Finite Group Databasebeta

← #193319

Subgroups — #193319

Order 512.

1124 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#34368C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21981C211yesmaximal, characteristic, complemented
4#21200C211yesmaximal, characteristic
5C2×#1543C211yesmaximal, characteristic
6#8927C214yesmaximal, complemented
7#8921C214yesmaximal, complemented
8#7896C211yesmaximal, characteristic, complemented
9#7896C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#417—22no—
14C2×C4×M16C2211yescharacteristic
15#2379—22no—
16#2369C2212yes—
17#2366C2212yes—
18#2354—21no—
19#2354—21no—
20D4×C4⋊C4—21no—
21D4×C4⋊C4—21no—
22#2324—22no—
23#2324—22no—
24#2311—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#1959—22no—
29#1957C2212yes—
30#1956C2212yes—
31#1948C2214yes—
32#1947C2214yes—
33#1853—22no—
34#1852—22no—
35#1845—21no—
36#1844—21no—
37#1834—21no—
38#1834—21no—
39#1543C418yescomplemented
40C2×#333C2212yes—
41C2×#331C2212yes—
42#1279C2214yes—
43#1279C2214yes—
44C22×#41D412yes—
45C22×M16—21no—
46#532—22no—
47#532—22no—
48#532C2×C414yes—
49#532Q814yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×#127C2311yescharacteristic
52C2×#124D412yes—
53C2×#121—21no—
54C2×#121—21no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×#117—22no—
60C2×#117—21no—
61C2×#117—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#431C2×C414yescomplemented
70#431C2×C414yescomplemented
71C2×#107—24no—
72C2×#106—24no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75#417—48no—
76C2×#103—24no—
77C2×#102—24no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C4×M16—22no—
81C4×M16—22no—
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2312yes—
86#391—24no—
87#390—24no—
88#390—22no—
89#390—22no—
90#389—22no—
91#389—22no—
92#388—24no—
93#388—24no—
94#386—24no—
95#386—24no—
96#385—24no—
97#385—24no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#384D411yescharacteristic
104#384D411yescharacteristic
105#384C2312yes—
106#382—21no—
107#382—21no—
108#380—22no—
109#380—22no—
110#380—21no—
111#380—21no—
112#380D412yes—
113#380D412yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2312yes—
119C4×C4⋊C4C2312yes—
120#340—24no—
121#340—24no—
122#333C2×C418yescomplemented
123#331C2×C418yescomplemented
124C2×#41—24no—
125C2×#41—44no—
126C2×#41—44no—
127C2×#41C4⋊C418yes—
128C22×Q8C2×D412yes—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×D412yes—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—42no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C22×C412yes—
148C4⋊Q8C2×Q812yes—
149#127—21no—
150#127—21no—
151#127C22×C412yes—
152#127C2×Q812yes—
153#124—24no—
154#124C4⋊C418yes—
155#121—24no—
156#121—24no—
157#121—24no—
158#121—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—84no—
164C4×D4—84no—
165#117—82no—
166#117—82no—
167#117—84no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—84no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×Q812yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42C2411yescharacteristic
211#107—84no—
212#106—84no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8C22×C414yes—
216C4⋊C8C22×C414yes—
217#103—48no—
218#102—48no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221#97—42no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—22no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24—22no—
233C24#12011yescharacteristic
234#41—416no—
235#41—88no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8C2×C4⋊C414yes—
240C2×D4—28no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4C2×C4⋊C414yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4C2×C4⋊C412yes—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4C22⋊Q812yes—
283C22×C4C22⋊Q812yes—
284C22×C4#12312yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—48no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C4⋊C4—81no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—84no—
320C4⋊C4—82no—
321C22⋊C4—28no—
322C22⋊C4—28no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—84no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C42—44no—
329C42—44no—
330C42—84no—
331C42C2×C4⋊C412yes—
332C42C4×D412yes—
333C42C4×D412yes—
334C42C22⋊Q812yes—
335C42C22⋊Q812yes—
336C42#12312yes—
337C42C23×C411yescharacteristic
338C42C22×Q811yescharacteristic
339C42C2×#4111yescharacteristic
340C42C2×#4111yescharacteristic
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23#38512yes—
355C23C2×#12011yescharacteristic
356Q8—44no—
357Q8—82no—
358D4—44no—
359D4—44no—
360D4—44no—
361D4—44no—
362D4—44no—
363D4—82no—
364D4—88no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—416no—
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—42no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—168no—
397C2×C4—164no—
398C2×C4#38512yes—
399C2×C4#38512yes—
400C2×C4#38512yes—
401C2×C4#45312yes—
402C2×C4#45512yes—
403C2×C4C22×C4⋊C411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×C22⋊Q811yescharacteristic
408C2×C4C2×#12311yescharacteristic
409C2×C4#57612yes—
410C2×C4#57712yes—
411C2×C4#57712yes—
412C2×C4#57812yes—
413C8—44no—
414C8—44no—
415C8—44no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—44no—
424C22—44no—
425C22—44no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22C2×#38511yescharacteristic
431C22#300811yescharacteristic
432C22#300911yescharacteristic
433C22#301011yescharacteristic
434C22#301111yescharacteristic
435C22C2×#45311yescharacteristic
436C22C2×#45511yescharacteristic
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—44no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—162no—
447C4#185614yes—
448C4#295112yes—
449C4#295212yes—
450C4#308312yes—
451C4#308412yes—
452C2—42no—
453C2—84no—
454C2#1871611yescharacteristic
455C2#1871711yescharacteristic
456C2#1872211yescharacteristic
457C2#1872511yescharacteristic
458C2C2×#185611yescharacteristic
459C2#2088111yescharacteristic
460C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. #21982C2semidirect
3. #21981C2semidirect
6. #8927C2semidirect
7. #8921C2semidirect
8. #7896C2semidirect
9. #7896C2semidirect
39. #1543C4semidirect
69. #431C2×C4semidirect
70. #431C2×C4semidirect
122. #333C2×C4semidirect
123. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343682
#219822
#219812
#212002
C2×#15432
#89272
#89212
#78962