Finite Group Databasebeta

← #195570

Subgroups — #195570

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9060C214yesmaximal, complemented
7#9059C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229C418yescomplemented
14C2×#498C2212yes—
15C2×#487C2212yes—
16C2×#483—22no—
17C2×#475—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#419—21no—
22C2×#419—21no—
23#2370—22no—
24#2369C2212yes—
25#2368—22no—
26#2365C2212yes—
27#2360—22no—
28#2296—21no—
29#2295—21no—
30#2295—22no—
31C42×Q8—21no—
32C4×#117—21no—
33C4×#117C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1988—21no—
36#1988—21no—
37#1988C2214yes—
38#1981—22no—
39#1981C2214yes—
40C2×#340—22no—
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#498—24no—
78#498C2×C418yescomplemented
79#487—24no—
80#487C2×C418yescomplemented
81#483—216no—
82#475—216no—
83#431—22no—
84#431C2×C414yes—
85C2×#106—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—
95#419—44no—
96#419—44no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×C22⋊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—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2312yes—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—22no—
129#376—21no—
130C2×#95—21no—
131C43—22no—
132C43—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—22no—
148C23×C4—41no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—42no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—44no—
159C22×C8—42no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8C22×C414yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C2×D412yes—
177C4⋊Q8C2×D412yes—
178#125—24no—
179#125—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8C22⋊C418yes—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8C22⋊C418yes—
190#117—24no—
191#117—24no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117SD1614yes—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—24no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242#106—216no—
243C4⋊C8—24no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250#103—216no—
251#103—48no—
252#103—48no—
253C22⋊C8—24no—
254C4×C8—22no—
255C4×C8C22×C414yes—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—22no—
265C24#12011yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—44no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8C2×C22⋊C414yes—
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—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
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×SD1612yes—
312C22×C4#13512yes—
313C22×C4AGL(1,ℤ/8)12yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318M16—84no—
319C2×C8—24no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—84no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—84no—
351C4⋊C4C2×C22⋊C414yes—
352C4⋊C4C2×SD1614yes—
353C4⋊C4#13714yes—
354C22⋊C4—28no—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C42—28no—
358C42—24no—
359C42—48no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—82no—
368C42—84no—
369C42C2×C22⋊C412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12112yes—
373C42#12112yes—
374C42#12312yes—
375C42C23×C411yescharacteristic
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—22no—
380C23—22no—
381C23#38612yes—
382C23#44812yes—
383C23#44912yes—
384C23C2×#12011yescharacteristic
385Q8—44no—
386Q8—44no—
387Q8—42no—
388Q8—42no—
389Q8—44no—
390Q8—44no—
391Q8—82no—
392Q8—88no—
393Q8—82no—
394Q8—82no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4#38612yes—
427C2×C4C4×SD1614yes—
428C2×C4#44014yes—
429C2×C4#44812yes—
430C2×C4#44912yes—
431C2×C4#46014yes—
432C2×C4#46414yes—
433C2×C4C22×C22⋊C411yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12311yescharacteristic
439C2×C4C22×SD1611yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4C2×#13712yes—
444C2×C4#57512yes—
445C8—44no—
446C8—82no—
447C8—82no—
448C22—24no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22#188014yes—
453C22C2×#38611yescharacteristic
454C22C2×#44811yescharacteristic
455C22C2×#44911yescharacteristic
456C22#300712yes—
457C22#302212yes—
458C4—22no—
459C4—21no—
460C4—21no—
461C4—22no—
462C4—44no—
463C4—44no—
464C4—44no—
465C4—44no—
466C4—82no—
467C4—82no—
468C4—84no—
469C4#188014yes—
470C4C2×C4×SD1611yescharacteristic
471C4C2×#44011yescharacteristic
472C4C4×#13712yes—
473C4C2×#46011yescharacteristic
474C4C2×#46411yescharacteristic
475C4#303911yescharacteristic
476C4#304011yescharacteristic
477C2—24no—
478C2#1880012yes—
479C2#1880412yes—
480C2C2×#188011yescharacteristic
481C2#2092711yescharacteristic
482C2#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
78. #498C2×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×#32292
#341452
C2×#19882
#212592
C2×#15502
#90602
#90592
#79012
#78912