Finite Group Databasebeta

← #403388

Subgroups — #403388

Order 512.

2514 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2061C214yesmaximal, complemented
3#20846C212yesmaximal
4#13881C211yesmaximal, characteristic
5#11392C214yesmaximal, complemented
6#10733C212yesmaximal
7#10727C211yesmaximal, characteristic
8C2×#544—216no—
9C2×#544—24no—
10C2×#544—22no—
11C2×#544C2214yes—
12C22×C4⋊Q8—22no—
13C22×#125—22no—
14C22×C22⋊Q8—22no—
15C22×#117C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C2×#428C2214yes—
18C2×#407C2212yes—
19C2×#382C2212yes—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#2061—216no—
23#1854—24no—
24#1853—22no—
25#1852—22no—
26C2×#343C2214yes—
27#1736C2214yes—
28C2×#339C2218yes—
29#1442—22no—
30C4⋊#95C2212yes—
31C22⋊#95—22no—
32C23×Q8—22no—
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35C22×M16C2314yes—
36#544—232no—
37#544—232no—
38#544—232no—
39#544—216no—
40#544—28no—
41#544D4116yes—
42C2×C4⋊Q8—232no—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8—28no—
45C2×#125—232no—
46C2×#125—28no—
47C2×#125—28no—
48C2×C22⋊Q8—232no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8—28no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—216no—
54C2×#117D414yes—
55C2×#117C2314yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—28no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—22no—
71C22×C42C2312yes—
72#428—216no—
73C2×#107—24no—
74C2×#106—24no—
75#407—24no—
76#407Q818yescomplemented
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79#382—28no—
80#380—22no—
81#380—22no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4Q814yescomplemented
84C2×#95—21no—
85C2×#95—24no—
86C2×#95—21no—
87C2×#95—24no—
88#343—28no—
89#340—24no—
90#340—24no—
91#339—216no—
92C25C2×D411yescharacteristic
93C22×Q8—216no—
94C22×Q8—28no—
95C22×Q8—84no—
96C22×Q8—84no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—28no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—44no—
108C23×C4—82no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D414yes—
113C23×C4C2×Q814yes—
114C23×C4C2×Q812yes—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C2×M16—24no—
118C2×M16—28no—
119C2×M16—28no—
120C2×M16C2×Q818yescomplemented
121C22×C8—24no—
122C4⋊Q8—264no—
123C4⋊Q8—232no—
124C4⋊Q8—232no—
125#125—264no—
126#125—232no—
127#125—232no—
128C22⋊Q8—264no—
129C22⋊Q8—232no—
130C22⋊Q8—232no—
131#117—216no—
132#117—216no—
133#117—28no—
134#117—28no—
135#117—28no—
136#117C2×D4116yes—
137C2×C4⋊C4—216no—
138C2×C4⋊C4—216no—
139C2×C4⋊C4—216no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—232no—
142C2×C4⋊C4—216no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4C2×D418yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×D414yes—
158C2×C22⋊C4—216no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—216no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42C2×Q814yescomplemented
174#107—48no—
175#106—48no—
176C22⋊C8—28no—
177C22⋊C8—28no—
178#95—24no—
179#95—42no—
180#95—48no—
181#95—84no—
182C24—22no—
183C24—22no—
184C24—22no—
185C24—22no—
186C24—41no—
187C24—41no—
188C24#12011yescharacteristic
189C24#12011yescharacteristic
190C24C22⋊Q812yes—
191C24C4⋊Q812yes—
192C24C22×D411yescharacteristic
193C2×Q8—232no—
194C2×Q8—832no—
195C2×Q8—88no—
196C2×Q8—88no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—216no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—816no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4#12012yes—
241C22×C4#12018yes—
242C22×C4C22⋊Q814yes—
243C22×C4C22⋊Q814yes—
244C22×C4C22⋊Q812yes—
245C22×C4C4⋊Q814yes—
246C22×C4C4⋊Q814yes—
247C22×C4C4⋊Q812yes—
248C22×C4C22×D412yes—
249C22×C4C22×D414yes—
250C22×C4C22×Q812yes—
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C42−1+414yes—
254M16—216no—
255M16—48no—
256C2×C8—28no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—216no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—232no—
263C4⋊C4—264no—
264C4⋊C4—216no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4#12018yes—
270C4⋊C4#12018yes—
271C22⋊C4—216no—
272C22⋊C4—216no—
273C22⋊C4—48no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C42—216no—
277C42—216no—
278C42—28no—
279C42—44no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—24no—
289C23—24no—
290C23—21no—
291C23—21no—
292C23—24no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—42no—
299C23—42no—
300C23—44no—
301C23—44no—
302C23—44no—
303C23—44no—
304C23#39512yes—
305C23#39512yes—
306C23#45412yes—
307C23#45512yes—
308C23C2×#12012yes—
309C23C2×#12011yescharacteristic
310C23C2×C22⋊Q811yescharacteristic
311C23C2×C4⋊Q811yescharacteristic
312C23#53612yes—
313Q8—832no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—416no—
326C2×C4—416no—
327C2×C4—416no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—84no—
339C2×C4—88no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4#39514yes—
353C2×C4C2×#12014yes—
354C2×C4C2×C22⋊Q812yes—
355C2×C4C2×C4⋊Q812yes—
356C2×C4#53614yes—
357C2×C4#54814yes—
358C2×C4D4×Q818yes—
359C2×C4D4×Q814yes—
360C8—48no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—48no—
379C22—48no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22#206114yescomplemented
385C22C2×#39511yescharacteristic
386C22#240912yes—
387C22#243412yes—
388C22#261411yescharacteristic
389C22#284611yescharacteristic
390C22C2×#45511yescharacteristic
391C22C2×#45411yescharacteristic
392C22#301911yescharacteristic
393C22#302011yescharacteristic
394C4—24no—
395C4—48no—
396C4—48no—
397C4—48no—
398C4—84no—
399C4—161no—
400C4—161no—
401C4#240914yes—
402C4#243414yes—
403C2—22no—
404C2—24no—
405C2—22no—
406C2—42no—
407C2#1487411yescharacteristic
408C2C2×#206111yescharacteristic
409C2#2356111yescharacteristic
410C2#2965111yescharacteristic
411C2#2965211yescharacteristic
412C2#3288911yescharacteristic
413C2#3289111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2061C2semidirect
5. #11392C2semidirect
76. #407Q8semidirect
83. C4×C22⋊C4Q8semidirect
120. C2×M16C2×Q8semidirect
173. C2×C42C2×Q8semidirect
384. C22#2061semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#20612
#208462
#138812
#113922
#107332
#107272