Finite Group Databasebeta

← #190425

Subgroups — #190425

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#8710C212yesmaximal, complemented
5#8709C212yesmaximal, complemented
6#8708C214yesmaximal, complemented
7#7899C214yesmaximal, complemented
8C23×C42C2211yescharacteristic
9C2×#431C2212yes—
10C2×#423C2212yes—
11C2×#407—21no—
12#2285C411yescharacteristic
13#2285C411yescharacteristic
14#2285C411yescharacteristic
15#2285C411yescharacteristic
16#2285C2211yescharacteristic
17#2285C2211yescharacteristic
18#2283—21no—
19#2283—21no—
20#2283—21no—
21#2283—21no—
22#2283C412yes—
23#2283C2212yes—
24#2280—21no—
25#2280—21no—
26#2280C412yes—
27C4×#117—21no—
28C4×#117—21no—
29C4×#117C2212yes—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1779—21no—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40#1728—21no—
41#1279C2218yescomplemented
42#1279C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—22no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117C2×C412yes—
52C2×#117C2×C412yes—
53C22×C42—28no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#431—28no—
67#423—28no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2314yes—
71#407—44no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C4×C8C2314yes—
78#384C2×C411yescharacteristic
79#384C2×C411yescharacteristic
80#384C2×C411yescharacteristic
81#384C2×C411yescharacteristic
82#383—22no—
83#383—22no—
84#382—21no—
85#382—21no—
86#382—21no—
87#382—21no—
88#382C2×C412yes—
89#382C2×C412yes—
90#381—22no—
91#381—22no—
92#381—22no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380C2×C412yes—
102#380C2×C412yes—
103#380C2×C412yes—
104#380C2×C412yes—
105#379—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2312yes—
117C4×C4⋊C4C2312yes—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124#376—22no—
125#376—22no—
126#376—22no—
127C43—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132C25C22⋊C411yescharacteristic
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—44no—
162C2×M16—44no—
163C2×M16—44no—
164C22×C8—24no—
165C22×C8—24no—
166#117—28no—
167#117—44no—
168#117—44no—
169#117—44no—
170#117—44no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4118yes—
224C2×C42C2411yescharacteristic
225C4⋊C8—28no—
226C4⋊C8—44no—
227C4⋊C8—44no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232C22⋊C8—44no—
233C4×C8—28no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24C2×C22⋊C411yescharacteristic
254C24C2×C22⋊C411yescharacteristic
255C24C2×C22⋊C411yescharacteristic
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
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—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—416no—
314C22×C4—416no—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#11714yes—
325C22×C4#12314yes—
326C22×C4#12314yes—
327C22×C4#13114yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4#13712yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333M16—88no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—44no—
371C42—44no—
372C42—42no—
373C42—42no—
374C42—44no—
375C42—44no—
376C42—44no—
377C42#11714yes—
378C42#12314yes—
379C42#12314yes—
380C42C2×#4114yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23#38112yes—
407C23#40812yes—
408C23#41712yes—
409C23C22×C22⋊C411yescharacteristic
410C2×C4—28no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4#38112yes—
466C2×C4#38112yes—
467C2×C4#38112yes—
468C2×C4#40812yes—
469C2×C4#41712yes—
470C2×C4#42112yes—
471C2×C4#42112yes—
472C2×C4C2×#11712yes—
473C2×C4C2×#12312yes—
474C2×C4C2×#12312yes—
475C2×C4C2×#13112yes—
476C2×C4#56812yes—
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×#13711yescharacteristic
479C2×C4#57512yes—
480C8—84no—
481C8—84no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—416no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22C2×#38111yescharacteristic
505C22C2×#40811yescharacteristic
506C22#287911yescharacteristic
507C22C2×#41711yescharacteristic
508C22#289611yescharacteristic
509C22C2×#42111yescharacteristic
510C22C2×#42111yescharacteristic
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4#292414yes—
526C4#309412yes—
527C4#309512yes—
528C2—42no—
529C2—42no—
530C2—42no—
531C2#1747711yescharacteristic
532C2#1748111yescharacteristic
533C2#1749311yescharacteristic
534C2#1749811yescharacteristic
535C2#2068511yescharacteristic
536C2#2070211yescharacteristic
537C2#2070511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8710C2semidirect
5. #8709C2semidirect
6. #8708C2semidirect
7. #7899C2semidirect
41. #1279C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#340122
#207142
#206262
#87102
#87092
#87082
#78992