Finite Group Databasebeta

← #195594

Subgroups — #195594

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#9060C214yesmaximal, complemented
7#9059C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229C418yescomplemented
14C2×#498C2212yes—
15C2×#487C2212yes—
16C2×#483—22no—
17C2×#475—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103—21no—
22C22×#103—21no—
23#2370—22no—
24#2369C2212yes—
25#2365C2212yes—
26#2355—21no—
27#2355—22no—
28Q8×C4⋊C4—21no—
29#2329—22no—
30#2317—21no—
31C4×C22⋊Q8—22no—
32#2285C2211yescharacteristic
33#2274—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990—21no—
36#1990—21no—
37#1988C2214yes—
38#1981—22no—
39#1981C2214yes—
40#1823—22no—
41#1542—24no—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533D414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8D412yes—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#498—24no—
77#498C2×C418yescomplemented
78#487—24no—
79#487C2×C418yescomplemented
80#483—216no—
81#475—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yes—
86C2×#106—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—44no—
97C2×#103—44no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2312yes—
102#391—24no—
103#391—22no—
104#389—24no—
105#389—22no—
106#389—22no—
107#389—22no—
108#387—24no—
109#387—24no—
110#387—24no—
111#385—24no—
112#384—22no—
113#384D411yescharacteristic
114#384D411yescharacteristic
115#382—22no—
116#382—21no—
117#380—21no—
118#380D412yes—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—24no—
132C2×#95—21no—
133#340—24no—
134#340—24no—
135#333—24no—
136#332—24no—
137C22×Q8—21no—
138C22×Q8—24no—
139C22×Q8—22no—
140C22×Q8—41no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8C2×D412yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—42no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8C22×C414yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C2×D412yes—
176C4⋊Q8C2×D412yes—
177#125—24no—
178#125—24no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8C22⋊C418yes—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—84no—
187C4×Q8C22⋊C418yes—
188#117—24no—
189#117—24no—
190#117—42no—
191#117—42no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4SD1614yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239#106—216no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246#103—216no—
247#103—48no—
248#103—48no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251C4×C8—22no—
252C4×C8—42no—
253C4×C8C22×C414yes—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
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—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8C2×C22⋊C414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12312yes—
313C22×C4C2×SD1612yes—
314C22×C4#13512yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4#13712yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—84no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4C2×C22⋊C414yes—
359C4⋊C4C2×SD1614yes—
360C22⋊C4—28no—
361C22⋊C4—44no—
362C42—28no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—82no—
367C42—84no—
368C42—84no—
369C42C2×C22⋊C412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12112yes—
373C42#12112yes—
374C42#12312yes—
375C42C23×C411yescharacteristic
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23#38612yes—
384C23#44812yes—
385C23#44912yes—
386C23C2×#12011yescharacteristic
387Q8—44no—
388Q8—44no—
389Q8—44no—
390Q8—44no—
391Q8—44no—
392Q8—82no—
393Q8—88no—
394Q8—82no—
395Q8—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
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—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—88no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#38612yes—
431C2×C4#38612yes—
432C2×C4#38612yes—
433C2×C4C4×SD1614yes—
434C2×C4#44014yes—
435C2×C4#44812yes—
436C2×C4#44912yes—
437C2×C4#46014yes—
438C2×C4#46414yes—
439C2×C4C22×C22⋊C411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12311yescharacteristic
445C2×C4C22×SD1611yescharacteristic
446C2×C4C2×#13511yescharacteristic
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C2×C4#57712yes—
450C2×C4#57812yes—
451C8—44no—
452C8—82no—
453C8—82no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22#188014yes—
461C22C2×#38611yescharacteristic
462C22C2×#44811yescharacteristic
463C22C2×#44911yescharacteristic
464C22#300911yescharacteristic
465C22#301111yescharacteristic
466C22#302211yescharacteristic
467C22#302211yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—44no—
473C4—44no—
474C4—44no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4#188014yes—
480C4C2×C4×SD1611yescharacteristic
481C4C2×#44011yescharacteristic
482C4#294712yes—
483C4C2×#46011yescharacteristic
484C4C2×#46411yescharacteristic
485C4#304612yes—
486C2—24no—
487C2#1890411yescharacteristic
488C2#1890811yescharacteristic
489C2#1892411yescharacteristic
490C2#1892811yescharacteristic
491C2C2×#188011yescharacteristic
492C2#2093111yescharacteristic
493C2#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. #9059C2semidirect
13. #3229C4semidirect
77. #498C2×C4semidirect
79. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#342132
#218332
C2×#19812
C2×#15422
#90602
#90592
#78932
#78922