Finite Group Databasebeta

← #183022

Subgroups — #183022

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#20533C211yesmaximal, characteristic
3#17718C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#17326C211yesmaximal, characteristic
6#8258C214yesmaximal, complemented
7#8252C214yesmaximal, complemented
8#8098C211yesmaximal, characteristic
9#8076C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986C414yescomplemented
12#2985—22no—
13#2984C414yescomplemented
14#2983—21no—
15#2982—21no—
16#2962—22no—
17#2960C2214yes—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C8×C4⋊C4—22no—
26C8×C4⋊C4C2214yes—
27C4×C22⋊C8—22no—
28#1554C414yescomplemented
29#1548C412yescomplemented
30#1546C412yescomplemented
31#1542C2214yes—
32#1503C414yes—
33#1503C2212yes—
34#1503C2212yes—
35C2×#328C414yes—
36#1454—22no—
37#1452C2212yes—
38#1451—21no—
39#1450C2212yes—
40C8×M16—21no—
41#1303C2214yes—
42#1301C2212yes—
43#1301C2212yes—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533C2×C412yes—
50#533C2×C412yes—
51#533C2312yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×#117—21no—
60C22×C4⋊C4—21no—
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63#446—24no—
64#446—28no—
65#446—24no—
66#446C2×C418yescomplemented
67C8×Q8—24no—
68C8×Q8—24no—
69C8×Q8C2×C418yescomplemented
70#433—28no—
71#432—21no—
72#432—21no—
73#432—24no—
74#432C2×C412yes—
75#432C2×C412yes—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—24no—
83#431C2×C412yes—
84#431C2×C412yes—
85#431C2×C412yes—
86#431C2×C412yes—
87#431C2×C412yes—
88#431C2×C412yes—
89#431D411yescharacteristic
90#431D411yescharacteristic
91#431D412yes—
92#431C2312yes—
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95#424—24no—
96#423C2×C412yes—
97#423C2312yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8D412yes—
100C2×C4⋊C8C2312yes—
101#407—21no—
102#407—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8C2×C412yes—
105C2×C22⋊C8D412yes—
106#405—24no—
107C4×M16—21no—
108C4×M16—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—24no—
112C2×C4×C8C2312yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2312yes—
117C2×#95C2×C412yes—
118#336—24no—
119#333C2×C418yescomplemented
120#332C2×C418yescomplemented
121#328C2×C418yes—
122#328C2×C418yes—
123#322—22no—
124#322—22no—
125#322—22no—
126C82—22no—
127C22×Q8—21no—
128C22×Q8C4212yes—
129C23×C4—22no—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D414yes—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C4212yes—
143C4⋊Q8C4212yes—
144C4⋊Q8C22×C412yes—
145C4⋊Q8C22×C412yes—
146#125—22no—
147#125—22no—
148#125—22no—
149#125—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8C4214yes—
157C22⋊Q8C22×C414yes—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C4214yes—
165C4×Q8C22×C414yes—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C4212yes—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—82no—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196C4⋊C8—28no—
197C4⋊C8—28no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8C22⋊C412yes—
205C4⋊C8C22⋊C412yes—
206C4⋊C8C22⋊C414yes—
207C4⋊C8C22×C414yes—
208C4⋊C8C22×C414yes—
209C4⋊C8C22×C414yes—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8C22⋊C412yes—
218C22⋊C8C22⋊C412yes—
219C22⋊C8C22⋊C414yes—
220C22⋊C8C22×C412yes—
221C22⋊C8C22×C412yes—
222#97—28no—
223#97—22no—
224#97—22no—
225C4×C8—28no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—21no—
231C4×C8—22no—
232C4×C8—21no—
233C4×C8—84no—
234C4×C8C22×C414yes—
235C4×C8C2×D412yes—
236C4×C8C2×D412yes—
237#95—22no—
238#95C22×C414yes—
239C24C4×D411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8C2×C4214yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—22no—
253C22×C4—81no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4C2×C4211yescharacteristic
257C22×C4C2×C4211yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#11711yescharacteristic
261C22×C4#11711yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274M16—42no—
275M16—42no—
276M16—44no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—82no—
287C2×C8C2×C22⋊C414yes—
288C2×C8C2×C22⋊C414yes—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4C2×C4214yes—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C42—24no—
306C42—24no—
307C42—44no—
308C42—82no—
309C42C2×C4211yescharacteristic
310C42C2×C4211yescharacteristic
311C42C2×C22⋊C411yescharacteristic
312C42C2×C22⋊C411yescharacteristic
313C42#11711yescharacteristic
314C42#11711yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42C23×C411yescharacteristic
322C42C23×C411yescharacteristic
323C42C22×D411yescharacteristic
324C42C2×#4111yescharacteristic
325C23—21no—
326C23—21no—
327C23—22no—
328C23—41no—
329C23C4×C22⋊C411yescharacteristic
330C23C4×C22⋊C411yescharacteristic
331C23C2×C4×D411yescharacteristic
332Q8—44no—
333Q8—44no—
334Q8—44no—
335Q8—82no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—82no—
360C2×C4—88no—
361C2×C4C4×C22⋊C411yescharacteristic
362C2×C4C4×C22⋊C411yescharacteristic
363C2×C4C4×C22⋊C411yescharacteristic
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4C4×C22⋊C411yescharacteristic
367C2×C4#44314yes—
368C2×C4C22×C4211yescharacteristic
369C2×C4C22×C22⋊C411yescharacteristic
370C2×C4C2×#11711yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C2×C4×D411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4#57712yes—
375C2×C4#57812yes—
376C8—24no—
377C8—24no—
378C8—42no—
379C8—42no—
380C8—44no—
381C8—44no—
382C8—44no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22#176412yes—
393C22C2×C4×C22⋊C411yescharacteristic
394C22#294911yescharacteristic
395C22#295011yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—48no—
405C4—44no—
406C4—44no—
407C4#176412yes—
408C4#289812yes—
409C4#294712yes—
410C4C2×#44312yes—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2#2040111yescharacteristic
416C2#2041711yescharacteristic
417C2C2×#176411yescharacteristic
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×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8258C2semidirect
7. #8252C2semidirect
11. #2986C4semidirect
13. #2984C4semidirect
28. #1554C4semidirect
29. #1548C4semidirect
30. #1546C4semidirect
66. #446C2×C4semidirect
69. C8×Q8C2×C4semidirect
119. #333C2×C4semidirect
120. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#205332
#177182
C2×#15032
#173262
#82582
#82522
#80982
#80762