no—
111C2×#107—21no—
112#423—22no—
113#423—22no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8D412yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2312yes—
122#419—44no—
123C2×#103—21no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—22no—
128C2×#103—22no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×#97—22no—
134C2×#97—21no—
135C2×C4×C8—22no—
136C2×C4×C8—21no—
137C2×C4×C8C2312yes—
138#384C2312yes—
139#382—21no—
140#380—21no—
141#380—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#333—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×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×Q16—24no—
169C2×Q16—24no—
170C2×Q16—42no—
171C2×Q16—42no—
172C2×Q16—42no—
173C2×Q16—42no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8C22×C412yes—
190C4⋊Q8C22×C412yes—
191#125—24no—
192#125—42no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—24no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8C22⋊C414yes—
212C4×Q8C22⋊C414yes—
213C4×Q8C22×C412yes—
214C4×Q8C22×C412yes—
215C4×Q8C2×D412yes—
216C4×Q8C2×D412yes—
217#117—24no—
218#117—42no—
219#117—42no—
220#117—42no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4C22⋊C414yes—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42#4111yescharacteristic
257C2×C42#4112yes—
258C2×C42C2411yescharacteristic
259#107—24no—
260#107—24no—
261C4⋊C8—28no—
262C4⋊C8—28no—
263C4⋊C8—24no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—42no—
270C4⋊C8C22×C412yes—
271C4⋊C8C22×C412yes—
272C4⋊C8C2×D414yes—
273#103—24no—
274#103—24no—
275#103—24no—
276#103—24no—
277#103—44no—
278#103—44no—
279C22⋊C8—24no—
280C22⋊C8—24no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283#97—28no—
284#97—24no—
285C4×C8—28no—
286C4×C8—24no—
287C4×C8—22no—
288C4×C8—22no—
289#95—41no—
290#95—41no—
291C24#12111yescharacteristic
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—24no—
297C2×Q8—28no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—82no—
316C2×Q8C2×C22⋊C412yes—
317C2×Q8C2×C22⋊C412yes—
318C2×Q8AGL(1,ℤ/8)12yes—
319C2×Q8#13712yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12012yes—
342C22×C4#12112yes—
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12312yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350Q16—48no—
351Q16—48no—
352M16—84no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4C2×C22⋊C412yes—
392C4⋊C4C2×C22⋊C412yes—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—24no—
401C42—44no—
402C42—82no—
403C42C2×C22⋊C412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42#12012yes—
407C42#12112yes—
408C42#12312yes—
409C42C23×C411yescharacteristic
410C42C22×D411yescharacteristic
411C42C22×D411yescharacteristic
412C42C2×#4111yescharacteristic
413C23—22no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23#38612yes—
418C23C2×#12111yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—22no—
422Q8—22no—
423Q8—22no—
424Q8—48no—
425Q8—41no—
426Q8—41no—
427Q8—42no—
428Q8—42no—
429Q8—82no—
430Q8—82no—
431Q8—81no—
432Q8—81no—
433Q8#41714yes—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—48no—
454C2×C4—48no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#38612yes—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4C22×C22⋊C411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×#12011yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57814yes—
481C8—42no—
482C8—42no—
483C8—42no—
484C8—42no—
485C8—44no—
486C8—82no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22C2×#38611yescharacteristic
493C22#304412yes—
494C22#305511yescharacteristic
495C22#305611yescharacteristic
496C22#306712yes—
497C4—21no—
498C4—21no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—24no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—82no—
508C4—162no—
509C4C2×#41711yescharacteristic
510C4#289611yescharacteristic
511C4#295012yes—
512C4#301112yes—
513C4#305511yescharacteristic
514C4#305611yescharacteristic
515C2—42no—
516C2#1947812yes—
517C2#1947912yes—
518C2#2093511yescharacteristic
519C2#2097111yescharacteristic
520C2#2100311yescharacteristic
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. #9226C2semidirect
7. #9224C2semidirect
8. #9218C2semidirect
9. #9216C2semidirect
15. #3104C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#29842
#218352
#206472
C2×#15592
#92262
#92242
#92182
#92162
#80462
#78802