Finite Group Databasebeta

← #205096

Subgroups — #205096

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic
4#21360C211yesmaximal, characteristic
5#17562C211yesmaximal, characteristic
6#17545C211yesmaximal, characteristic
7#17324C211yesmaximal, characteristic
8#9720C214yesmaximal, complemented
9#9718C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3230C412yescomplemented
14#3230C412yescomplemented
15#3209C412yescomplemented
16#3209C412yescomplemented
17#3189—22no—
18#3170C2212yes—
19#3170C2212yes—
20#3104—22no—
21#3104—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2283C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1995—22no—
28#1994—22no—
29#1986C2214yes—
30#1986C2214yes—
31#1949—22no—
32#1854—22no—
33#1826—22no—
34#1826—22no—
35C4×#107C2212yes—
36C4×#106C2212yes—
37#1515—22no—
38#1511—22no—
39#1508—22no—
40#1506—22no—
41#1503C2214yes—
42#1503C2214yes—
43#1454—22no—
44#1453—22no—
45#1451C2212yes—
46C8×M16C2212yes—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533C2×C412yes—
60#533C2×C412yes—
61#533C2312yes—
62#533C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×#125—21no—
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#500C2×C412yes—
82#500C2×C412yes—
83#494C2×C412yes—
84#494C2×C412yes—
85#487C2×C412yes—
86#487C2×C412yes—
87#487C2×C412yes—
88#487C2×C412yes—
89#484—24no—
90#484—24no—
91#477—24no—
92#477—24no—
93#467—24no—
94#467—24no—
95#462—24no—
96#462—24no—
97#451—24no—
98#451—24no—
99C4×Q16—24no—
100C4×Q16—24no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431C2×C412yes—
110#431D412yes—
111#431D412yes—
112#431C2312yes—
113C2×#107—22no—
114C2×#106—22no—
115#423—21no—
116#423D412yes—
117C2×C4⋊C8D412yes—
118#407D412yes—
119#407D412yes—
120C2×C22⋊C8D412yes—
121C4×M16—21no—
122C4×M16C2312yes—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#383—21no—
126#382—21no—
127#381—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136#376—21no—
137C43—21no—
138#340—22no—
139#340—22no—
140#339—28no—
141#339—28no—
142#328—28no—
143#328—28no—
144#322—22no—
145#322—24no—
146C82—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C2×Q16—24no—
155C2×Q16—24no—
156C2×M16—22no—
157C2×M16C2×D414yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C412yes—
176#125—22no—
177#125—22no—
178#125—22no—
179#125—22no—
180#125—41no—
181#125—41no—
182#125—41no—
183#125—41no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216#117—22no—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—44no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225C2×C4⋊C4—21no—
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—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42C2411yescharacteristic
261#107—24no—
262#107—24no—
263#107—44no—
264#106—44no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8C22⋊C414yes—
270C4⋊C8C22⋊C414yes—
271#103—24no—
272#103—24no—
273#103—24no—
274#103—24no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8C22⋊C414yes—
279C22⋊C8C22⋊C414yes—
280C22⋊C8C2×D414yes—
281#97—22no—
282#97—22no—
283C4×C8—22no—
284C4×C8—22no—
285C4×C8C22×C412yes—
286C4×C8C22×C412yes—
287C4×C8C2×D412yes—
288C4×C8C2×D412yes—
289#95—22no—
290#95—22no—
291#95—41no—
292#95—41no—
293#95—41no—
294#95—41no—
295C24#12711yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8—82no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—81no—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12411yescharacteristic
328C22×C4#12411yescharacteristic
329C22×C4#12711yescharacteristic
330C22×C4#12711yescharacteristic
331C22×C4#12711yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335Q16—84no—
336Q16—84no—
337M16—28no—
338M16—82no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8C2×C22⋊C414yes—
344C2×C8C2×C22⋊C414yes—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—82no—
379C42—44no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—44no—
385C42—44no—
386C42—82no—
387C42C2×C22⋊C411yescharacteristic
388C42C2×C22⋊C411yescharacteristic
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C42#12111yescharacteristic
397C42#12411yescharacteristic
398C42#12411yescharacteristic
399C42C23×C411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—22no—
404C23—41no—
405C23—41no—
406C23#39011yescharacteristic
407C23#39011yescharacteristic
408C23C2×#12711yescharacteristic
409Q8—44no—
410Q8—44no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415Q8—82no—
416Q8—82no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#39011yescharacteristic
449C2×C4#39011yescharacteristic
450C2×C4#39011yescharacteristic
451C2×C4#39011yescharacteristic
452C2×C4#39011yescharacteristic
453C2×C4#39011yescharacteristic
454C2×C4C22×C22⋊C411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12411yescharacteristic
460C2×C4#57814yes—
461C8—24no—
462C8—24no—
463C8—44no—
464C8—44no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—82no—
474C22C2×#39011yescharacteristic
475C22#315612yes—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—162no—
491C4#290112yes—
492C4#295012yes—
493C4#305912yes—
494C2—22no—
495C2—41no—
496C2—41no—
497C2#2123211yescharacteristic
498C2#2124411yescharacteristic
499C2#2125311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC24#127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9720C2semidirect
9. #9718C2semidirect
13. #3230C4semidirect
14. #3230C4semidirect
15. #3209C4semidirect
16. #3209C4semidirect

Maximal subgroups

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

subgroupindex
#609432
#218452
#218442
#213602
#175622
#175452
#173242
#97202
#97182