Finite Group Databasebeta

← #200527

Subgroups — #200527

Order 512.

1508 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#60248C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic, complemented
4#17741C212yesmaximal
5#9406C214yesmaximal, complemented
6#9398C214yesmaximal, complemented
7#8098C212yesmaximal, complemented
8C22×#127C2211yescharacteristic
9C22×#124—21no—
10C22×C4×Q8C2211yescharacteristic
11#2984—24no—
12#2983—22no—
13#2983C2212yescomplemented
14#2982C2212yescomplemented
15Q8×M16—24no—
16#2920—22no—
17#2920C2212yes—
18#2920C2212yes—
19#2548—21no—
20#2548—21no—
21#2548—22no—
22#2548C414yes—
23#2548C2212yes—
24#2548C2212yes—
25C2×#394C414yes—
26C2×#390—22no—
27C2×#390—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1821—21no—
30#1801C2212yescomplemented
31#1801C2212yescomplemented
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1555—24no—
36#1545—24no—
37#1545C2218yes—
38#1303C2218yescomplemented
39C23×Q8D411yescharacteristic
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2×C412yes—
43C2×#127—22no—
44C2×#127—22no—
45C2×#127—21no—
46C2×#127—21no—
47C2×#127—22no—
48C2×#127—22no—
49C2×#127C2×C412yes—
50C2×#127C2312yes—
51C2×#124—24no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—48no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D412yes—
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—24no—
69C22×C22⋊C4—24no—
70C22×C42—21no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73#446—216no—
74#446—28no—
75C8×Q8—28no—
76#432—24no—
77#432—24no—
78#432—22no—
79#432—22no—
80#432—22no—
81#432D414yes—
82#431—24no—
83#431—24no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431C2314yes—
88#423—24no—
89#423D414yes—
90#423C2314yes—
91C2×#101—22no—
92C2×#101—22no—
93#407—22no—
94#407—22no—
95C4×M16—24no—
96C4×M16—22no—
97C4×M16—22no—
98#394—24no—
99#394—24no—
100#394C2×C418yes—
101#394C2×C418yes—
102#390—24no—
103#390—22no—
104#390—22no—
105#390—24no—
106#390—24no—
107#390—24no—
108#390—24no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2312yes—
113C2×#95C2×C412yes—
114#343—22no—
115#343—22no—
116#331—216no—
117C25—22no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—42no—
122C22×Q8C22⋊C412yes—
123C22×Q8C2×D411yescharacteristic
124C22×Q8C2×D411yescharacteristic
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—48no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—41no—
140C22×D4—41no—
141C22×D4—84no—
142C22×D4C22×C414yes—
143C23×C4—22no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151#127—24no—
152#127—24no—
153#127—24no—
154#127—24no—
155#127—42no—
156#127—42no—
157#127—42no—
158#127—42no—
159#124—416no—
160#124—48no—
161#124—48no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—24no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—416no—
193C2×C22⋊C4—88no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213C4⋊C8—28no—
214C4⋊C8—28no—
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8C2×D418yes—
219#101—44no—
220#101—44no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#97—28no—
226#97—24no—
227#97—42no—
228C4×C8—28no—
229C4×C8—24no—
230C4×C8—42no—
231#95—22no—
232#95C22×C414yes—
233C24—24no—
234C24—44no—
235C24—48no—
236C24—44no—
237C24—48no—
238C24—42no—
239C24—42no—
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—42no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8AGL(1,ℤ/8)14yes—
252C2×D4—28no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—48no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—88no—
267C2×D4—84no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—88no—
271C2×D4—88no—
272C2×D4—84no—
273C2×D4—88no—
274C2×D4—84no—
275C2×D4—84no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12311yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—48no—
310M16—44no—
311M16—44no—
312C2×C8—28no—
313C2×C8—44no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C22⋊C4—416no—
324C22⋊C4—416no—
325C22⋊C4—88no—
326C22⋊C4—88no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—44no—
336C42—44no—
337C42C4×D412yes—
338C42C4×D412yes—
339C42#12012yes—
340C42#12112yes—
341C42#12112yes—
342C42#12312yes—
343C42C22×D412yes—
344C42C2×#4112yes—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—22no—
349C23—48no—
350C23—48no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—84no—
355C23—84no—
356C23—84no—
357C23—84no—
358C23—816no—
359C23—816no—
360C23—84no—
361C23—84no—
362C23#38611yescharacteristic
363C23#38611yescharacteristic
364C23C22×C22⋊C411yescharacteristic
365Q8—28no—
366Q8—48no—
367Q8—44no—
368D4—88no—
369D4—84no—
370D4—84no—
371D4—88no—
372D4—88no—
373D4—84no—
374D4—84no—
375D4—88no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—82no—
405C2×C4—88no—
406C2×C4#38611yescharacteristic
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4#38611yescharacteristic
410C2×C4#41814yes—
411C2×C4#45414yes—
412C2×C4C2×C4×D412yes—
413C2×C4C2×#12011yescharacteristic
414C2×C4C2×#12112yes—
415C2×C4C2×#12311yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)12yes—
417C2×C4#57512yes—
418C8—48no—
419C8—84no—
420C22—24no—
421C22—24no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—88no—
429C22—88no—
430C22—84no—
431C22—84no—
432C22—84no—
433C22—168no—
434C22—168no—
435C22#188812yes—
436C22#191412yes—
437C22C2×#38611yescharacteristic
438C22C2×#41811yescharacteristic
439C22#289611yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—22no—
445C4—24no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4#188812yes—
451C4C2×#45412yes—
452C4#305514yes—
453C2—21no—
454C2—21no—
455C2—22no—
456C2—22no—
457C2—164no—
458C2C2×#188811yescharacteristic
459C2#2097311yescharacteristic
460C2C2×#191411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
3. #20727C2semidirect
5. #9406C2semidirect
6. #9398C2semidirect
7. #8098C2semidirect
13. #2983C22semidirect
14. #2982C22semidirect
30. #1801C22semidirect
31. #1801C22semidirect
38. #1303C22semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25482
#207272
#177412
#94062
#93982
#80982