Finite Group Databasebeta

← #480253

Subgroups — #480253

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2310C211yesmaximal, characteristic
2#17357C212yesmaximal
3#11504C214yesmaximal, complemented
4#11500C214yesmaximal
5#11428C214yesmaximal, complemented
6C24×Q8C411yescharacteristic
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9#2920—22no—
10#2310—22no—
11#2310—22no—
12#2310C412yes—
13#2310C412yes—
14#2310C2212yes—
15#2310C2212yes—
16C2×#391—22no—
17C2×#391—22no—
18C2×#381C412yes—
19C2×#381C2212yes—
20#1464—22no—
21#1464C2214yes—
22#1463—22no—
23#1462C2214yes—
24#1458—22no—
25#1458C2214yes—
26#1317C22116yes—
27C23×Q8—28no—
28C23×Q8—21no—
29C23×Q8—21no—
30C23×Q8—21no—
31C23×Q8—21no—
32C23×Q8—42no—
33C23×Q8—42no—
34C23×Q8—44no—
35C23×Q8C2×C412yes—
36C23×Q8C2×C412yes—
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C412yes—
39C2×#117—21no—
40C2×#117—21no—
41C2×#117—42no—
42C2×#117—42no—
43C2×#117—44no—
44C2×#117C2×C412yes—
45C2×#117C2×C412yes—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117D412yes—
49C2×#117D412yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42D412yes—
59#432—24no—
60#432—42no—
61#431—24no—
62#431—42no—
63#423—42no—
64#407—24no—
65C4×M16—42no—
66#391—24no—
67#391—24no—
68#391—24no—
69#391—24no—
70#391—24no—
71#391—24no—
72#391—24no—
73#391—24no—
74#381—24no—
75#381—24no—
76#381C2×C414yes—
77#381C2×C414yes—
78#381C2×C414yes—
79#381C2314yes—
80C2×#95—24no—
81C2×#95—24no—
82#324—24no—
83#324—24no—
84#323—24no—
85#323C2318yes—
86C25C22×C411yescharacteristic
87C22×Q8—28no—
88C22×Q8—28no—
89C22×Q8—28no—
90C22×Q8—28no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—28no—
96C22×Q8—24no—
97C22×Q8—44no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—44no—
101C22×Q8—44no—
102C22×Q8—432no—
103C22×Q8—432no—
104C22×Q8—44no—
105C22×Q8—44no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—44no—
109C22×Q8—44no—
110C22×Q8—44no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—44no—
114C22×Q8M1618yescomplemented
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—28no—
122C23×C4—22no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C414yes—
133C23×C4C2×D412yes—
134C2×M16—44no—
135#117—41no—
136#117—41no—
137#117—42no—
138#117—42no—
139#117—42no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—41no—
144#117—41no—
145#117—44no—
146#117—44no—
147#117—44no—
148#117—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22×C414yes—
164C2×C22⋊C4C22×C411yescharacteristic
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C22×C411yescharacteristic
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2411yescharacteristic
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—44no—
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C4⋊C8—44no—
181C4⋊C8—44no—
182C22⋊C8—28no—
183C22⋊C8—44no—
184#97—44no—
185C4×C8—44no—
186#95—28no—
187#95—28no—
188#95—28no—
189#95—28no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24—44no—
194C24C2×C22⋊C411yescharacteristic
195C24C2×C22⋊C411yescharacteristic
196C24C23×C411yescharacteristic
197C2×Q8—28no—
198C2×Q8—28no—
199C2×Q8—28no—
200C2×Q8—28no—
201C2×Q8—416no—
202C2×Q8—416no—
203C2×Q8—416no—
204C2×Q8—432no—
205C2×Q8—416no—
206C2×Q8—416no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—416no—
212C2×Q8—416no—
213C2×Q8—432no—
214C2×Q8—416no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—216no—
221C22×C4—216no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—432no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4#9912yes—
260C22×C4#9912yes—
261C22×C4#10014yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×M1614yes—
270C22×C4C2×M1618yes—
271C22×C4C22×D412yes—
272M16—88no—
273C2×C8—48no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—44no—
297C42—44no—
298C23—21no—
299C23—21no—
300C23—24no—
301C23—28no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—48no—
309C23—48no—
310C23—44no—
311C23—44no—
312C23C2×#9912yes—
313C23C2×#10012yes—
314C23C22×C22⋊C411yescharacteristic
315C23C22×M1612yes—
316Q8—416no—
317Q8—48no—
318Q8—48no—
319Q8—416no—
320Q8—416no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—416no—
342C2×C4—48no—
343C2×C4—416no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—88no—
358C2×C4#40712yes—
359C2×C4#40712yes—
360C2×C4C2×#9911yescharacteristic
361C2×C4C2×#9911yescharacteristic
362C2×C4C2×#9912yes—
363C2×C4C2×#10012yes—
364C2×C4C2×#10012yes—
365C8—88no—
366C22—24no—
367C22—24no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—416no—
378C22—42no—
379C22—42no—
380C22#212314yes—
381C22C2×#40711yescharacteristic
382C22C22×#9911yescharacteristic
383C22C22×#10011yescharacteristic
384C4—22no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—48no—
393C4—84no—
394C4#146214yes—
395C2—21no—
396C2—21no—
397C2—24no—
398C2—42no—
399C2—42no—
400C2C2×#146211yescharacteristic
401C2C2×#212312yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11504C2semidirect
5. #11428C2semidirect
114. C22×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23102
#173572
#115042
#115002
#114282