Finite Group Databasebeta

← #192163

Subgroups — #192163

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4C2×#1461C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#8855C218yesmaximal
7#8014C211yesmaximal, characteristic
8#8014C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2356—22no—
13#2356—22no—
14#2355—21no—
15#2355—21no—
16#2355—21no—
17#2355—21no—
18#2313C2214yes—
19#2310—21no—
20C2×#391—21no—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2283—22no—
25#2274—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1948—22no—
28#1947—22no—
29#1853C2214yes—
30#1852C2214yes—
31#1845—21no—
32#1844—21no—
33#1834—21no—
34#1834—21no—
35#1461C418yescomplemented
36C2×#324C2214yes—
37#1441C2214yes—
38C4⋊#95—22no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8D411yescharacteristic
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8C2×C414yes—
50C2×C4×Q8Q814yes—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42—21no—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62#431—22no—
63#431—22no—
64#431C2×C414yes—
65#431C2×C414yes—
66C2×#107—24no—
67C2×#106—24no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2312yes—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#391—22no—
75#391—22no—
76#391—22no—
77#391—22no—
78#391—44no—
79#387—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384D411yescharacteristic
88#384D411yescharacteristic
89#384D411yescharacteristic
90#384D411yescharacteristic
91#383—24no—
92#382—22no—
93#382—22no—
94#382—21no—
95#382—21no—
96#381—24no—
97#380—21no—
98#380—21no—
99#380C2312yes—
100#380C2312yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C22⋊C4—24no—
114#376—24no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95C2312yes—
118#340—24no—
119#340—24no—
120#324C2×C4116yes—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C4⋊C414yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146#117—42no—
147#117—42no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C4⋊C414yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×Q814yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C4⋊C418yes—
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q812yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206#107—84no—
207#106—84no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8C22×C414yes—
213C22⋊C8C22×C414yes—
214C4×C8—24no—
215C4×C8—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—48no—
229#95—84no—
230C24C22×D411yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—28no—
234C2×Q8—28no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—44no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C2×C4⋊C414yes—
272C22×C4C2×C4⋊C412yes—
273C22×C4C2×C4⋊C414yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12312yes—
280C22×C4#12312yes—
281C22×C4#13512yes—
282C22×C4#13512yes—
283C22×C4AGL(1,ℤ/8)12yes—
284C22×C4#13712yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×Q811yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C42—28no—
327C42—28no—
328C42—44no—
329C42—44no—
330C42—84no—
331C42—84no—
332C42—84no—
333C42C4×D412yes—
334C42C4×D412yes—
335C42C4×D412yes—
336C42C4×D412yes—
337C42C22⋊Q812yes—
338C42C22⋊Q812yes—
339C42C22⋊Q812yes—
340C42C22⋊Q812yes—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23C22×C4⋊C411yescharacteristic
346C23C2×#12011yescharacteristic
347C23C2×#12311yescharacteristic
348C23C2×#13511yescharacteristic
349C23C2×#13511yescharacteristic
350C23C2×AGL(1,ℤ/8)11yescharacteristic
351C23C2×#13711yescharacteristic
352Q8—48no—
353Q8—48no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#38512yes—
403C2×C4#38512yes—
404C2×C4#38512yes—
405C2×C4#38512yes—
406C2×C4#42714yes—
407C2×C4#42814yes—
408C2×C4#44912yes—
409C2×C4#44912yes—
410C2×C4#45212yes—
411C2×C4#45212yes—
412C2×C4#45412yes—
413C2×C4#45612yes—
414C2×C4#48214yes—
415C2×C4#48514yes—
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C22⋊Q811yescharacteristic
420C8—44no—
421C8—44no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22C2×#38511yescharacteristic
435C22C2×#42711yescharacteristic
436C22C2×#42811yescharacteristic
437C22C2×#44911yescharacteristic
438C22C2×#45211yescharacteristic
439C22C2×#45411yescharacteristic
440C22C2×#45611yescharacteristic
441C22#302711yescharacteristic
442C22#302811yescharacteristic
443C22C2×#48211yescharacteristic
444C22C2×#48511yescharacteristic
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—44no—
451C4—44no—
452C4—84no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—162no—
458C4#185414yes—
459C4#185414yes—
460C4#185714yes—
461C2—21no—
462C2—21no—
463C2—21no—
464C2—21no—
465C2#1839111yescharacteristic
466C2#1839211yescharacteristic
467C2#1839811yescharacteristic
468C2#1839911yescharacteristic
469C2C2×#185411yescharacteristic
470C2C2×#185711yescharacteristic
471C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#219822
#219812
C2×#14612
#139262
#88552
#80142