Finite Group Databasebeta

← #181182

Subgroups — #181182

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#60125C211yesmaximal, characteristic
3#34184C211yesmaximal, characteristic
4#7961C214yesmaximal, complemented
5#7959C214yesmaximal, complemented
6#7957C212yesmaximal, complemented
7C2×#1286C212yesmaximal, complemented
8C2×#520—21no—
9C2×C4×#41C2211yescharacteristic
10#2989—24no—
11#2983—21no—
12#2975—21no—
13#2973—22no—
14#2965C418yescomplemented
15#2925—24no—
16#2925C418yescomplemented
17#2921—22no—
18#2920C2214yes—
19#2920C2214yes—
20C2×#431C2212yes—
21C2×#423C2212yes—
22#2879—21no—
23#2873—21no—
24#2351—21no—
25#2351C412yes—
26#2349—22no—
27#2349—21no—
28#2349C414yes—
29#2349C412yes—
30#2317—21no—
31#2317—22no—
32#2285C2211yescharacteristic
33#2285C2212yes—
34C2×#382C2211yescharacteristic
35C2×#382C2212yes—
36#1286C2218yes—
37#1286C2214yes—
38#1286C2214yes—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41#520—48no—
42C4×#41C2×C414yes—
43C4×#41C2×C414yes—
44C4×#41D414yes—
45C4×#41Q814yes—
46C2×C4×Q8—22no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×#117—22no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D412yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2312yes—
63#446—28no—
64#436—28no—
65#435—216no—
66#433—216no—
67#432—24no—
68#432—24no—
69#432—24no—
70#432C2×C418yescomplemented
71#431—22no—
72#431—22no—
73#431—24no—
74#431C2×C418yescomplemented
75#431C2×C414yescomplemented
76#431C2×C418yescomplemented
77#431C2314yes—
78#423—22no—
79#423—22no—
80#423—24no—
81#423C2×C414yescomplemented
82#423C2314yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8C2×C414yescomplemented
86#408—28no—
87#407—24no—
88#407—24no—
89C2×C22⋊C8—22no—
90#405—28no—
91C4×M16—24no—
92C4×M16C2×C418yescomplemented
93C2×C4×C8C2×C414yescomplemented
94#389—24no—
95#389—22no—
96#389—24no—
97#389—22no—
98#385—24no—
99#385—24no—
100#385—24no—
101#384—22no—
102#384—21no—
103#382—22no—
104#382—21no—
105#382—21no—
106#382—22no—
107#382C2×C412yes—
108#382C2×C414yes—
109#382C2312yes—
110#382C2311yescharacteristic
111#382C2311yescharacteristic
112#380—21no—
113#380—22no—
114#380C2×C412yes—
115#380C2×C414yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C414yes—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95C2×C412yes—
123C2×#95C2×C414yes—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41C22⋊C414yes—
127C2×#41C4⋊C414yes—
128C22×Q8C22⋊C411yescharacteristic
129C22×D4C22⋊C412yes—
130C22×D4C22⋊C411yescharacteristic
131C23×C4—24no—
132C23×C4—22no—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—28no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—24no—
145C22×C8—24no—
146C4×Q8—48no—
147C4×Q8C4214yes—
148C4×Q8C22⋊C414yes—
149C4×Q8C4⋊C414yes—
150C4×Q8C4⋊C414yes—
151C4×D4—416no—
152C4×D4—48no—
153C4×D4C4218yes—
154C4×D4C4214yes—
155C4×D4C22⋊C418yes—
156C4×D4C22⋊C414yes—
157C4×D4C4⋊C418yes—
158C4×D4C4⋊C414yes—
159C4×D4C4⋊C418yes—
160C4×D4C4⋊C414yes—
161#117—48no—
162#117—44no—
163#117C22⋊C414yes—
164#117C22⋊C418yes—
165#117C4⋊C418yes—
166#117C4⋊C414yes—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C412yes—
192C2×C42C22×C414yes—
193C2×C42C22×C412yes—
194C2×C42C22×C414yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D414yes—
201C2×C42C2×Q814yes—
202C2×C42C2×Q812yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—28no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8C22×C418yes—
211C22⋊C8—24no—
212C22⋊C8—28no—
213C22⋊C8—28no—
214C22⋊C8—28no—
215C22⋊C8—24no—
216#97—28no—
217#97—28no—
218C4×C8—24no—
219C4×C8—28no—
220C4×C8C22×C418yes—
221#95—24no—
222#95—22no—
223#95—24no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95C22×C414yes—
229#95C22×C412yes—
230#95C22×C412yes—
231C24C2×C22⋊C412yes—
232C24C2×C22⋊C411yescharacteristic
233#41—28no—
234#41—28no—
235#41—216no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8#9514yes—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4#9518yes—
244C2×D4#9514yes—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—81no—
253C22×C4—82no—
254C22×C4—81no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—84no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4C2×C4214yes—
261C22×C4C2×C4212yes—
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C4⋊C414yes—
273C22×C4C2×C4⋊C414yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4C2×C4⋊C414yes—
276C22×C4C2×C4⋊C412yes—
277C22×C4C2×C4⋊C412yes—
278C22×C4AGL(1,ℤ/8)14yes—
279C22×C4#13714yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283M16—44no—
284M16—44no—
285M16—48no—
286M16—88no—
287C2×C8—28no—
288C2×C8—28no—
289C2×C8—48no—
290C2×C8—84no—
291C4⋊C4—48no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—84no—
300C4⋊C4#9518yes—
301C4⋊C4#9514yes—
302C4⋊C4#9518yes—
303C4⋊C4#9514yes—
304C22⋊C4—48no—
305C22⋊C4—44no—
306C22⋊C4#9518yes—
307C22⋊C4#9518yes—
308C22⋊C4#9518yes—
309C42—48no—
310C42—44no—
311C42C2×C4214yes—
312C42C2×C4212yes—
313C42C2×C22⋊C414yes—
314C42C2×C22⋊C412yes—
315C42C2×C22⋊C414yes—
316C42C2×C22⋊C412yes—
317C42C2×C4⋊C412yes—
318C42C2×C4⋊C414yes—
319C42C2×C4⋊C414yes—
320C42C2×C4⋊C412yes—
321C42C2×C4⋊C414yes—
322C42C2×C4⋊C412yes—
323C42C23×C411yescharacteristic
324C42C23×C411yescharacteristic
325C42C22×D411yescharacteristic
326C42C22×Q811yescharacteristic
327C23—22no—
328C23—22no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—42no—
333C23C2×#9514yes—
334C23C2×#9512yes—
335C23#41812yes—
336C23#41912yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—28no—
339D4—216no—
340D4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—164no—
369C2×C4—168no—
370C2×C4C2×#9514yes—
371C2×C4C2×#9512yes—
372C2×C4C2×#9514yes—
373C2×C4C2×#9514yes—
374C2×C4C2×#9512yes—
375C2×C4C2×#9512yes—
376C2×C4#41714yes—
377C2×C4#41714yes—
378C2×C4#41812yes—
379C2×C4#41912yes—
380C2×C4#42818yes—
381C2×C4C22×C4211yescharacteristic
382C2×C4C22×C22⋊C411yescharacteristic
383C2×C4C22×C22⋊C411yescharacteristic
384C2×C4C22×C4⋊C411yescharacteristic
385C2×C4C22×C4⋊C411yescharacteristic
386C2×C4C22×C4⋊C411yescharacteristic
387C2×C4C2×AGL(1,ℤ/8)12yes—
388C2×C4C2×#13712yes—
389C8—48no—
390C8—48no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—22no—
395C22—24no—
396C22—24no—
397C22—24no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—44no—
402C22#173614yes—
403C22C22×#9511yescharacteristic
404C22C2×#41811yescharacteristic
405C22C2×#41911yescharacteristic
406C4—21no—
407C4—21no—
408C4—22no—
409C4—24no—
410C4—24no—
411C4—21no—
412C4—21no—
413C4—22no—
414C4—24no—
415C4—82no—
416C4—84no—
417C4—82no—
418C4#173614yes—
419C4C2×#41712yes—
420C4C2×#42812yes—
421C2—21no—
422C2—21no—
423C2—24no—
424C2—44no—
425C2#2033811yescharacteristic
426C2C2×#173611yescharacteristic
427C2#2036011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7961C2semidirect
5. #7959C2semidirect
6. #7957C2semidirect
7. C2×#1286C2semidirect
14. #2965C4semidirect
16. #2925C4semidirect
70. #432C2×C4semidirect
74. #431C2×C4semidirect
75. #431C2×C4semidirect
76. #431C2×C4semidirect
81. #423C2×C4semidirect
85. C2×C4⋊C8C2×C4semidirect
92. C4×M16C2×C4semidirect
93. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#601252
#341842
#79612
#79592
#79572
C2×#12862