Finite Group Databasebeta

← #196937

Subgroups — #196937

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35237C211yesmaximal, characteristic
2C2×#2319C211yesmaximal, characteristic, complemented
3#21788C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6#9158C214yesmaximal, complemented
7#9155C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2847C2214yes—
15#2631—22no—
16#2563C412yescomplemented
17#2541C412yescomplemented
18#2535C414yescomplemented
19#2504—21no—
20#2433—21no—
21#2404—22no—
22#2328—22no—
23#2319—24no—
24#2319—24no—
25C2×#390—21no—
26C2×#389—21no—
27C2×#388—22no—
28#2283C2211yescharacteristic
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1955—22no—
34#1954—22no—
35#1779C2214yes—
36#1505—24no—
37#1502—24no—
38C2×#327C2214yes—
39C2×#327C2212yes—
40C2×#327C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—21no—
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47#532—22no—
48#532—22no—
49#532—22no—
50#532—22no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#127—22no—
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#121—21no—
56C2×#121C2×C418yes—
57C2×#121C2312yes—
58C2×#120—24no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C412yes—
67C2×C4×D4D412yes—
68C2×C4×D4D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432—22no—
80#431—22no—
81#431—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C22⋊C8—28no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89#397—24no—
90#393—24no—
91#392—24no—
92#392—24no—
93#390—28no—
94#390—22no—
95#390—22no—
96#389—28no—
97#388—216no—
98#386—28no—
99#386—24no—
100#386—28no—
101#383—21no—
102#382—22no—
103#382C2×C412yes—
104#381—21no—
105#381—24no—
106#381—24no—
107#380—24no—
108#380—24no—
109#380C2×C418yes—
110#380C2312yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D414yescomplemented
114C4×C22⋊C4—21no—
115#376—22no—
116#376—21no—
117#376D414yescomplemented
118C2×#95—21no—
119C2×#95—22no—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123#327—28no—
124#327—24no—
125#327—24no—
126C25C2×D412yes—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—82no—
130C2×#41—82no—
131C22×Q8—21no—
132C22×D4—22no—
133C22×D4—28no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—44no—
139C22×D4C22⋊C412yes—
140C22×D4C22⋊C412yes—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—84no—
151C23×C4C22×C414yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D414yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161#127—22no—
162#127—22no—
163#127—48no—
164#124—42no—
165#124—42no—
166#121—22no—
167#121—22no—
168#121—22no—
169#121—22no—
170#121—42no—
171#121—42no—
172#121—88no—
173#120—84no—
174#120—84no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—42no—
186C4×D4—42no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—84no—
208C2×C22⋊C4—84no—
209C2×C22⋊C4C22⋊C414yes—
210C2×C22⋊C4C22⋊C414yes—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—44no—
215C2×C42—82no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—88no—
231#97—42no—
232C4×C8—42no—
233#95—28no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—28no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95C2×D412yes—
244#95C2×D412yes—
245#95C2×D414yes—
246C24—22no—
247C24—22no—
248C24—24no—
249C24—24no—
250C24—22no—
251C24—44no—
252C24—44no—
253C24C2×C22⋊C414yes—
254C24C22×D411yescharacteristic
255#41—88no—
256#41—84no—
257#41—84no—
258C2×Q8—42no—
259C2×Q8—82no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—44no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—416no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—82no—
279C2×D4—88no—
280C2×D4—88no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#12012yes—
320C22×C4#12012yes—
321C22×C4#12312yes—
322C22×C4#12312yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—88no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—88no—
349C22⋊C4—88no—
350C22⋊C4—88no—
351C22⋊C4—88no—
352C42—48no—
353C42—44no—
354C42—84no—
355C42C4×D412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12112yes—
360C42#12112yes—
361C42#12112yes—
362C42#12112yes—
363C23—22no—
364C23—22no—
365C23—28no—
366C23—28no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—88no—
382C23—88no—
383C23C22×C22⋊C411yescharacteristic
384C23C2×#12011yescharacteristic
385C23C2×#12311yescharacteristic
386C23#57612yes—
387C23#57712yes—
388Q8—84no—
389D4—48no—
390D4—48no—
391D4—82no—
392D4—82no—
393D4—84no—
394D4—84no—
395D4—82no—
396D4—82no—
397D4—84no—
398D4—1616no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—84no—
437C2×C4—88no—
438C2×C4—84no—
439C2×C4#38612yes—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4#38612yes—
443C2×C4#42114yes—
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12111yescharacteristic
448C8—82no—
449C8—82no—
450C22—24no—
451C22—24no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—44no—
457C22—48no—
458C22—48no—
459C22—44no—
460C22—44no—
461C22—84no—
462C22—84no—
463C22—88no—
464C22—82no—
465C22—88no—
466C22—82no—
467C22—82no—
468C22C2×#38611yescharacteristic
469C22#288312yes—
470C22#289712yes—
471C22C2×#42112yes—
472C22#300811yescharacteristic
473C22#301011yescharacteristic
474C22#309612yes—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—44no—
483C4—82no—
484C4—84no—
485C4—82no—
486C4—82no—
487C4—84no—
488C4#189714yes—
489C2—21no—
490C2—21no—
491C2—22no—
492C2—84no—
493C2—162no—
494C2#1759914yes—
495C2#2091411yescharacteristic
496C2C2×#189711yescharacteristic
497C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35237C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2319C2semidirect
3. #21788C2semidirect
4. C2×#1505C2semidirect
5. C2×#1502C2semidirect
6. #9158C2semidirect
7. #9155C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
16. #2563C4semidirect
17. #2541C4semidirect
18. #2535C4semidirect
113. C4×C4⋊C4D4semidirect
117. #376D4semidirect

Maximal subgroups

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

subgroupindex
#352372
C2×#23192
#217882
C2×#15052
C2×#15022
#91582
#91552
#80212
#80132