Finite Group Databasebeta

← #196977

Subgroups — #196977

Order 512.

1462 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35262C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic, complemented
3#20685C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5C2×#1507C211yesmaximal, characteristic
6#9156C214yesmaximal, complemented
7#9149C214yesmaximal, complemented
8#8016C211yesmaximal, characteristic, complemented
9#8012C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2854C2214yes—
15#2624C414yescomplemented
16#2504—21no—
17#2501—21no—
18#2500—22no—
19#2482—22no—
20#2405—22no—
21#2346—24no—
22#2346—24no—
23#2317C412yescomplemented
24#2309C412yescomplemented
25C2×#389—21no—
26C2×#388—22no—
27C2×#382C2212yescomplemented
28#2280C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×#376C2212yescomplemented
33#1818—22no—
34#1817C2214yescomplemented
35#1729—22no—
36#1519—24no—
37#1507—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1294C2212yescomplemented
42#1294C2212yescomplemented
43#1294C2214yescomplemented
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—21no—
48C23×C8—21no—
49C2×#123—24no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121C2×C412yes—
57C2×#121C2312yes—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117C2×C412yes—
63C2×#117C2×C412yes—
64C2×#117C2×C412yes—
65C2×#117D412yes—
66C2×#117D412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42—22no—
77C22×C42—21no—
78C22×C42C2312yes—
79#407—22no—
80#407—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87#399—24no—
88#397—24no—
89#394—28no—
90#390—22no—
91#390—22no—
92#390C2×C418yes—
93#389—28no—
94#389C2×C418yes—
95#388—216no—
96#388—28no—
97#386—24no—
98#385—24no—
99#384—22no—
100#382—22no—
101#382—24no—
102#382—24no—
103#381—21no—
104#380—22no—
105#380C2×C412yes—
106#380D414yes—
107#380C2312yes—
108#379—21no—
109#379—24no—
110C4×C4⋊C4—28no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4D414yes—
115#376—24no—
116#376—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2311yescharacteristic
120C2×#95C2312yes—
121C2×#95C2311yescharacteristic
122#336—28no—
123#327—24no—
124#327—24no—
125#327—28no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—44no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—82no—
137C22×D4—82no—
138C22×D4C22×C412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C2×M16—44no—
157C2×M16—42no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—44no—
161C22×C8—42no—
162#123—84no—
163#123—84no—
164#121—24no—
165#121—24no—
166#121—22no—
167#121—22no—
168#121—22no—
169#121—22no—
170#121—42no—
171#121—42no—
172#121—42no—
173#121—42no—
174#117—24no—
175#117—24no—
176#117—24no—
177#117—24no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C414yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4C22⋊C414yes—
217C2×C22⋊C4C22⋊C414yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—28no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—44no—
226C2×C42—48no—
227C2×C42C22×C414yes—
228C2×C42C2×D414yes—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—42no—
234C22⋊C8—42no—
235#95—28no—
236#95—28no—
237#95—22no—
238#95—28no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95C2×D412yes—
244#95C2×D414yes—
245#95C2×D412yes—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—42no—
250C24—42no—
251C24—44no—
252C24—42no—
253C24—82no—
254C24—82no—
255C24C2×C22⋊C411yescharacteristic
256C24C2×C22⋊C411yescharacteristic
257C24#12012yes—
258C24#12312yes—
259C24C22×D411yescharacteristic
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—84no—
271C2×D4—84no—
272C2×D4—84no—
273C2×D4—88no—
274C2×D4—84no—
275C2×D4—84no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—82no—
309C22×C4C2×C22⋊C414yes—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12012yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12312yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329M16—84no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—84no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—48no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—88no—
356C22⋊C4—88no—
357C22⋊C4—82no—
358C22⋊C4—84no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—48no—
364C42—416no—
365C42—44no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—84no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23—84no—
390C23—168no—
391C23#38612yes—
392C23#38612yes—
393C23C22×C22⋊C411yescharacteristic
394C23C2×#12011yescharacteristic
395C23C2×#12311yescharacteristic
396C23#57612yes—
397C23#57712yes—
398D4—82no—
399D4—82no—
400D4—88no—
401D4—84no—
402D4—82no—
403D4—82no—
404D4—84no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—84no—
447C2×C4—88no—
448C2×C4—84no—
449C2×C4#38612yes—
450C2×C4#38612yes—
451C2×C4#41014yes—
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—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—44no—
471C22—44no—
472C22—82no—
473C22—82no—
474C22—84no—
475C22—82no—
476C22—88no—
477C22—82no—
478C22—82no—
479C22—168no—
480C22C2×#38611yescharacteristic
481C22C2×#41012yes—
482C22#289712yes—
483C22#290112yes—
484C22#300811yescharacteristic
485C22#301011yescharacteristic
486C22#309612yes—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—22no—
492C4—44no—
493C4—44no—
494C4—44no—
495C4—82no—
496C4—84no—
497C4—82no—
498C4—84no—
499C4#189914yes—
500C2—21no—
501C2—21no—
502C2—22no—
503C2—42no—
504C2—42no—
505C2—162no—
506C2#1761014yes—
507C2#2091411yescharacteristic
508C2#2098011yescharacteristic
509C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2346C2semidirect
3. #20685C2semidirect
6. #9156C2semidirect
7. #9149C2semidirect
8. #8016C2semidirect
9. #8012C2semidirect
15. #2624C4semidirect
23. #2317C4semidirect
24. #2309C4semidirect
27. C2×#382C22semidirect
32. C2×#376C22semidirect
34. #1817C22semidirect
41. #1294C22semidirect
42. #1294C22semidirect
43. #1294C22semidirect

Maximal subgroups

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

subgroupindex
#352622
C2×#23462
#206852
C2×#15192
C2×#15072
#91562
#91492
#80162
#80122