Finite Group Databasebeta

← #195572

Subgroups — #195572

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#21385C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9065C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8—21no—
13#3230—24no—
14#3230C418yescomplemented
15C2×#500C2212yes—
16C2×#487C2212yes—
17C2×#484—22no—
18C2×#477—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2365—22no—
23#2365—22no—
24#2365—22no—
25#2365C2214yes—
26C2×#391—21no—
27C2×#389—21no—
28C2×#387—22no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1989—22no—
33#1988C2212yes—
34#1987C2212yes—
35#1987C2212yes—
36#1981C2212yes—
37#1872—21no—
38#1872—21no—
39C4×#103—22no—
40#1736—21no—
41#1736—21no—
42#1549—24no—
43C2×#332C2214yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×M16—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—48no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#500—24no—
83#500C2×C418yescomplemented
84#487—24no—
85#487C2×C418yescomplemented
86#484—216no—
87#477—216no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431C2×C414yes—
92C2×#107—24no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99C2×#103—24no—
100C2×#103—22no—
101C2×#103—22no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2312yes—
106#391—24no—
107#391—24no—
108#389—24no—
109#389—24no—
110#387—28no—
111#387—28no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2312yes—
121C4×C4⋊C4C2312yes—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—21no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#332—28no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—44no—
140C22×Q8—44no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C22⋊C418yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C414yes—
172C4⋊Q8C2×D414yes—
173#125—24no—
174#125—42no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—416no—
182C4×Q8—48no—
183C4×Q8—48no—
184C4×Q8—42no—
185C4×Q8—42no—
186#117—24no—
187#117—42no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C22⋊C418yes—
212C2×C4⋊C4Q1614yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—41no—
219C2×C42—28no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—48no—
225C2×C42—44no—
226C2×C42—41no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C22×C412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42C2411yescharacteristic
237#107—216no—
238C4⋊C8—24no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—42no—
244#103—216no—
245#103—44no—
246#103—44no—
247C22⋊C8—24no—
248C22⋊C8—42no—
249C4×C8—22no—
250C4×C8—42no—
251C4×C8C22×C414yes—
252#95—28no—
253#95—28no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258C24#12011yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—28no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—416no—
267C2×Q8—416no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C2×C22⋊C412yes—
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4#12312yes—
311C22×C4C2×Q1612yes—
312C22×C4#13512yes—
313C22×C4#13712yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318M16—84no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—416no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4C2×C22⋊C418yes—
355C4⋊C4C2×Q1614yes—
356C4⋊C4AGL(1,ℤ/8)14yes—
357C22⋊C4—44no—
358C42—44no—
359C42—416no—
360C42—42no—
361C42—44no—
362C42—42no—
363C42—44no—
364C42—44no—
365C42C2×C22⋊C412yes—
366C42C4×D412yes—
367C42C4×D412yes—
368C42#12112yes—
369C42#12112yes—
370C42#12312yes—
371C42C23×C411yescharacteristic
372C42C22×D411yescharacteristic
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23#38612yes—
380C23#45112yes—
381C23#45212yes—
382C23C2×#12011yescharacteristic
383Q8—416no—
384Q8—44no—
385Q8—48no—
386Q8—48no—
387Q8—44no—
388Q8—82no—
389Q8—82no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—416no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4#38612yes—
427C2×C4#38612yes—
428C2×C4#38612yes—
429C2×C4C4×Q1614yes—
430C2×C4#44114yes—
431C2×C4#45112yes—
432C2×C4#45212yes—
433C2×C4#46214yes—
434C2×C4#46414yes—
435C2×C4C22×C22⋊C411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C22×Q1611yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)12yes—
444C2×C4C2×#13711yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4#57512yes—
447C8—44no—
448C8—82no—
449C8—82no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22#187814yes—
457C22C2×#38611yescharacteristic
458C22C2×#45111yescharacteristic
459C22C2×#45211yescharacteristic
460C22#300311yescharacteristic
461C22#300511yescharacteristic
462C22#302011yescharacteristic
463C22#302011yescharacteristic
464C4—22no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—44no—
469C4—44no—
470C4—48no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4#187814yes—
476C4C2×C4×Q1611yescharacteristic
477C4C2×#44111yescharacteristic
478C4C4×AGL(1,ℤ/8)12yes—
479C4C2×#46211yescharacteristic
480C4C2×#46411yescharacteristic
481C4#303612yes—
482C2—24no—
483C2#1875911yescharacteristic
484C2#1876311yescharacteristic
485C2#1877911yescharacteristic
486C2#1878311yescharacteristic
487C2C2×#187811yescharacteristic
488C2#2092511yescharacteristic
489C2#2095411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9065C2semidirect
7. #9058C2semidirect
14. #3230C4semidirect
83. #500C2×C4semidirect
85. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#23652
#213852
#213122
C2×#15492
#90652
#90582
#79012
#78922