Finite Group Databasebeta

← #192785

Subgroups — #192785

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 448 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×#1803C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8891C214yesmaximal, complemented
7#8883C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7893C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103—21no—
15#2356—22no—
16#2356—22no—
17#2355—21no—
18#2355—21no—
19#2355C2212yes—
20#2355C2212yes—
21#2313—22no—
22#2310—21no—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2283—22no—
27#2274—21no—
28#1948C2214yes—
29#1947—22no—
30#1853—22no—
31#1852—22no—
32#1845C2212yes—
33#1844—21no—
34#1834—21no—
35#1834C2212yes—
36#1803—22no—
37#1803C2214yes—
38C2×#340—21no—
39#1479C418yescomplemented
40C2×Q8⋊C8C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43#1279C2214yes—
44C23×Q8D411yescharacteristic
45C23×C8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C414yes—
51C2×C4×Q8D412yes—
52C2×C4×Q8Q814yes—
53C2×C4×Q8C2312yes—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63#431—22no—
64#431—22no—
65#431C2×C414yes—
66#431C2×C414yes—
67C2×#107—24no—
68C2×#106—24no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—24no—
76C2×#103—44no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#384C2312yes—
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#383—24no—
98#382—22no—
99#382—22no—
100#382—21no—
101#382—21no—
102#381—24no—
103#380—21no—
104#380—21no—
105#380D412yes—
106#380D412yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—24no—
118#376—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122Q8⋊C8C2×C4116yescomplemented
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8C4⋊C414yes—
130C22×Q8C2×D412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8C4⋊C418yes—
151C4×Q8C22×C414yes—
152C4×Q8C2×Q814yes—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C4⋊C414yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42C22×C412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×Q812yes—
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42C2411yescharacteristic
205#107—84no—
206#106—84no—
207C4⋊C8—24no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C4⋊C8C22×C414yes—
213#103—48no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C4×C8—24no—
217C4×C8—22no—
218C4×C8C22×C414yes—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229C24#12011yescharacteristic
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8C2×C4⋊C414yes—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4C2×C4⋊C412yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4C22⋊Q812yes—
270C22×C4C22⋊Q812yes—
271C22×C4#12312yes—
272C22×C4#13512yes—
273C22×C4#13512yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—84no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4C2×C4⋊C414yes—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C42—24no—
319C42—24no—
320C42—28no—
321C42—42no—
322C42—42no—
323C42—44no—
324C42—84no—
325C42—84no—
326C42C2×C4⋊C412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42#12312yes—
332C42C23×C411yescharacteristic
333C42C22×Q811yescharacteristic
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#38512yes—
341C23#44912yes—
342C23#45212yes—
343C23C2×#12011yescharacteristic
344Q8—24no—
345Q8—24no—
346Q8—42no—
347Q8—42no—
348Q8—48no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—164no—
383C2×C4#38512yes—
384C2×C4#38512yes—
385C2×C4#38512yes—
386C2×C4#42714yes—
387C2×C4#42814yes—
388C2×C4#44912yes—
389C2×C4#45212yes—
390C2×C4#48214yes—
391C2×C4#48514yes—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12311yescharacteristic
398C2×C4C2×#13511yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4#57712yes—
403C2×C4#57812yes—
404C8—44no—
405C8—44no—
406C8—82no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22#185414yes—
414C22C2×#38511yescharacteristic
415C22C2×#44911yescharacteristic
416C22C2×#45211yescharacteristic
417C22#301011yescharacteristic
418C22#301111yescharacteristic
419C22#302111yescharacteristic
420C22#302211yescharacteristic
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—44no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—162no—
432C4#185414yes—
433C4C2×#42711yescharacteristic
434C4C2×#42811yescharacteristic
435C4#294712yes—
436C4#308212yes—
437C4C2×#48211yescharacteristic
438C4C2×#48511yescharacteristic
439C2—24no—
440C2#1889611yescharacteristic
441C2#1889911yescharacteristic
442C2#1890611yescharacteristic
443C2#1890911yescharacteristic
444C2C2×#185411yescharacteristic
445C2#2086511yescharacteristic
446C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8891C2semidirect
7. #8883C2semidirect
39. #1479C4semidirect
122. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#219822
#219812
C2×#18032
C2×#14792
#88912
#88832
#80392
#78932