Finite Group Databasebeta

← #235772

Subgroups — #235772

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic, complemented
2C22×#486C211yesmaximal, characteristic
3#35285C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic, complemented
5C2×#1545C211yesmaximal, characteristic
6C2×#1493C211yesmaximal, characteristic, complemented
7#10063C218yesmaximal, complemented
8#7890C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494—21no—
15C2×#487—21no—
16C2×#486—24no—
17C2×#486—24no—
18C2×#486—22no—
19C2×#486C2214yes—
20C2×#462C2214yescomplemented
21C2×#451—22no—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#102—22no—
25C22×C4×C8C2211yescharacteristic
26#2858—21no—
27#2858—21no—
28#2641—22no—
29#2548—22no—
30#2548C2214yes—
31#2435—22no—
32#2435—22no—
33#2309—22no—
34C4×#117C2211yescharacteristic
35#1988C2214yescomplemented
36#1987—22no—
37#1736—22no—
38#1545—24no—
39C2×#331C2214yes—
40#1493—24no—
41C2×Q8⋊C8C2214yescomplemented
42#1279C2214yescomplemented
43C23×Q8—21no—
44C23×D4—21no—
45C23×D4—21no—
46C23×D4D412yes—
47C22×Q16—24no—
48C22×M16—21no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#127—22no—
65C2×#127—22no—
66C2×#127—22no—
67C2×#127—21no—
68C2×#127—21no—
69C2×#127—22no—
70C2×#127D412yes—
71C2×#127C2312yes—
72C2×#125—21no—
73C2×#124—24no—
74C2×#124—24no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D412yes—
77C2×C4×Q8—21no—
78C2×C4×Q8C2312yes—
79C2×#117—21no—
80C2×#117D412yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C42C2311yescharacteristic
84#494—28no—
85#487—28no—
86#486—216no—
87#486—28no—
88#486—28no—
89#462—28no—
90#462—28no—
91#451—216no—
92#431—22no—
93#431D414yes—
94#423—22no—
95#423—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99C2×#103—24no—
100C2×#102—216no—
101C2×#102—28no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8D414yes—
107C2×C4×C8C2312yes—
108#394—28no—
109#394—28no—
110#390—24no—
111#390—24no—
112#390—24no—
113#390—28no—
114#390—24no—
115#379—28no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C43—21no—
122#340—28no—
123#331—28no—
124Q8⋊C8—28no—
125C25—22no—
126C22×Q8—21no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—84no—
132C22×Q8C2×D412yes—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—48no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—84no—
148C22×D4C2×D414yes—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×Q16—216no—
154C2×Q16—88no—
155C2×M16—42no—
156C2×M16—44no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—44no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—24no—
165C4⋊Q8—22no—
166C4⋊Q8—24no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8C2×D414yes—
176#127—24no—
177#127—24no—
178#127—22no—
179#127—22no—
180#127—22no—
181#127—22no—
182#127—42no—
183#127—42no—
184#127—42no—
185#127—42no—
186#125—42no—
187#125—42no—
188#124—88no—
189#124—88no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—24no—
201#117—24no—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—216no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239C4⋊C8C2×D414yes—
240#103—216no—
241#102—232no—
242C22⋊C8—24no—
243C4×C8—24no—
244C4×C8—24no—
245C4×C8—22no—
246C4×C8C2×D414yes—
247#95—22no—
248#95—22no—
249C24—24no—
250C24—44no—
251C24—48no—
252C24—84no—
253C24—84no—
254C24#12011yescharacteristic
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—28no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—88no—
265C2×Q8—84no—
266C2×Q8—88no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—48no—
270C2×D4—416no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—84no—
284C2×D4—88no—
285C2×D4—88no—
286C2×D4—88no—
287C2×D4—88no—
288C2×D4—84no—
289C2×D4—84no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—84no—
298C22×C4#12012yes—
299C22×C4#12012yes—
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12712yes—
307C22×C4#13514yes—
308C22×C4#13714yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312Q16—816no—
313M16—84no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—44no—
317C2×C8—48no—
318C2×C8—44no—
319C2×C8—42no—
320C2×C8—44no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—816no—
340C22⋊C4—816no—
341C42—42no—
342C42—42no—
343C42—44no—
344C42—44no—
345C42—44no—
346C42#12012yes—
347C42#12012yes—
348C42#12112yes—
349C42#12112yes—
350C42#12112yes—
351C42#12712yes—
352C42C22×D411yescharacteristic
353C42C22×D411yescharacteristic
354C42C22×D411yescharacteristic
355C42C2×#4111yescharacteristic
356C23—22no—
357C23—22no—
358C23—44no—
359C23—48no—
360C23—48no—
361C23—48no—
362C23—88no—
363C23—88no—
364C23—88no—
365C23—1616no—
366C23#39212yes—
367C23#45214yes—
368C23C2×#12011yescharacteristic
369Q8—44no—
370Q8—48no—
371Q8—44no—
372Q8—88no—
373Q8—84no—
374Q8—84no—
375D4—48no—
376D4—84no—
377D4—88no—
378D4—82no—
379D4—82no—
380D4—84no—
381D4—88no—
382D4—84no—
383D4—88no—
384D4—84no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—88no—
400C2×C4—84no—
401C2×C4#39212yes—
402C2×C4#39212yes—
403C2×C4#39212yes—
404C2×C4#45214yes—
405C2×C4#45514yes—
406C2×C4#46418yes—
407C2×C4#49514yes—
408C2×C4#49714yes—
409C2×C4C2×#12011yescharacteristic
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×#12111yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12711yescharacteristic
415C2×C4C2×#13512yes—
416C2×C4C2×#13712yes—
417C2×C4#57614yes—
418C8—44no—
419C8—44no—
420C8—82no—
421C22—22no—
422C22—22no—
423C22—28no—
424C22—84no—
425C22—84no—
426C22—84no—
427C22—816no—
428C22—1616no—
429C22#201514yes—
430C22C2×#39211yescharacteristic
431C22C2×#45212yes—
432C22#300812yes—
433C22C2×#45512yes—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—82no—
438C4—84no—
439C4—82no—
440C4—84no—
441C4#200414yes—
442C4#201514yes—
443C4#300812yes—
444C4C2×#46412yes—
445C4#305812yes—
446C4C2×#49511yescharacteristic
447C4C2×#49711yescharacteristic
448C2—24no—
449C2—164no—
450C2#1863514yes—
451C2#2338711yescharacteristic
452C2C2×#200411yescharacteristic
453C2C2×#201511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×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
1. C2×#3209C2semidirect
4. #21313C2semidirect
6. C2×#1493C2semidirect
7. #10063C2semidirect
8. #7890C2semidirect
20. C2×#462C22semidirect
35. #1988C22semidirect
41. C2×Q8⋊C8C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#4862
#352852
#213132
C2×#15452
C2×#14932
#100632
#78902