Finite Group Databasebeta

← #181137

Subgroups — #181137

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#34179C211yesmaximal, characteristic
3#34109C211yesmaximal, characteristic
4#7946C212yesmaximal
5#7945C214yesmaximal, complemented
6#7942C212yesmaximal
7#7941C214yesmaximal, complemented
8C22×#117C2211yescharacteristic
9#2924—24no—
10#2924C418yescomplemented
11#2921—22no—
12#2920C2214yes—
13#2905—21no—
14C2×#407C2212yes—
15#2873—21no—
16#2347—21no—
17#2347C418yescomplemented
18#2346—21no—
19#2346—24no—
20#2345—21no—
21#2344—21no—
22#2343—22no—
23#2342—22no—
24#2316—21no—
25#2312—22no—
26#2298C414yescomplemented
27#2296C412yescomplemented
28#2283C2214yes—
29#2283C2212yes—
30#2280C2212yes—
31#2280C2211yescharacteristic
32#2274—21no—
33C4×#117C412yescomplemented
34C4×#117C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1284C2214yes—
37#1284C2218yes—
38#1284C2214yes—
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117C2×C412yes—
44C2×#117C2×C414yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C412yes—
48C2×#117C2×C414yes—
49C2×#117D412yes—
50C2×#117D411yescharacteristic
51C2×#117D414yes—
52C2×#117D411yescharacteristic
53C2×#117Q814yes—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4D412yes—
58C22×C42C2×C412yes—
59C22×C42C2312yes—
60#433—216no—
61#432—24no—
62#432C2×C418yescomplemented
63#431—24no—
64#431C2×C418yescomplemented
65#424—28no—
66#423—24no—
67#407—22no—
68#407—22no—
69#407C2×C414yes—
70#407C2314yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2×C414yes—
73#405—28no—
74C4×M16—24no—
75#389—28no—
76#389—22no—
77#389—22no—
78#388—216no—
79#388—24no—
80#388—24no—
81#387—24no—
82#387—24no—
83#385—24no—
84#383—22no—
85#383C2×C418yescomplemented
86#382—22no—
87#382—24no—
88#381—22no—
89#381—21no—
90#381C2314yes—
91#380—24no—
92#380—22no—
93#380—21no—
94#380—22no—
95#380—21no—
96#380C2×C414yes—
97#379—22no—
98#379—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2×C414yes—
105C4×C4⋊C4C2×C418yescomplemented
106C4×C22⋊C4—24no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C414yes—
115C4×C22⋊C4C2311yescharacteristic
116C4×C22⋊C4C2311yescharacteristic
117C4×C22⋊C4C2312yes—
118#376—22no—
119#376—24no—
120#376C2×C414yes—
121C2×#95—21no—
122C43C2×C414yes—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—44no—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×Q812yes—
137C23×C4C2411yescharacteristic
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—48no—
141C22×C8—24no—
142#117—48no—
143#117—48no—
144#117—44no—
145#117C4214yes—
146#117C4214yes—
147#117C4218yes—
148#117C22⋊C414yes—
149#117C22⋊C414yes—
150#117C22⋊C418yes—
151#117C4⋊C414yes—
152#117C4⋊C414yes—
153#117C4⋊C414yes—
154#117C4⋊C414yes—
155#117C4⋊C418yes—
156#117C4⋊C418yes—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C4⋊C414yes—
171C2×C4⋊C4C4⋊C414yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C418yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C4⋊C418yes—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—28no—
193C2×C42—44no—
194C2×C42C22×C414yes—
195C2×C42C22×C412yes—
196C2×C42C22×C412yes—
197C2×C42C22×C414yes—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C414yes—
201C2×C42C2×D412yes—
202C2×C42C2×D414yes—
203C2×C42C2×D412yes—
204C2×C42C2×Q814yes—
205C4⋊C8—28no—
206C4⋊C8—28no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8C22×C418yes—
211#97—28no—
212C4×C8—28no—
213#95—22no—
214#95—24no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—28no—
219#95—22no—
220#95—24no—
221#95—21no—
222#95—21no—
223#95—24no—
224#95—22no—
225#95C22×C418yes—
226C24—42no—
227C24—42no—
228C24—81no—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C412yes—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C4⋊C412yes—
233C24C22×D411yescharacteristic
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4C2×C4212yes—
254C22×C4C2×C4212yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C414yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C4⋊C412yes—
267C22×C4C2×C4⋊C412yes—
268C22×C4C2×C4⋊C412yes—
269C22×C4C2×C4⋊C412yes—
270C22×C4C2×C4⋊C412yes—
271C22×C4C2×C4⋊C414yes—
272C22×C4C2×C4⋊C414yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×Q811yescharacteristic
279M16—48no—
280M16—48no—
281C2×C8—28no—
282C2×C8—48no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—84no—
287C4⋊C4—88no—
288C4⋊C4#9518yes—
289C4⋊C4#9518yes—
290C4⋊C4#9514yes—
291C4⋊C4#9514yes—
292C4⋊C4#9514yes—
293C4⋊C4#9514yes—
294C22⋊C4—24no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—48no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C22⋊C4—82no—
302C22⋊C4—84no—
303C22⋊C4#9518yes—
304C22⋊C4#9518yes—
305C22⋊C4#9518yes—
306C22⋊C4#9518yes—
307C42—42no—
308C42—42no—
309C42—44no—
310C42—48no—
311C42—48no—
312C42—44no—
313C42—88no—
314C42—84no—
315C42C2×C4214yes—
316C42C2×C4214yes—
317C42C2×C22⋊C414yes—
318C42C2×C22⋊C414yes—
319C42C2×C4⋊C414yes—
320C42C2×C4⋊C414yes—
321C42C2×C4⋊C414yes—
322C42C2×C4⋊C414yes—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—24no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—88no—
339C23C2×#9512yes—
340C23C2×#9512yes—
341C23C22×C22⋊C411yescharacteristic
342C23C22×C22⋊C411yescharacteristic
343C23C22×C4⋊C411yescharacteristic
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—88no—
382C2×C4C2×#9514yes—
383C2×C4C2×#9514yes—
384C2×C4C2×#9514yes—
385C2×C4C2×#9514yes—
386C2×C4C2×#9512yes—
387C2×C4C2×#9512yes—
388C2×C4#41014yes—
389C2×C4C22×C4211yescharacteristic
390C2×C4C22×C22⋊C411yescharacteristic
391C2×C4C22×C4⋊C411yescharacteristic
392C2×C4C22×C4⋊C411yescharacteristic
393C8—48no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—44no—
408C22—44no—
409C22—82no—
410C22—84no—
411C22—84no—
412C22C22×#9511yescharacteristic
413C22C2×#41012yes—
414C22#290112yes—
415C4—24no—
416C4—24no—
417C4—24no—
418C4—24no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—21no—
424C4—44no—
425C4—84no—
426C4—88no—
427C4—82no—
428C4#174014yes—
429C2—22no—
430C2—22no—
431C2—41no—
432C2—42no—
433C2—42no—
434C2#2033811yescharacteristic
435C2#2034311yescharacteristic
436C2C2×#174011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #7945C2semidirect
7. #7941C2semidirect
10. #2924C4semidirect
17. #2347C4semidirect
26. #2298C4semidirect
27. #2296C4semidirect
33. C4×#117C4semidirect
62. #432C2×C4semidirect
64. #431C2×C4semidirect
85. #383C2×C4semidirect
105. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#341792
#341092
#79462
#79452
#79422
#79412