Finite Group Databasebeta

← #195575

Subgroups — #195575

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9060C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230C418yescomplemented
14C2×#500C2212yes—
15C2×#487C2212yes—
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103—22no—
22#2370—22no—
23#2369C2212yes—
24#2368—22no—
25#2365C2212yes—
26#2360—22no—
27#2296—21no—
28#2295—21no—
29#2295—22no—
30C42×Q8—21no—
31C4×#117—21no—
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1988C2214yes—
35#1987—22no—
36#1981—21no—
37#1981—21no—
38#1981C2214yes—
39#1736—21no—
40#1736—21no—
41#1550—24no—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×M16—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8D412yes—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#500—24no—
78#500C2×C418yescomplemented
79#487—24no—
80#487C2×C418yescomplemented
81#484—216no—
82#477—216no—
83#431—22no—
84#431C2×C414yes—
85C2×#107—24no—
86#423—22no—
87#423—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—48no—
99C2×C22⋊C8—21no—
100C2×C4×C8C2312yes—
101#391—24no—
102#391—22no—
103#389—24no—
104#389—22no—
105#389—22no—
106#389—22no—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4C2312yes—
123C4×C4⋊C4C2312yes—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—22no—
128#376—21no—
129C2×#95—21no—
130C43—22no—
131C43—21no—
132#340—24no—
133#340—24no—
134#333—24no—
135#332—24no—
136C22×Q8—21no—
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8—41no—
142C22×Q8—42no—
143C22×Q8C2×D412yes—
144C23×C4—22no—
145C23×C4—41no—
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—
155C22×C8—44no—
156C22×C8—42no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8C22×C414yes—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C2×D412yes—
174C4⋊Q8C2×D412yes—
175#125—24no—
176#125—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8C22⋊C418yes—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8C22⋊C418yes—
187#117—24no—
188#117—24no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117Q1614yes—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—24no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C22×C412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239#107—216no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247#103—216no—
248#103—48no—
249#103—48no—
250C22⋊C8—24no—
251C4×C8—22no—
252C4×C8C22×C414yes—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—21no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—22no—
262C24#12011yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—44no—
268C2×Q8—416no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—84no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8C2×C22⋊C414yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C4×D412yes—
296C22×C4C4×D412yes—
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4C2×Q1612yes—
304C22×C4#13512yes—
305C22×C4#13712yes—
306C22×C4#13712yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310M16—84no—
311C2×C8—24no—
312C2×C8—44no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—84no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—84no—
343C4⋊C4C2×C22⋊C414yes—
344C4⋊C4C2×Q1614yes—
345C4⋊C4AGL(1,ℤ/8)14yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C42—28no—
350C42—24no—
351C42—48no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42—84no—
361C42C2×C22⋊C412yes—
362C42C4×D412yes—
363C42C4×D412yes—
364C42#12112yes—
365C42#12112yes—
366C42#12312yes—
367C42C23×C411yescharacteristic
368C42C22×D411yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—22no—
372C23—22no—
373C23#38612yes—
374C23#45112yes—
375C23#45212yes—
376C23C2×#12011yescharacteristic
377Q8—44no—
378Q8—44no—
379Q8—42no—
380Q8—42no—
381Q8—48no—
382Q8—82no—
383Q8—88no—
384Q8—82no—
385Q8—82no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—88no—
412C2×C4—88no—
413C2×C4—84no—
414C2×C4#38612yes—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4C4×Q1614yes—
418C2×C4#44114yes—
419C2×C4#45112yes—
420C2×C4#45212yes—
421C2×C4#46214yes—
422C2×C4#46414yes—
423C2×C4C22×C22⋊C411yescharacteristic
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12311yescharacteristic
429C2×C4C22×Q1611yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)12yes—
432C2×C4C2×#13711yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4#57512yes—
435C8—44no—
436C8—82no—
437C8—82no—
438C22—24no—
439C22—24no—
440C22—22no—
441C22—22no—
442C22#187814yes—
443C22C2×#38611yescharacteristic
444C22C2×#45111yescharacteristic
445C22C2×#45211yescharacteristic
446C22#300612yes—
447C22#302212yes—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—22no—
452C4—44no—
453C4—44no—
454C4—44no—
455C4—44no—
456C4—82no—
457C4—82no—
458C4—84no—
459C4#187814yes—
460C4C2×C4×Q1611yescharacteristic
461C4C2×#44111yescharacteristic
462C4C4×AGL(1,ℤ/8)12yes—
463C4C2×#46211yescharacteristic
464C4C2×#46411yescharacteristic
465C4#303811yescharacteristic
466C4#304011yescharacteristic
467C2—24no—
468C2#1880112yes—
469C2#1880512yes—
470C2C2×#187811yescharacteristic
471C2#2092811yescharacteristic
472C2#2095511yescharacteristic
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. #9060C2semidirect
7. #9058C2semidirect
13. #3230C4semidirect
78. #500C2×C4semidirect
80. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#341452
#213132
C2×#19812
C2×#15502
#90602
#90582
#79012
#78912