Finite Group Databasebeta

← #183010

Subgroups — #183010

Order 512.

995 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2C4⋊C4×M16C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#17314C211yesmaximal, characteristic
6#8256C214yesmaximal, complemented
7#8252C214yesmaximal, complemented
8C2×#1303C211yesmaximal, characteristic
9#8079C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11Q8×M16—22no—
12Q8×M16C414yescomplemented
13#2962—22no—
14#2962C414yescomplemented
15#2961—22no—
16#2960C2214yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2905—21no—
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2873—21no—
23C2×C4×M16—21no—
24C2×C4×M16C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1771—22no—
27#1770C2212yes—
28C8×C4⋊C4C2212yes—
29#1728—21no—
30#1550C414yescomplemented
31#1549C412yescomplemented
32#1548C412yescomplemented
33#1542C2214yes—
34#1503C414yes—
35#1503C2212yes—
36#1503C2212yes—
37C2×#328C414yes—
38#1450C2214yes—
39C8×M16—22no—
40C8×M16—22no—
41#1303C2212yes—
42#1303C2212yes—
43#1301C2214yes—
44C22×M16—21no—
45C22×M16—21no—
46C22×M16D411yescharacteristic
47C22×M16D411yescharacteristic
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533C2×C412yes—
54#533C2×C412yes—
55#533C2312yes—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#446—28no—
68#446—28no—
69#446—24no—
70#446C2×C418yescomplemented
71C8×Q8—24no—
72C8×Q8C2×C418yescomplemented
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431C2×C412yes—
78#431C2×C412yes—
79#431C2312yes—
80#431C2312yes—
81#424—24no—
82#424—24no—
83#423—24no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423C2×C412yes—
89#423C2×C412yes—
90#423C2×C412yes—
91#423C2312yes—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C412yes—
96C2×C4⋊C8C2×C412yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8—21no—
99C2×C22⋊C8C2×C412yes—
100#405—24no—
101#405—24no—
102C4×M16—24no—
103C4×M16—44no—
104C4×M16C2×C412yes—
105C4×M16C2×C412yes—
106C4×M16D412yes—
107C4×M16D412yes—
108C2×#97—22no—
109C2×#97D412yes—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—22no—
113C2×C4×C8C2312yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C2×#95C2×C412yes—
120#336—22no—
121#336—22no—
122#333C2×C418yescomplemented
123#332C2×C418yescomplemented
124#328C2×C418yes—
125#328C2×C418yes—
126#322—24no—
127C82—24no—
128C22×Q8—21no—
129C22×Q8C4212yes—
130C23×C4—22no—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—42no—
141C2×M16—42no—
142C2×M16—42no—
143C2×M16—42no—
144C2×M16C22⋊C412yes—
145C2×M16C22⋊C412yes—
146C2×M16C22⋊C412yes—
147C2×M16C22⋊C412yes—
148C2×M16SD1614yes—
149C2×M16Q1614yes—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8C4212yes—
161C4⋊Q8C4212yes—
162C4⋊Q8C22×C412yes—
163C4⋊Q8C22×C412yes—
164#125—22no—
165#125—22no—
166#125—22no—
167#125—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8C4214yes—
175C22⋊Q8C22×C414yes—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8C4214yes—
183C4×Q8C22×C414yes—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C4212yes—
198C2×C22⋊C4—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217C4⋊C8—28no—
218C4⋊C8—28no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8C22×C414yes—
229C4⋊C8C22×C414yes—
230C4⋊C8C22×C414yes—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8C22×C414yes—
235#97—28no—
236#97—44no—
237#97C22⋊C414yes—
238#97C22⋊C414yes—
239C4×C8—28no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—44no—
245C4×C8C22×C414yes—
246C4×C8C2×D414yes—
247#95—22no—
248#95C22×C412yes—
249#95C22×C412yes—
250C24C4×D411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8C2×C4214yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4C2×C4211yescharacteristic
267C22×C4C2×C4211yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4#11711yescharacteristic
271C22×C4#11711yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C2×SD1612yes—
282C22×C4C2×Q1612yes—
283C22×C4#13512yes—
284C22×C4#13512yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4111yescharacteristic
288M16—24no—
289M16—24no—
290M16—42no—
291M16—42no—
292M16—44no—
293M16—44no—
294M16—44no—
295M16#10318yes—
296M16#10318yes—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—44no—
306C2×C8—82no—
307C2×C8—82no—
308C2×C8C2×C22⋊C414yes—
309C2×C8C2×C22⋊C414yes—
310C2×C8C2×SD1614yes—
311C2×C8C2×Q1614yes—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4C2×C4214yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C42—24no—
330C42—24no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—44no—
336C42C2×C4211yescharacteristic
337C42C2×C4211yescharacteristic
338C42C2×C22⋊C411yescharacteristic
339C42C2×C22⋊C411yescharacteristic
340C42#11711yescharacteristic
341C42#11711yescharacteristic
342C42C4×D411yescharacteristic
343C42C4×D411yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42C4×D411yescharacteristic
347C42C4×D411yescharacteristic
348C42C23×C411yescharacteristic
349C42C23×C411yescharacteristic
350C42C22×D411yescharacteristic
351C42C2×#4111yescharacteristic
352C23—41no—
353C23—41no—
354C23C4×C22⋊C411yescharacteristic
355C23C4×C22⋊C411yescharacteristic
356C23C4×SD1612yes—
357C23C4×Q1612yes—
358C23C2×C4×D411yescharacteristic
359Q8—44no—
360Q8—44no—
361Q8—82no—
362Q8—82no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—88no—
386C2×C4C4×C22⋊C411yescharacteristic
387C2×C4C4×C22⋊C411yescharacteristic
388C2×C4C4×C22⋊C411yescharacteristic
389C2×C4C4×C22⋊C411yescharacteristic
390C2×C4C4×C22⋊C411yescharacteristic
391C2×C4C4×C22⋊C411yescharacteristic
392C2×C4C2×#10312yes—
393C2×C4C2×#10312yes—
394C2×C4#41612yes—
395C2×C4#41612yes—
396C2×C4C4×SD1612yes—
397C2×C4C4×SD1612yes—
398C2×C4C4×SD1612yes—
399C2×C4C4×Q1612yes—
400C2×C4C4×Q1612yes—
401C2×C4C4×Q1612yes—
402C2×C4C22×C4211yescharacteristic
403C2×C4C22×C22⋊C411yescharacteristic
404C2×C4C2×#11711yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C22×SD1611yescharacteristic
409C2×C4C22×Q1611yescharacteristic
410C2×C4C2×#13511yescharacteristic
411C2×C4C2×#13511yescharacteristic
412C8—24no—
413C8—44no—
414C8—44no—
415C8—44no—
416C8—44no—
417C8C2×#10318yes—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—42no—
425C22—42no—
426C22C4×#10312yes—
427C22C4×#10312yes—
428C22C2×C4×C22⋊C411yescharacteristic
429C22C2×C4×SD1611yescharacteristic
430C22C2×C4×Q1611yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—44no—
440C4—44no—
441C4—82no—
442C4C4×#10312yes—
443C4C4×#10312yes—
444C4C22×#10311yescharacteristic
445C4C2×#41611yescharacteristic
446C4C2×C4×SD1611yescharacteristic
447C4C2×C4×Q1611yescharacteristic
448C4#295612yes—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—22no—
453C2#2040111yescharacteristic
454C2C2×C4×#10311yescharacteristic
455C2#2042111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C2×C4×C22⋊C4
prime radicalC24C4×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8256C2semidirect
7. #8252C2semidirect
12. Q8×M16C4semidirect
14. #2962C4semidirect
30. #1550C4semidirect
31. #1549C4semidirect
32. #1548C4semidirect
70. #446C2×C4semidirect
72. C8×Q8C2×C4semidirect
122. #333C2×C4semidirect
123. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601922
C4⋊C4×M162
#177082
C2×#15032
#173142
#82562
#82522
C2×#13032
#80792