Finite Group Databasebeta

← #205084

Subgroups — #205084

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#21358C211yesmaximal, characteristic
3#21306C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6C2×#1503C211yesmaximal, characteristic
7#17314C211yesmaximal, characteristic
8#9718C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3249—22no—
12#3230C412yescomplemented
13#3230C412yescomplemented
14#3209C412yescomplemented
15#3209C412yescomplemented
16#3189—22no—
17#3170C2214yes—
18#3104—22no—
19#3104—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1986—22no—
26#1986—22no—
27#1986C2214yes—
28#1986C2214yes—
29#1947—22no—
30#1852—22no—
31C4×#107C2214yes—
32#1733—22no—
33C2×#339—22no—
34#1520—22no—
35#1518—22no—
36#1503—22no—
37#1503C2214yes—
38#1503C2214yes—
39C2×#328—22no—
40#1450—22no—
41C8×M16—22no—
42C8×M16C2214yes—
43C22×M16D412yes—
44#533—21no—
45#533—21no—
46#533—21no—
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#533C2×C412yes—
57#533C2×C412yes—
58#533C2312yes—
59#533C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125—21no—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#500C2×C414yescomplemented
81#494C2×C414yescomplemented
82#487C2×C414yescomplemented
83#487C2×C414yescomplemented
84#484—24no—
85#484—24no—
86#477—24no—
87#477—24no—
88#467—28no—
89#462—24no—
90#462—24no—
91#451—24no—
92#451—24no—
93C4×Q16—28no—
94#431—21no—
95#431—21no—
96#431C2×C412yes—
97#431C2×C412yes—
98#431D412yes—
99#431D412yes—
100#431D412yes—
101#431C2312yes—
102C2×#107—24no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D412yes—
105C2×C22⋊C8D412yes—
106C2×C22⋊C8D412yes—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16D412yes—
110C4×M16C2312yes—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2312yes—
115#384—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2312yes—
124C4×C4⋊C4C2312yes—
125C2×#95—21no—
126#340—24no—
127#339—28no—
128#339—28no—
129#328—28no—
130#328—28no—
131#322—24no—
132C82—24no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×Q16—28no—
142C2×M16—22no—
143C2×M16—24no—
144C2×M16Q1618yescomplemented
145C2×M16C2×D414yes—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C2×D414yes—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C22×C412yes—
162C4⋊Q8C22×C412yes—
163C4⋊Q8C22×C412yes—
164C4⋊Q8C22×C412yes—
165#125—22no—
166#125—22no—
167#125—22no—
168#125—22no—
169#125—41no—
170#125—41no—
171#125—41no—
172#125—41no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
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—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—44no—
240C2×C42—41no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42C2411yescharacteristic
253#107—28no—
254#107—48no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8C22⋊C414yes—
258C4⋊C8C22⋊C414yes—
259#103—28no—
260#103—28no—
261C22⋊C8—22no—
262C22⋊C8C22⋊C414yes—
263C22⋊C8C22⋊C414yes—
264C22⋊C8C2×D414yes—
265#97—22no—
266#97—22no—
267C4×C8—22no—
268C4×C8—22no—
269C4×C8—22no—
270C4×C8—22no—
271C4×C8C22×C414yes—
272C4×C8C2×D414yes—
273#95—22no—
274#95—22no—
275#95—41no—
276#95—41no—
277C24#12711yescharacteristic
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—82no—
289C2×Q8—82no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12711yescharacteristic
314C22×C4#12711yescharacteristic
315C22×C4#12711yescharacteristic
316C22×C4C2×Q1614yes—
317C22×C4#13514yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321Q16—88no—
322M16—24no—
323M16—28no—
324M16C2×Q1618yes—
325C2×C8—24no—
326C2×C8—28no—
327C2×C8—24no—
328C2×C8—44no—
329C2×C8—82no—
330C2×C8C2×C22⋊C418yes—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—82no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—44no—
371C42—44no—
372C42—82no—
373C42C2×C22⋊C411yescharacteristic
374C42C2×C22⋊C411yescharacteristic
375C42C4×D411yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42#12411yescharacteristic
384C42#12411yescharacteristic
385C42C23×C411yescharacteristic
386C42C22×D411yescharacteristic
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C23—41no—
390C23—41no—
391C23#39011yescharacteristic
392C23#39011yescharacteristic
393C23#49412yes—
394C23#49512yes—
395C23C2×#12711yescharacteristic
396Q8—44no—
397Q8—44no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402Q8—82no—
403Q8—82no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—88no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4#39011yescharacteristic
432C2×C4#39011yescharacteristic
433C2×C4#39011yescharacteristic
434C2×C4#39011yescharacteristic
435C2×C4#39011yescharacteristic
436C2×C4#39011yescharacteristic
437C2×C4C4×Q1614yes—
438C2×C4C4×Q1614yes—
439C2×C4#46714yes—
440C2×C4#46714yes—
441C2×C4#49412yes—
442C2×C4#49512yes—
443C2×C4C22×C22⋊C411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12111yescharacteristic
448C2×C4C2×#12411yescharacteristic
449C2×C4C22×Q1612yes—
450C2×C4C2×#13512yes—
451C8—28no—
452C8—28no—
453C8—44no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22#196912yes—
463C22#196912yes—
464C22C2×#39011yescharacteristic
465C22C2×#49411yescharacteristic
466C22C2×#49511yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—84no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—162no—
480C4#196912yes—
481C4#196912yes—
482C4#290112yes—
483C4C2×C4×Q1612yes—
484C4C2×#46712yes—
485C2—21no—
486C2—21no—
487C2—22no—
488C2—22no—
489C2C2×#196911yescharacteristic
490C2#2124411yescharacteristic
491C2#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. #9718C2semidirect
12. #3230C4semidirect
13. #3230C4semidirect
14. #3209C4semidirect
15. #3209C4semidirect
80. #500C2×C4semidirect
81. #494C2×C4semidirect
82. #487C2×C4semidirect
83. #487C2×C4semidirect
144. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#213582
#213062
C2×#19862
#175752
C2×#15032
#173142
#97182