Finite Group Databasebeta

← #189948

Subgroups — #189948

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#8729C212yesmaximal, complemented
5#8728C212yesmaximal, complemented
6#8727C212yesmaximal, complemented
7#8726C212yesmaximal, complemented
8#8045C212yesmaximal, complemented
9#8045C212yesmaximal, complemented
10C23×C42C2211yescharacteristic
11C2×#423C2212yes—
12C22×C4⋊C8C2212yes—
13C2×#407—21no—
14C22×C22⋊C8—21no—
15#2285C2211yescharacteristic
16#2285C2211yescharacteristic
17#2285C2211yescharacteristic
18#2285C2211yescharacteristic
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283—21no—
22#2283—21no—
23#2283—21no—
24#2283—21no—
25#2283C412yes—
26#2283C412yes—
27#2280—21no—
28#2280—21no—
29#2280C412yes—
30C4×#117—21no—
31C4×#117—21no—
32C4×#117C412yes—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36#1797—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41#1728—21no—
42C2×#336—21no—
43#1296C2214yescomplemented
44#1296C2214yescomplemented
45#1296C2214yescomplemented
46#1296C2214yescomplemented
47C24×C4C2×C411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C22×M16—22no—
51C23×C8—22no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—21no—
56C2×#117C2×C412yes—
57C2×#117C2×C412yes—
58C22×C42—28no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#423—28no—
72C2×C4⋊C8—28no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2314yes—
76C2×C4⋊C8C2314yes—
77#407—44no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—44no—
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#383—22no—
88#383—22no—
89#382—21no—
90#382—21no—
91#382—21no—
92#382—21no—
93#382C2×C412yes—
94#382C2×C412yes—
95#381—22no—
96#381—22no—
97#381—22no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2×C412yes—
107#380C2×C412yes—
108#380C2×C412yes—
109#380C2×C412yes—
110#379—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#376—22no—
130#376—22no—
131#376—22no—
132C43—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137C25C22⋊C411yescharacteristic
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×M16—44no—
168C2×M16—44no—
169C2×M16—44no—
170C22×C8—24no—
171C22×C8—24no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8—44no—
175#117—44no—
176#117—44no—
177#117—42no—
178#117—42no—
179#117—42no—
180#117—42no—
181#117—44no—
182#117—44no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4118yes—
234C2×C42C2411yescharacteristic
235C4⋊C8—28no—
236C4⋊C8—28no—
237C4⋊C8—44no—
238C4⋊C8—44no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—41no—
261C24—41no—
262C24C2×C22⋊C411yescharacteristic
263C24C2×C22⋊C411yescharacteristic
264C24C2×C22⋊C411yescharacteristic
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—416no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4#11714yes—
336C22×C4#12314yes—
337C22×C4#12314yes—
338C22×C4#13514yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4#13712yes—
342C22×C4#13712yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346M16—88no—
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—88no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—44no—
389C42—44no—
390C42—44no—
391C42—44no—
392C42—44no—
393C42—44no—
394C42#11714yes—
395C42#12314yes—
396C42#12314yes—
397C42C2×#4114yes—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—28no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23#38112yes—
422C23#41312yes—
423C23#41612yes—
424C23#41712yes—
425C23#41712yes—
426C23C22×C22⋊C411yescharacteristic
427C2×C4—28no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4#38112yes—
484C2×C4#38112yes—
485C2×C4#38112yes—
486C2×C4#41312yes—
487C2×C4#41612yes—
488C2×C4#41712yes—
489C2×C4#41712yes—
490C2×C4#48218yes—
491C2×C4#48518yes—
492C2×C4C2×#11712yes—
493C2×C4C2×#12312yes—
494C2×C4C2×#12312yes—
495C2×C4C2×#13512yes—
496C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
497C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
498C2×C4C2×#13711yescharacteristic
499C2×C4C2×#13711yescharacteristic
500C2×C4#57512yes—
501C8—84no—
502C8—84no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—416no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22#182514yes—
525C22C2×#38111yescharacteristic
526C22C2×#41311yescharacteristic
527C22C2×#41311yescharacteristic
528C22C2×#41611yescharacteristic
529C22C2×#41711yescharacteristic
530C22C2×#41711yescharacteristic
531C22#289611yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4#182514yes—
546C4C2×#48212yes—
547C4C2×#48512yes—
548C4#309412yes—
549C4#309512yes—
550C2—24no—
551C2—42no—
552C2—42no—
553C2#1771611yescharacteristic
554C2#1771911yescharacteristic
555C2#1775211yescharacteristic
556C2#1775311yescharacteristic
557C2#2069411yescharacteristic
558C2C2×#182511yescharacteristic
559C2#2070211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8729C2semidirect
5. #8728C2semidirect
6. #8727C2semidirect
7. #8726C2semidirect
8. #8045C2semidirect
9. #8045C2semidirect
43. #1296C22semidirect
44. #1296C22semidirect
45. #1296C22semidirect
46. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#340122
#206262
C2×#17972
#87292
#87282
#87272
#87262
#80452