Finite Group Databasebeta

← #196467

Subgroups — #196467

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic, complemented
2#35262C211yesmaximal, characteristic
3C2×#1510C211yesmaximal, characteristic
4C22×#327C211yesmaximal, characteristic
5#13958C211yesmaximal, characteristic, complemented
6#9118C214yesmaximal, complemented
7#9114C214yesmaximal, complemented
8#8024C211yesmaximal, characteristic, complemented
9#8020C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C22×#117C2211yescharacteristic
12#2924—24no—
13#2924—24no—
14C2×#433—22no—
15C2×#432C2212yescomplemented
16C2×#431C2212yescomplemented
17C2×#424—21no—
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#405—21no—
21#2854C2214yes—
22#2624C412yescomplemented
23#2624C412yescomplemented
24#2504—21no—
25#2501—21no—
26#2500—22no—
27#2482—21no—
28#2482—21no—
29#2405—22no—
30#2317C412yescomplemented
31#2309C412yescomplemented
32#2280C2211yescharacteristic
33C2×#381C2211yescharacteristic
34C22×#95C2211yescharacteristic
35#1510—24no—
36C2×#327—24no—
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2214yes—
40#1445—21no—
41#1445—21no—
42#1434C2214yescomplemented
43C22⋊#95—22no—
44#1294C2214yescomplemented
45#1294C2212yescomplemented
46#1294C2212yescomplemented
47C23×D4—21no—
48C23×D4C2×C411yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—21no—
52C23×C8—21no—
53C2×#123—24no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C412yes—
61C2×#121C2312yes—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117C2×C412yes—
67C2×#117C2×C412yes—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117D412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42C2312yes—
80#433—216no—
81#432—24no—
82#432—24no—
83#431—24no—
84#431—24no—
85#424—28no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88#407—22no—
89#407D414yes—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D414yes—
94C2×C22⋊C8C2311yescharacteristic
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2312yes—
97#405—28no—
98C2×#97—22no—
99C2×C4×C8—22no—
100#399—22no—
101#399—22no—
102#397—22no—
103#397—22no—
104#394—24no—
105#394—24no—
106#390—22no—
107#390—22no—
108#390C2×C414yescomplemented
109#390C2×C414yescomplemented
110#389C2×C414yescomplemented
111#389C2×C414yescomplemented
112#388—24no—
113#388—24no—
114#386—24no—
115#385—22no—
116#385—22no—
117#384—22no—
118#382—22no—
119#381—21no—
120#381—22no—
121#381—22no—
122#380C2×C412yes—
123#380C2312yes—
124#379—21no—
125#379—22no—
126#379—22no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—24no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—24no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137#327—24no—
138#327—24no—
139#327—28no—
140C25—21no—
141C25C2×D411yescharacteristic
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—82no—
151C22×D4—82no—
152C22×D4C22×C412yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—82no—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—24no—
174C2×M16—44no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—44no—
180#123—84no—
181#123—84no—
182#121—24no—
183#121—24no—
184#121—22no—
185#121—22no—
186#121—22no—
187#121—22no—
188#121—42no—
189#121—42no—
190#121—42no—
191#121—42no—
192#117—24no—
193#117—24no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—24no—
201#117—22no—
202#117—22no—
203#117—24no—
204#117—44no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C414yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C4⋊C8—28no—
251C4⋊C8—28no—
252C22⋊C8—24no—
253C22⋊C8—24no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—22no—
257C22⋊C8C2×D412yes—
258C22⋊C8C2×D412yes—
259C22⋊C8C2×D414yes—
260#97—28no—
261C4×C8—28no—
262#95—22no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—48no—
271#95—42no—
272#95—42no—
273#95—84no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—22no—
278C24—42no—
279C24—42no—
280C24—44no—
281C24—82no—
282C24—82no—
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285C24#12112yes—
286C24#12112yes—
287C24C22×D411yescharacteristic
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—42no—
298C2×D4—84no—
299C2×D4—84no—
300C2×D4—84no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4—88no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—48no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—82no—
336C22×C4—816no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12312yes—
354C22×C4#12312yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361M16—48no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—22no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—48no—
368C2×C8—44no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—82no—
396C22⋊C4—84no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—44no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—42no—
419C23—42no—
420C23—48no—
421C23—48no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—84no—
428C23—84no—
429C23—84no—
430C23—168no—
431C23#38612yes—
432C23#38612yes—
433C23#41014yes—
434C23C22×C22⋊C411yescharacteristic
435C23C2×#12111yescharacteristic
436C23C2×#12111yescharacteristic
437C23#57612yes—
438C23#57712yes—
439D4—82no—
440D4—82no—
441D4—84no—
442D4—88no—
443D4—82no—
444D4—82no—
445D4—84no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—816no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4#38612yes—
495C2×C4#38612yes—
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×#12011yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C8—42no—
501C8—42no—
502C8—44no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—24no—
507C22—24no—
508C22—28no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—44no—
516C22—48no—
517C22—44no—
518C22—44no—
519C22—82no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—168no—
525C22#189914yes—
526C22C2×#38611yescharacteristic
527C22C2×#41012yes—
528C22#289712yes—
529C22#290112yes—
530C22#305711yescharacteristic
531C22#305811yescharacteristic
532C22#306212yes—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—22no—
538C4—22no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—84no—
544C4—84no—
545C4—82no—
546C4—162no—
547C2—21no—
548C2—22no—
549C2—24no—
550C2—21no—
551C2—42no—
552C2—162no—
553C2#1760414yes—
554C2#2090311yescharacteristic
555C2#2097511yescharacteristic
556C2C2×#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
1. C2×#2924C2semidirect
5. #13958C2semidirect
6. #9118C2semidirect
7. #9114C2semidirect
8. #8024C2semidirect
9. #8020C2semidirect
15. C2×#432C22semidirect
16. C2×#431C22semidirect
22. #2624C4semidirect
23. #2624C4semidirect
30. #2317C4semidirect
31. #2309C4semidirect
42. #1434C22semidirect
44. #1294C22semidirect
45. #1294C22semidirect
46. #1294C22semidirect
108. #390C2×C4semidirect
109. #390C2×C4semidirect
110. #389C2×C4semidirect
111. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
#352622
C2×#15102
C22×#3272
#139582
#91182
#91142
#80242
#80202