Finite Group Databasebeta

← #380293

Subgroups — #380293

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2#20846C211yesmaximal, characteristic
3C2×#1826C211yesmaximal, characteristic
4#20702C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#11340C214yesmaximal, complemented
7#11276C214yesmaximal, complemented
8#10733C211yesmaximal, characteristic
9#10727C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#428C2212yes—
14C2×#407C2211yescharacteristic
15C2×#407C2211yescharacteristic
16#2317—22no—
17#2317—22no—
18#2317—22no—
19#2317C2214yes—
20C2×#389—21no—
21C2×#389—21no—
22C2×#385—22no—
23C2×#385—22no—
24C2×#384—21no—
25C2×#384C2212yes—
26C2×#382—21no—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1854—22no—
30#1853—21no—
31#1852—21no—
32#1826C418yescomplemented
33#1825—22no—
34#1824—21no—
35#1823—21no—
36C2×#343C2214yes—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#339C2214yes—
41C2×#339C2214yes—
42#1444—22no—
43C4⋊#95C2212yes—
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2312yes—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—44no—
55C2×#117C2×C414yes—
56C2×#117D412yes—
57C2×#117D412yes—
58C2×#117Q814yes—
59C2×#117C2312yes—
60C2×#117C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D412yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2312yes—
84#428—44no—
85C2×#107—22no—
86C2×#106—22no—
87#407—22no—
88#407—22no—
89#407—22no—
90#407C2×C414yes—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#385—28no—
100#385—28no—
101#385—48no—
102#384—24no—
103#384—24no—
104#384—44no—
105#382—24no—
106#382—24no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4C2×C414yes—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—24no—
117C2×#95—21no—
118#343—44no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#339—44no—
126#339C2×C4116yescomplemented
127C25C2×D411yescharacteristic
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—82no—
142C23×C4C22×C412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D414yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—22no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—24no—
159C2×M16—24no—
160C2×M16C22×C414yes—
161C22×C8—22no—
162C22×C8—22no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—24no—
168#117—22no—
169#117—22no—
170#117—24no—
171#117—24no—
172#117—42no—
173#117—42no—
174#117—44no—
175#117—44no—
176#117C4⋊C418yescomplemented
177#117C22×C414yes—
178#117C2×Q814yes—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4—84no—
210C2×C4⋊C4—84no—
211C2×C4⋊C4C4⋊C414yescomplemented
212C2×C4⋊C4C4⋊C414yescomplemented
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42C22×C414yes—
241#107—44no—
242#106—44no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#95—28no—
248#95—28no—
249#95—24no—
250#95—48no—
251#95—41no—
252#95—41no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24C4×D412yes—
260C24#12011yescharacteristic
261C24#12011yescharacteristic
262C24C22⋊Q812yes—
263C24C22×D411yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—816no—
321C22×C4C2×C4⋊C412yes—
322C22×C4C2×C4⋊C412yes—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4#12011yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4C22⋊Q812yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)14yes—
340C22×C42−1+414yes—
341M16—28no—
342M16—44no—
343M16—44no—
344M16—44no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4—816no—
372C4⋊C4C2×C4⋊C414yes—
373C4⋊C4C2×C4⋊C414yes—
374C22⋊C4—216no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—28no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—24no—
383C42—24no—
384C42—28no—
385C42—28no—
386C42—44no—
387C42—44no—
388C42—44no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—24no—
398C23—21no—
399C23—21no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23#38512yes—
414C23#38512yes—
415C23#45312yes—
416C23#45612yes—
417C23C2×C4×D411yescharacteristic
418C23C2×#12011yescharacteristic
419C23C2×C22⋊Q811yescharacteristic
420C23#53412yes—
421C23#53612yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—88no—
464C2×C4—88no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#38512yes—
475C2×C4#38512yes—
476C2×C4C22×C4⋊C411yescharacteristic
477C2×C4C2×C4×D411yescharacteristic
478C2×C4#52014yes—
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4#53412yes—
482C2×C4#53512yes—
483C2×C4#53612yes—
484C2×C4#53612yes—
485C2×C4#53914yes—
486C2×C4#53914yes—
487C2×C4#54414yes—
488C8—44no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—21no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—48no—
508C22—48no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#185914yes—
514C22C2×#38511yescharacteristic
515C22#232212yes—
516C22#261311yescharacteristic
517C22#261411yescharacteristic
518C22#262712yes—
519C22C2×#45311yescharacteristic
520C22C2×#45611yescharacteristic
521C22#301911yescharacteristic
522C22#302011yescharacteristic
523C4—22no—
524C4—22no—
525C4—48no—
526C4—44no—
527C4—44no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4—161no—
532C4—161no—
533C4#231712yes—
534C4#232212yes—
535C4#262312yes—
536C4#262712yes—
537C2—22no—
538C2—24no—
539C2—22no—
540C2—42no—
541C2#1426311yescharacteristic
542C2C2×#185911yescharacteristic
543C2#2086411yescharacteristic
544C2#2961011yescharacteristic
545C2#2961311yescharacteristic
546C2#2965011yescharacteristic
547C2#2965311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11340C2semidirect
7. #11276C2semidirect
32. #1826C4semidirect
126. #339C2×C4semidirect
176. #117C4⋊C4semidirect
211. C2×C4⋊C4C4⋊C4semidirect
212. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
#208462
C2×#18262
#207022
#138802
#113402
#112762
#107332
#107272