Finite Group Databasebeta

← #192245

Subgroups — #192245

Order 512.

925 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#20627C211yesmaximal, characteristic
3#17820C212yesmaximal, complemented
4#17357C211yesmaximal, characteristic
5#8863C214yesmaximal, complemented
6#8862C212yesmaximal, complemented
7#8861C212yesmaximal, complemented
8#8078C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10#2924—24no—
11#2924—24no—
12#2921—21no—
13#2921C2212yes—
14#2920—21no—
15#2920C2212yes—
16#2920C2212yes—
17#2905C2211yescharacteristic
18#2905C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#407—21no—
22C2×#407—21no—
23#2873—21no—
24#1798—21no—
25#1798—21no—
26#1798C2212yes—
27#1798C2212yes—
28#1728—21no—
29#1574—24no—
30#1571C2214yescomplemented
31#1570C2214yescomplemented
32#1569—24no—
33#1464C412yes—
34#1464C412yes—
35#1463C412yes—
36#1462C412yes—
37#1458C2212yes—
38#1458C2212yes—
39#1301C2214yescomplemented
40#1301C2214yescomplemented
41C24×C4D411yescharacteristic
42C22×M16—21no—
43C22×M16—21no—
44C22×M16—21no—
45C22×M16—21no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D411yescharacteristic
53C2×#117D411yescharacteristic
54C2×#117Q811yescharacteristic
55C2×#117Q811yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4Q811yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4Q811yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64#433—24no—
65#433—24no—
66#433—24no—
67#433—24no—
68#432—22no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432C2×C414yes—
73#432D414yes—
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431D414yes—
79#431C2314yes—
80#424—24no—
81#424—24no—
82#423—21no—
83#423—21no—
84#423—22no—
85#423—22no—
86#423—22no—
87#423—22no—
88#423—22no—
89#423C2312yes—
90#423C2312yes—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95#407—22no—
96#407—22no—
97#407—22no—
98#407—22no—
99#407—44no—
100#407C2×C414yes—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105#405—24no—
106#405—24no—
107C4×M16—22no—
108C4×M16—22no—
109C4×M16—22no—
110#336—22no—
111#336—22no—
112#335—28no—
113#334—28no—
114#324C2×C414yes—
115#324C2×C414yes—
116#323C2×C414yes—
117#323C2×C414yes—
118C25C4⋊C411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4C4⋊C411yescharacteristic
128C23×C4C4⋊C411yescharacteristic
129C23×C4C4⋊C411yescharacteristic
130C23×C4C4⋊C411yescharacteristic
131C23×C4C4⋊C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×Q811yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—42no—
144C2×M16—42no—
145C2×M16—42no—
146C2×M16—42no—
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—41no—
151C2×M16—41no—
152C2×M16—41no—
153C2×M16—41no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C4⋊C411yescharacteristic
179C2×C4⋊C4C4⋊C411yescharacteristic
180C2×C4⋊C4C4⋊C411yescharacteristic
181C2×C4⋊C4C4⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C2×Q811yescharacteristic
185C2×C4⋊C4C2×Q811yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C4⋊C411yescharacteristic
191C2×C22⋊C4C4⋊C411yescharacteristic
192C2×C22⋊C4C4⋊C411yescharacteristic
193C2×C22⋊C4C4⋊C411yescharacteristic
194C2×C22⋊C4C22×C411yescharacteristic
195C2×C22⋊C4C22×C411yescharacteristic
196C2×C22⋊C4C2×Q811yescharacteristic
197C2×C22⋊C4C2×Q811yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C4⋊C411yescharacteristic
203C2×C42C4⋊C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q811yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—24no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8C2×D414yes—
227C4⋊C8C2×D414yes—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8C22×C418yes—
236#97—24no—
237#97—24no—
238#97—24no—
239C4×C8—24no—
240C4×C8—24no—
241C4×C8—24no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—41no—
246C24—81no—
247C24—81no—
248C24C2×C4⋊C411yescharacteristic
249C24C2×C4⋊C411yescharacteristic
250C24C2×C4⋊C411yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4C2×C4⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301C22×C4C2×C4⋊C411yescharacteristic
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×Q811yescharacteristic
315M16—44no—
316M16—44no—
317M16—42no—
318M16—42no—
319M16—42no—
320M16—42no—
321M16—84no—
322M16—82no—
323M16—82no—
324M16—82no—
325M16—82no—
326C2×C8—28no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42C4×D412yes—
362C42C4×D412yes—
363C42#12012yes—
364C42C22⋊Q812yes—
365C42C22⋊Q812yes—
366C42#12312yes—
367C42C22×D412yes—
368C42C2×#4112yes—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—81no—
387C23—81no—
388C23—84no—
389C23—84no—
390C23#38511yescharacteristic
391C23#38511yescharacteristic
392C23#42812yes—
393C23C22×C4⋊C411yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4#38511yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4#38512yes—
446C2×C4#38512yes—
447C2×C4#42812yes—
448C2×C4C2×C4×D412yes—
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×C22⋊Q812yes—
451C2×C4C2×#12311yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4#57511yescharacteristic
455C2×C4#57511yescharacteristic
456C8—48no—
457C8—44no—
458C8—44no—
459C8—82no—
460C8—82no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—81no—
480C22—81no—
481C22—84no—
482C22—84no—
483C22C2×#38511yescharacteristic
484C22C2×#42811yescharacteristic
485C22#291111yescharacteristic
486C4—21no—
487C4—21no—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4#300611yescharacteristic
499C4#300711yescharacteristic
500C4#309411yescharacteristic
501C4#309511yescharacteristic
502C2—21no—
503C2—21no—
504C2—41no—
505C2—41no—
506C2—42no—
507C2—81no—
508C2#2084611yescharacteristic
509C2#2086011yescharacteristic
510C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17820C2semidirect
5. #8863C2semidirect
6. #8862C2semidirect
7. #8861C2semidirect
8. #8078C2semidirect
30. #1571C22semidirect
31. #1570C22semidirect
39. #1301C22semidirect
40. #1301C22semidirect

Maximal subgroups

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

subgroupindex
#601242
#206272
#178202
#173572
#88632
#88622
#88612
#80782