Finite Group Databasebeta

← #193215

Subgroups — #193215

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#8928C214yesmaximal, complemented
7#8927C214yesmaximal, complemented
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#428—21no—
13C2×#423C2211yescharacteristic
14C22×#103—22no—
15C22×C4×C8C2211yescharacteristic
16#2369C2212yes—
17#2369C2212yes—
18#2365—21no—
19#2365—21no—
20#2345—22no—
21Q8×C22⋊C4—22no—
22#2331—22no—
23#2316—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274—21no—
29C4×#117C2211yescharacteristic
30#1956—22no—
31#1956C2214yes—
32#1948—21no—
33#1948C2212yes—
34#1948C2214yes—
35#1947—21no—
36#1947C2212yes—
37#1853—22no—
38#1844—21no—
39#1834—21no—
40#1736—21no—
41#1548C418yescomplemented
42C2×#333C2214yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2214yes—
46C23×Q8D412yes—
47C22×M16—21no—
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8Q814yes—
56C2×C4⋊Q8C2312yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×#117—21no—
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#431—22no—
69#431C2×C414yes—
70#428—44no—
71C2×#107—22no—
72C2×#106—22no—
73C2×#106—24no—
74#423—22no—
75#423C2×C414yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—28no—
80C2×#103—48no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8C2312yes—
87#391—22no—
88#391—22no—
89#391—24no—
90#391—24no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#387—24no—
98#387—24no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#384C2312yes—
105#382—21no—
106#380—24no—
107#380—21no—
108#380D412yes—
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⋊C4—21no—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2312yes—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125C43—21no—
126#340—24no—
127#340—22no—
128#340—22no—
129#333C2×C4116yescomplemented
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8C4⋊C418yes—
135C22×Q8C2×D414yes—
136C23×C4—22no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—42no—
149C22×C8—42no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8C4⋊C418yes—
158C4⋊Q8C22×C414yes—
159C4⋊Q8C2×Q814yes—
160C4×Q8—84no—
161C4×Q8—84no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—42no—
166#117—42no—
167#117SD1614yes—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#107—44no—
219#106—44no—
220#106—84no—
221C4⋊C8—24no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8C22×C414yes—
225C4⋊C8C22×C414yes—
226#103—416no—
227C22⋊C8—24no—
228C4×C8—24no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—22no—
242C24#12011yescharacteristic
243C2×Q8—24no—
244C2×Q8—28no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—416no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—88no—
251C2×Q8C2×C4⋊C418yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C4⋊C412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4#12312yes—
279C22×C4C2×SD1612yes—
280C22×C4#13512yes—
281C22×C4#13712yes—
282C22×C4#13712yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—48no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—48no—
292C2×C8—42no—
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—21no—
302C4⋊C4—21no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4C2×SD1612yes—
325C4⋊C4C2×SD1612yes—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—84no—
338C42—84no—
339C42C2×C4⋊C412yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42C22⋊Q812yes—
343C42C22⋊Q812yes—
344C42#12312yes—
345C42C23×C411yescharacteristic
346C42C22×Q811yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#38512yes—
354C23#44812yes—
355C23#45212yes—
356C23C2×#12011yescharacteristic
357Q8—48no—
358Q8—416no—
359Q8—84no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—28no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—88no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—164no—
394C2×C4#38512yes—
395C2×C4#38512yes—
396C2×C4#38512yes—
397C2×C4C4×SD1614yes—
398C2×C4#44114yes—
399C2×C4#44812yes—
400C2×C4#45212yes—
401C2×C4#45312yes—
402C2×C4#45512yes—
403C2×C4#47514yes—
404C2×C4#47914yes—
405C2×C4C22×C4⋊C411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×#12311yescharacteristic
411C2×C4C22×SD1611yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4C2×#13711yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4#57712yes—
416C2×C4#57812yes—
417C8—44no—
418C8—44no—
419C8—44no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22#186514yes—
427C22C2×#38511yescharacteristic
428C22C2×#44811yescharacteristic
429C22C2×#45211yescharacteristic
430C22#301011yescharacteristic
431C22#301111yescharacteristic
432C22C2×#45311yescharacteristic
433C22C2×#45511yescharacteristic
434C4—21no—
435C4—21no—
436C4—22no—
437C4—48no—
438C4—42no—
439C4—42no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—82no—
444C4—162no—
445C4#185614yes—
446C4#186514yes—
447C4C2×C4×SD1611yescharacteristic
448C4C2×#44111yescharacteristic
449C4#294712yes—
450C4C2×#47511yescharacteristic
451C4C2×#47911yescharacteristic
452C4#308212yes—
453C2—24no—
454C2#1862211yescharacteristic
455C2#1862711yescharacteristic
456C2#1863211yescharacteristic
457C2#1863711yescharacteristic
458C2C2×#185611yescharacteristic
459C2C2×#186511yescharacteristic
460C2#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. #8928C2semidirect
7. #8927C2semidirect
41. #1548C4semidirect
129. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342162
#219812
C2×#19562
#211712
C2×#15482
#89282
#89272
#78972
#78902