Finite Group Databasebeta

← #192722

Subgroups — #192722

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8892C214yesmaximal, complemented
7#8891C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#428—21no—
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2360—22no—
16#2360—22no—
17#2355—21no—
18#2355C2212yes—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4C2212yes—
21#2330—22no—
22#2317—21no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#2274—21no—
28C4×#117C2211yescharacteristic
29#1990—21no—
30#1956—21no—
31#1948—21no—
32#1948C2212yes—
33#1947—21no—
34#1947C2212yes—
35#1872—22no—
36#1834—22no—
37#1834C2214yes—
38#1803C2214yes—
39#1736—21no—
40C2×#340—22no—
41#1493C418yescomplemented
42C2×Q8⋊C8C2214yes—
43#1296C2214yes—
44#1279C2212yes—
45#1279C2212yes—
46C22×M16—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533Q814yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8D412yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2312yes—
61C2×#117—21no—
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D412yes—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#428—44no—
74C2×#107—22no—
75C2×#106—22no—
76#423—22no—
77#423C2×C414yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—24no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8C2×C414yes—
87C2×C4×C8C2312yes—
88#391—22no—
89#391—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—24no—
94#389—22no—
95#389—24no—
96#387—24no—
97#387—24no—
98#385—24no—
99#384—22no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384D412yes—
106#384C2312yes—
107#384C2312yes—
108#382—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—24no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376—21no—
123C43—21no—
124#340—24no—
125#340—28no—
126#340—22no—
127#340—22no—
128Q8⋊C8C2×C4116yescomplemented
129C22×Q8—21no—
130C22×Q8C2×D412yes—
131C23×C4—22no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—24no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8—42no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8C4⋊C414yes—
148C4⋊Q8C4⋊C414yes—
149#125—24no—
150#125—24no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8C4⋊C418yes—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—84no—
160C4×Q8—84no—
161C4×Q8C22×C414yes—
162C4×Q8C2×Q814yes—
163#117—24no—
164#117—24no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117SD1614yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D414yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221#107—44no—
222#106—44no—
223C4⋊C8—24no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8C22×C414yes—
228#103—84no—
229C22⋊C8—42no—
230C4×C8—24no—
231C4×C8—22no—
232C4×C8—42no—
233C4×C8C22×C414yes—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24#12011yescharacteristic
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8C2×C4⋊C414yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4C2×C4⋊C412yes—
269C22×C4C4×D412yes—
270C22×C4C4×D412yes—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4C22⋊Q812yes—
275C22×C4C22⋊Q812yes—
276C22×C4#12312yes—
277C22×C4C2×SD1612yes—
278C22×C4#13512yes—
279C22×C4#13712yes—
280C22×C4#13712yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284M16—48no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—48no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4C2×C4⋊C414yes—
325C4⋊C4C2×SD1612yes—
326C4⋊C4C2×SD1612yes—
327C22⋊C4—28no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—84no—
339C42—84no—
340C42C2×C4⋊C412yes—
341C42C4×D412yes—
342C42C4×D412yes—
343C42C22⋊Q812yes—
344C42C22⋊Q812yes—
345C42#12312yes—
346C42C23×C411yescharacteristic
347C42C22×Q811yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23#38512yes—
355C23#44812yes—
356C23#45212yes—
357C23C2×#12011yescharacteristic
358Q8—28no—
359Q8—44no—
360Q8—48no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—164no—
394C2×C4—164no—
395C2×C4#38512yes—
396C2×C4#38512yes—
397C2×C4#38512yes—
398C2×C4C4×SD1614yes—
399C2×C4#44114yes—
400C2×C4#44812yes—
401C2×C4#45212yes—
402C2×C4#45412yes—
403C2×C4#45512yes—
404C2×C4#47514yes—
405C2×C4#47914yes—
406C2×C4C22×C4⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4C22×SD1611yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C2×C4#57612yes—
417C2×C4#57812yes—
418C8—44no—
419C8—44no—
420C8—82no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22#186514yes—
428C22C2×#38511yescharacteristic
429C22C2×#44811yescharacteristic
430C22C2×#45211yescharacteristic
431C22#300811yescharacteristic
432C22#301111yescharacteristic
433C22C2×#45511yescharacteristic
434C22C2×#45411yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—44no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—84no—
444C4—81no—
445C4—162no—
446C4#186014yes—
447C4#186514yes—
448C4#291312yes—
449C4C2×C4×SD1611yescharacteristic
450C4C2×#44111yescharacteristic
451C4C2×#47511yescharacteristic
452C4C2×#47911yescharacteristic
453C4#309812yes—
454C2—24no—
455C2#1863211yescharacteristic
456C2#1863511yescharacteristic
457C2#1864211yescharacteristic
458C2#1864511yescharacteristic
459C2#2084711yescharacteristic
460C2C2×#186011yescharacteristic
461C2C2×#186511yescharacteristic
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. #8892C2semidirect
7. #8891C2semidirect
41. #1493C4semidirect
128. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342142
#218602
#211712
C2×#18342
C2×#14932
#88922
#88912
#80432
#78902