Finite Group Databasebeta

← #197239

Subgroups — #197239

Order 512.

1117 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#9178C218yesmaximal, complemented
7#8095C211yesmaximal, characteristic
8#7957C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3078—22no—
12#3078—22no—
13#3078C414yescomplemented
14#3078C2214yes—
15C2×#479—22no—
16C2×#477C412yescomplemented
17C2×#475C412yescomplemented
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22C2×#419—21no—
23#2356—22no—
24#2356C2214yes—
25#2355—22no—
26#2355—22no—
27#2313—22no—
28#2310—21no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#2283—22no—
32#2274—21no—
33#1520—22no—
34#1520—22no—
35#1518—22no—
36#1518C2214yes—
37#1503—22no—
38#1503C2214yes—
39#1479—22no—
40#1479C2214yes—
41C2×Q8⋊C8—22no—
42#1303C2214yes—
43#1286C2214yes—
44C23×Q8D411yescharacteristic
45C22×M16—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533C2312yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8D412yes—
72C2×C4×Q8C2312yes—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#479—28no—
81#479—28no—
82#477—24no—
83#477C2×C418yes—
84#475—24no—
85#475C2×C418yes—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431C2312yes—
93#431C2312yes—
94#428—24no—
95#428—24no—
96C2×#107—22no—
97C2×#106—22no—
98#423—21no—
99#423—21no—
100#423C2×C412yes—
101#423C2312yes—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2×C412yes—
106#419—24no—
107#419—24no—
108C2×#103—22no—
109C2×#103—22no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#391—24no—
115#391—24no—
116#387—28no—
117#387—28no—
118#384—21no—
119#384—21no—
120#383—24no—
121#382—24no—
122#382D412yes—
123#382C2312yes—
124#381—24no—
125#380—21no—
126#380—21no—
127#380D412yes—
128#380D412yes—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D412yes—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—24no—
137#376—24no—
138#328—28no—
139#328—28no—
140Q8⋊C8—28no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8C22⋊C412yes—
151C22×Q8C22⋊C412yes—
152C22×Q8C2×D412yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—22no—
163C22×C8—41no—
164C22×C8—41no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173#125—22no—
174#125—22no—
175#125—41no—
176#125—41no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8C22⋊C414yes—
206C4×Q8C22⋊C414yes—
207C4×Q8C22×C414yes—
208C4×Q8C2×D414yes—
209#117—22no—
210#117—22no—
211#117—41no—
212#117—41no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—42no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—41no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—28no—
253C2×C42—41no—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42C2411yescharacteristic
266#107—28no—
267#106—28no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8C22×C414yes—
276#103—28no—
277#103—28no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—41no—
281C22⋊C8—41no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—41no—
285C4×C8—41no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—22no—
290#95—22no—
291#95C2×D414yes—
292#95C2×D414yes—
293C24#12011yescharacteristic
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8C2×C22⋊C414yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—81no—
332C22×C4—82no—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12311yescharacteristic
347C22×C4#12311yescharacteristic
348C22×C4#13512yes—
349C22×C4#13512yes—
350C22×C4#13712yes—
351C22×C4#13712yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355M16—42no—
356M16—42no—
357M16—44no—
358C2×C8—24no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—82no—
363C2×C8—82no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4C2×C22⋊C414yes—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—44no—
404C42—88no—
405C42C2×C22⋊C411yescharacteristic
406C42C2×C22⋊C411yescharacteristic
407C42C4×D411yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42#12111yescharacteristic
414C42#12111yescharacteristic
415C42#12311yescharacteristic
416C42#12311yescharacteristic
417C42C23×C411yescharacteristic
418C42C22×D411yescharacteristic
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C23—41no—
422C23—41no—
423C23#38611yescharacteristic
424C23#38611yescharacteristic
425C23#45212yes—
426C23#45212yes—
427C23C2×#12011yescharacteristic
428Q8—24no—
429Q8—24no—
430Q8—42no—
431Q8—42no—
432Q8—44no—
433Q8—44no—
434Q8—82no—
435Q8—82no—
436Q8—82no—
437Q8—82no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4#38611yescharacteristic
470C2×C4#38611yescharacteristic
471C2×C4#38611yescharacteristic
472C2×C4#38611yescharacteristic
473C2×C4#38611yescharacteristic
474C2×C4#38611yescharacteristic
475C2×C4#41612yes—
476C2×C4#41612yes—
477C2×C4#41912yes—
478C2×C4#41912yes—
479C2×C4#45212yes—
480C2×C4#45212yes—
481C2×C4#48512yes—
482C2×C4#48512yes—
483C2×C4#48512yes—
484C2×C4#48512yes—
485C2×C4C22×C22⋊C411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C2×C4C2×#13511yescharacteristic
492C2×C4C2×#13511yescharacteristic
493C2×C4C2×#13711yescharacteristic
494C2×C4C2×#13711yescharacteristic
495C8—44no—
496C8—82no—
497C8—82no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—42no—
505C22—42no—
506C22#187312yes—
507C22#187312yes—
508C22C2×#38611yescharacteristic
509C22C2×#45211yescharacteristic
510C22C2×#45211yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—44no—
519C4—44no—
520C4—84no—
521C4—84no—
522C4—82no—
523C4—82no—
524C4#187312yes—
525C4#187312yes—
526C4#190512yes—
527C4C2×#41611yescharacteristic
528C4C2×#41911yescharacteristic
529C4C2×#48511yescharacteristic
530C4C2×#48511yescharacteristic
531C2—21no—
532C2—21no—
533C2—22no—
534C2—22no—
535C2C2×#187311yescharacteristic
536C2C2×#190511yescharacteristic
537C2#2100011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9178C2semidirect
13. #3078C4semidirect
16. C2×#477C4semidirect
17. C2×#475C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#341872
#175752
C2×#14792
#91782
#80952
#79572