Finite Group Databasebeta

← #183241

Subgroups — #183241

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34208C211yesmaximal, characteristic
2#20480C212yesmaximal, complemented
3C2×#1536C211yesmaximal, characteristic
4C2×#1525C211yesmaximal, characteristic
5#8226C214yesmaximal, complemented
6#8224C212yesmaximal, complemented
7C4×#329C212yesmaximal, complemented
8#8033C212yesmaximal, complemented
9C2×#532C2211yescharacteristic
10C2×#520C411yescharacteristic
11C2×#520C411yescharacteristic
12C2×#432C2212yes—
13C2×C4×M16C2212yes—
14#2364—24no—
15#2352—22no—
16#2352—22no—
17#2328C412yes—
18#2327C412yes—
19#2311C411yescharacteristic
20#2311C411yescharacteristic
21C4×#124C2212yes—
22C4×#124C2212yes—
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1770—22no—
26C8×C4⋊C4—22no—
27#1753—24no—
28#1745C2214yes—
29C4×#97C2214yes—
30#1536C418yescomplemented
31#1525C418yescomplemented
32C2×#329C2211yescharacteristic
33C2×#329C2212yes—
34C2×#329C2211yescharacteristic
35C2×#329C2212yes—
36C2×#329C2212yes—
37#1295C2214yes—
38#1295C2214yes—
39C22×#41C2×C411yescharacteristic
40C22×#41C2×C411yescharacteristic
41C22×M16—22no—
42#532—24no—
43#532C2×C414yes—
44#532C2×C414yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124C2311yescharacteristic
48C2×#124C2311yescharacteristic
49C2×#121—22no—
50#520—44no—
51#520—44no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4C2×C412yes—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68#432—24no—
69#432C2×C418yescomplemented
70C2×C4⋊C8—22no—
71C2×C22⋊C8—22no—
72C4×M16—24no—
73C4×M16C2×C418yescomplemented
74C2×#97—24no—
75C2×#97C2312yes—
76C2×#97C2312yes—
77C2×#97C2314yes—
78C2×C4×C8—22no—
79C2×C4×C8—24no—
80#391C2×C412yes—
81#391C2×C412yes—
82#390C2×C412yes—
83#390C2×C412yes—
84#388—24no—
85#388—24no—
86#387—24no—
87#387—24no—
88#386—22no—
89#386—22no—
90#386—24no—
91#386—24no—
92#382—22no—
93#382C2×C412yes—
94#382C2312yes—
95#380—22no—
96#380—22no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4D412yes—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4D414yes—
102#376C2×C412yes—
103C2×#95—22no—
104C43C2312yes—
105#336—24no—
106#336—24no—
107#329C2×C418yescomplemented
108#329C2×C418yescomplemented
109#329C2×C414yescomplemented
110#329C2×C414yescomplemented
111#329C2×C418yescomplemented
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—42no—
115C2×#41—42no—
116C2×#41—42no—
117C2×#41—42no—
118C2×#41C4214yes—
119C2×#41C4214yes—
120C22×Q8C22×C411yescharacteristic
121C22×Q8C22×C411yescharacteristic
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4C22×C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C2×M16—28no—
136C2×M16—44no—
137C2×M16—44no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—24no—
141C4⋊Q8—22no—
142C4⋊Q8C4214yes—
143#127—22no—
144#127C4214yes—
145#124—22no—
146#124—22no—
147#124C22×C412yes—
148#124C22×C412yes—
149#124C22×C412yes—
150#124C22×C412yes—
151#121—28no—
152#121—28no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×D4—28no—
156C4×D4—28no—
157C4×D4—44no—
158C4×D4—44no—
159C4×D4—48no—
160C4×D4—48no—
161#117—24no—
162#117—42no—
163#117—44no—
164#117—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D414yes—
195C2×C42C2×D414yes—
196C2×C42C2×D412yes—
197C2×C42#4114yes—
198C2×C42C2411yescharacteristic
199C4⋊C8—28no—
200C22⋊C8—28no—
201#97—22no—
202#97—22no—
203#97—24no—
204#97—48no—
205#97C22×C414yes—
206#97C22×C414yes—
207#97C22×C418yes—
208C4×C8—28no—
209C4×C8—48no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95C22⋊C414yes—
215#95C22⋊C414yes—
216C24—21no—
217C24—21no—
218C24C2×C22⋊C411yescharacteristic
219#41—48no—
220#41—48no—
221#41—48no—
222#41—48no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8C2×C4212yes—
230C2×Q8C2×C4212yes—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—28no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4C2×C4212yes—
247C2×D4C2×C4212yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C4212yes—
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4#11712yes—
277C22×C4C4×D414yes—
278C22×C4C4×D414yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281M16—416no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—28no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—28no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4#10014yes—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C42—24no—
312C42—44no—
313C42—44no—
314C42—42no—
315C42—44no—
316C42—44no—
317C42C2×C4212yes—
318C42#11712yes—
319C42C4×D414yes—
320C42C4×D414yes—
321C42C23×C411yescharacteristic
322C42C23×C411yescharacteristic
323C42C2×#4112yes—
324C23—21no—
325C23—21no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23C4×C22⋊C412yes—
339C23C22×C22⋊C411yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—42no—
343Q8—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—48no—
349D4—48no—
350D4—88no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—164no—
380C2×C4—164no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4C4×C22⋊C414yes—
384C2×C4C4×C22⋊C412yes—
385C2×C4C2×#10014yes—
386C2×C4C2×#10012yes—
387C2×C4#41312yes—
388C2×C4C22×C4211yescharacteristic
389C2×C4C2×#11711yescharacteristic
390C2×C4C2×C4×D412yes—
391C2×C4C2×C4×D412yes—
392C8—28no—
393C8—48no—
394C8—44no—
395C22—22no—
396C22—24no—
397C22—41no—
398C22—42no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—44no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22—82no—
408C22—81no—
409C22—81no—
410C22C2×C4×C22⋊C411yescharacteristic
411C22C22×#10011yescharacteristic
412C22C2×#41311yescharacteristic
413C22#290112yes—
414C22#290112yes—
415C4—24no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—24no—
421C4—44no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—82no—
427C4C4×#10014yes—
428C2—42no—
429C2—82no—
430C2—82no—
431C2#1768511yescharacteristic
432C2#1768712yes—
433C2#1768811yescharacteristic
434C2C2×C4×#10011yescharacteristic
435C2#2040512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20480C2semidirect
5. #8226C2semidirect
6. #8224C2semidirect
7. C4×#329C2semidirect
8. #8033C2semidirect
30. #1536C4semidirect
31. #1525C4semidirect
69. #432C2×C4semidirect
73. C4×M16C2×C4semidirect
107. #329C2×C4semidirect
108. #329C2×C4semidirect
109. #329C2×C4semidirect
110. #329C2×C4semidirect
111. #329C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342082
#204802
C2×#15362
C2×#15252
#82262
#82242
C4×#3292
#80332