Finite Group Databasebeta

← #196931

Subgroups — #196931

Order 512.

1535 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35247C211yesmaximal, characteristic
2C2×#2276C211yesmaximal, characteristic, complemented
3#21820C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic
5C2×#1500C211yesmaximal, characteristic
6#9149C212yesmaximal
7#9148C212yesmaximal
8#9145C212yesmaximal, complemented
9#9144C212yesmaximal, complemented
10#8022C211yesmaximal, characteristic, complemented
11#8014C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2855C2212yes—
18#2854C2212yes—
19#2637—22no—
20#2592C412yescomplemented
21#2584C412yescomplemented
22#2582C412yescomplemented
23#2538C412yescomplemented
24#2486—21no—
25#2413—22no—
26#2409—21no—
27#2323—22no—
28#2285C2211yescharacteristic
29#2276—24no—
30#2276—24no—
31C2×C4×C4⋊C4—22no—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×#376C2212yes—
36C2×#376C2211yescharacteristic
37#1985—22no—
38#1984—22no—
39#1817C2212yes—
40#1817C2212yes—
41#1509—24no—
42#1500—24no—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#327C2211yescharacteristic
47C2×#327C2211yescharacteristic
48C2×#327C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#529—22no—
55#529—22no—
56#529—22no—
57#529—22no—
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—21no—
62C2×#123—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×Q8—216no—
67C2×C4×Q8—28no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D412yes—
76C2×C4×Q8D412yes—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×#117—21no—
80C2×#117—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4D411yescharacteristic
85C22×C42—22no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#432—22no—
92#432—22no—
93#431—22no—
94#431—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C22⋊C8C2311yescharacteristic
98C2×C22⋊C8C2311yescharacteristic
99C2×C22⋊C8C2312yes—
100C2×#97—21no—
101C2×C4×C8—21no—
102#398—24no—
103#398—22no—
104#397—22no—
105#396—24no—
106#395—22no—
107#393—22no—
108#391—22no—
109#391C2×C414yes—
110#390C2×C414yes—
111#389—22no—
112#389—24no—
113#389C2×C414yes—
114#389C2×C414yes—
115#388—24no—
116#388—24no—
117#387—24no—
118#386—22no—
119#385—22no—
120#384C2×C412yes—
121#382—21no—
122#381—22no—
123#380—21no—
124#380—22no—
125#380C2312yes—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—216no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4D414yes—
134#376—22no—
135#376—24no—
136#376—24no—
137#376D414yes—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143#336—24no—
144#336—24no—
145#328—22no—
146#328—22no—
147#328—24no—
148#327—22no—
149#327—22no—
150#327—24no—
151C22×Q8—28no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C412yes—
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171#126—42no—
172#126—42no—
173#125—22no—
174#125—22no—
175#124—22no—
176#124—22no—
177#123—42no—
178#123—42no—
179#123—82no—
180#123—82no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—48no—
186#121—21no—
187#121—21no—
188#121—21no—
189#121—21no—
190C4×Q8—264no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—42no—
204C4×D4—42no—
205#117—42no—
206#117—42no—
207#117—81no—
208#117—81no—
209C2×C4⋊C4—216no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—24no—
240C2×C42—28no—
241C2×C42—216no—
242C2×C42—28no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—28no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—81no—
253C2×C42—81no—
254C2×C42—82no—
255C2×C42—82no—
256C2×C42C22×C414yes—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D414yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—22no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—22no—
274#97—42no—
275C4×C8—42no—
276#95—28no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95C2×D412yes—
288#95C2×D414yes—
289#95C2×D412yes—
290C24—21no—
291C24C22×D411yescharacteristic
292C2×Q8—216no—
293C2×Q8—216no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—28no—
297C2×Q8—24no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8#10418yes—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4C2×C22⋊C414yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C414yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4#12312yes—
356C22×C4#12312yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—44no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C4⋊C4—232no—
370C4⋊C4—216no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—44no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—82no—
388C4⋊C4—84no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—48no—
399C22⋊C4—84no—
400C42—232no—
401C42—216no—
402C42—24no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—42no—
407C42—84no—
408C42—82no—
409C42—88no—
410C42C4×D412yes—
411C42C4×D412yes—
412C42C4×D412yes—
413C42C4×D412yes—
414C42#12112yes—
415C42#12112yes—
416C42#12112yes—
417C42#12112yes—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23C22×C22⋊C411yescharacteristic
426C23C2×#12011yescharacteristic
427C23C2×#12311yescharacteristic
428C23#57611yescharacteristic
429C23#57711yescharacteristic
430C23#57711yescharacteristic
431C23#57811yescharacteristic
432Q8—216no—
433Q8—216no—
434Q8—81no—
435Q8—81no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—84no—
496C2×C4#38612yes—
497C2×C4#38612yes—
498C2×C4#38612yes—
499C2×C4#38612yes—
500C2×C4C2×#10418yes—
501C2×C4C2×#10414yes—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C8—81no—
507C8—81no—
508C8—82no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C2×#38611yescharacteristic
523C22#288412yes—
524C22#289711yescharacteristic
525C22#289811yescharacteristic
526C22C22×#10412yes—
527C22#301211yescharacteristic
528C22#301311yescharacteristic
529C22#309811yescharacteristic
530C22#309911yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—28no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—48no—
540C4—44no—
541C4—44no—
542C4—44no—
543C4—84no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4#189614yes—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—161no—
553C2#1754912yes—
554C2#1755012yes—
555C2#2091511yescharacteristic
556C2C2×#189611yescharacteristic
557C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2276C2semidirect
3. #21820C2semidirect
8. #9145C2semidirect
9. #9144C2semidirect
10. #8022C2semidirect
11. #8014C2semidirect
20. #2592C4semidirect
21. #2584C4semidirect
22. #2582C4semidirect
23. #2538C4semidirect

Maximal subgroups

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

subgroupindex
#352472
C2×#22762
#218202
C2×#15092
C2×#15002
#91492
#91482
#91452
#91442
#80222
#80142