Finite Group Databasebeta

← #196928

Subgroups — #196928

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35237C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic, complemented
3#21788C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6#9150C214yesmaximal, complemented
7#9147C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2847C2214yes—
16#2631—22no—
17#2563C412yescomplemented
18#2541C412yescomplemented
19#2535C414yescomplemented
20#2504—21no—
21#2433—21no—
22#2404—22no—
23#2328—22no—
24#2283C2211yescharacteristic
25#2275—24no—
26#2275—24no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—22no—
30C2×C4×C22⋊C4C2212yes—
31C2×#376C2211yescharacteristic
32C22×#95C2212yes—
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×C4—22no—
47C24×C4C2312yes—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8C2×C411yescharacteristic, complemented
53C2×#127—22no—
54C2×#127C2×C411yescharacteristic, complemented
55C2×#124—21no—
56C2×#121—21no—
57C2×#121C2×C418yescomplemented
58C2×#121C2312yes—
59C2×#120—24no—
60C2×C4×D4—216no—
61C2×C4×D4—28no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C412yescomplemented
70C2×C4×D4D412yes—
71C2×C4×D4D412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2311yescharacteristic
80#432—22no—
81#432—22no—
82#431—22no—
83#431—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C22⋊C8—28no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91#397—24no—
92#393—24no—
93#392—24no—
94#392—24no—
95#390—22no—
96#390—22no—
97#386—28no—
98#386—24no—
99#386—28no—
100#383—21no—
101#382—22no—
102#382C2×C412yescomplemented
103#381—21no—
104#380C2×C418yescomplemented
105#380C2312yes—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C412yescomplemented
109C4×C4⋊C4D414yescomplemented
110C4×C22⋊C4—216no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—24no—
114#376—21no—
115#376—22no—
116#376D414yescomplemented
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2311yescharacteristic
124#327—28no—
125#327—24no—
126#327—24no—
127C25C2×D412yes—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—82no—
131C2×#41—82no—
132C22×Q8—21no—
133C22×D4—28no—
134C22×D4—22no—
135C22×D4—28no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C22⋊C412yes—
141C22×D4C22⋊C412yes—
142C22×D4C22×C412yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—216no—
147C23×C4—28no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—84no—
154C23×C4—84no—
155C23×C4—84no—
156C23×C4C22×C414yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166#127—22no—
167#127—22no—
168#127—48no—
169#124—42no—
170#124—42no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—42no—
176#121—42no—
177#121—88no—
178#120—84no—
179#120—84no—
180C4×D4—264no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—42no—
192C4×D4—42no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C414yes—
202C2×C22⋊C4—216no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4C22⋊C414yes—
212C2×C22⋊C4C22⋊C414yes—
213C2×C42—28no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—28no—
217C2×C42—24no—
218C2×C42—82no—
219C2×C42—84no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—88no—
235#97—42no—
236C4×C8—42no—
237#95—28no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—28no—
242#95—24no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95C2×D412yes—
248#95C2×D414yes—
249#95C2×D412yes—
250C24—22no—
251C24—22no—
252C24—28no—
253C24—24no—
254C24—24no—
255C24—22no—
256C24—44no—
257C24C2×C22⋊C414yes—
258C24C22×D411yescharacteristic
259#41—88no—
260#41—84no—
261#41—84no—
262C2×Q8—42no—
263C2×Q8—82no—
264C2×D4—216no—
265C2×D4—216no—
266C2×D4—28no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—416no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—82no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4#10418yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—232no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—88no—
320C22×C4—832no—
321C22×C4—88no—
322C22×C4C2×C22⋊C414yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4#12012yes—
328C22×C4#12012yes—
329C22×C4#12312yes—
330C22×C4#12312yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—216no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C22⋊C4—232no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—44no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C42—216no—
359C42—44no—
360C42—84no—
361C42—88no—
362C42C4×D412yes—
363C42C4×D412yes—
364C42C4×D412yes—
365C42C4×D412yes—
366C42#12112yes—
367C42#12112yes—
368C42#12112yes—
369C42#12112yes—
370C23—22no—
371C23—22no—
372C23—28no—
373C23—28no—
374C23—28no—
375C23—28no—
376C23—24no—
377C23—24no—
378C23—416no—
379C23—416no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23C2×#10418yes—
390C23C22×C22⋊C411yescharacteristic
391C23C2×#12011yescharacteristic
392C23C2×#12311yescharacteristic
393C23#57612yes—
394C23#57712yes—
395Q8—84no—
396D4—216no—
397D4—216no—
398D4—82no—
399D4—82no—
400D4—84no—
401D4—84no—
402D4—82no—
403D4—82no—
404D4—84no—
405D4—1616no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—416no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—88no—
446C2×C4—832no—
447C2×C4#38612yes—
448C2×C4#38612yes—
449C2×C4#38612yes—
450C2×C4#38612yes—
451C2×C4C2×#10414yes—
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4C2×#12111yescharacteristic
456C8—82no—
457C8—82no—
458C22—24no—
459C22—24no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—216no—
465C22—28no—
466C22—48no—
467C22—48no—
468C22—416no—
469C22—48no—
470C22—44no—
471C22—44no—
472C22—88no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22C2×#38611yescharacteristic
477C22#288312yes—
478C22#289712yes—
479C22C22×#10412yes—
480C22#300811yescharacteristic
481C22#301011yescharacteristic
482C22#309612yes—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—44no—
491C4—48no—
492C4—44no—
493C4—44no—
494C4—84no—
495C4—82no—
496C4#189614yes—
497C2—21no—
498C2—21no—
499C2—22no—
500C2—48no—
501C2—162no—
502C2#1754714yes—
503C2#2091411yescharacteristic
504C2C2×#189611yescharacteristic
505C2#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×#2275C2semidirect
3. #21788C2semidirect
4. C2×#1505C2semidirect
5. C2×#1502C2semidirect
6. #9150C2semidirect
7. #9147C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
17. #2563C4semidirect
18. #2541C4semidirect
19. #2535C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
54. C2×#127C2×C4semidirect
57. C2×#121C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
102. #382C2×C4semidirect
104. #380C2×C4semidirect
108. C4×C4⋊C4C2×C4semidirect
109. C4×C4⋊C4D4semidirect
116. #376D4semidirect

Maximal subgroups

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

subgroupindex
#352372
C2×#22752
#217882
C2×#15052
C2×#15022
#91502
#91472
#80212
#80132