Finite Group Databasebeta

← #480403

Subgroups — #480403

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic
2#17365C212yesmaximal
3#11536C214yesmaximal, complemented
4#11511C212yesmaximal
5#11506C212yesmaximal
6#11482C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C22×C4×Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—22no—
11#2327—22no—
12#2327—22no—
13#2327C412yes—
14#2327C412yes—
15#2327C2212yes—
16#2327C2212yes—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#387—22no—
20C2×#387—22no—
21C2×#382C2211yescharacteristic
22C2×#376C411yescharacteristic
23#1467—22no—
24#1467C2214yes—
25#1463—22no—
26#1463C2214yes—
27#1461—22no—
28#1461C2214yes—
29#1324C2218yes—
30#1319C2218yes—
31C23×Q8C2×C411yescharacteristic
32C23×Q8C2×C411yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—28no—
35C2×C4⋊Q8C2×C412yes—
36C2×C4⋊Q8C2×C412yes—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—42no—
40C2×C4×Q8—42no—
41C2×C4×Q8—42no—
42C2×C4×Q8—42no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C42—21no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D412yes—
61#432—24no—
62#432—42no—
63#431—24no—
64#431—42no—
65#423—42no—
66#407—24no—
67C4×M16—42no—
68#391—22no—
69#391—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2312yes—
74#387—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—24no—
81#387—24no—
82#382—22no—
83#382C2×C412yes—
84#382C2312yes—
85#376—21no—
86#376—21no—
87#376C2×C412yes—
88#376C2×C412yes—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94#324—24no—
95#324—24no—
96#324—24no—
97#324C2318yes—
98C22×Q8—24no—
99C22×Q8—24no—
100C22×Q8—21no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8C22⋊C412yes—
108C22×Q8C22×C412yes—
109C22×Q8C22×C411yescharacteristic
110C22×Q8C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D412yes—
120C2×M16—44no—
121C4⋊Q8—216no—
122C4⋊Q8—24no—
123C4⋊Q8—24no—
124C4⋊Q8—24no—
125C4⋊Q8M1618yescomplemented
126C4×Q8—42no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—44no—
135C4×Q8—44no—
136C4×Q8—44no—
137C4×Q8—44no—
138C2×C4⋊C4—216no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C4⋊C8—44no—
185C4⋊C8—44no—
186C22⋊C8—28no—
187C22⋊C8—44no—
188#97—44no—
189C4×C8—44no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200C24C2×C22⋊C411yescharacteristic
201C2×Q8—24no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—42no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—48no—
213C2×Q8—48no—
214C2×Q8—48no—
215C2×Q8—48no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8C2×M1614yes—
220C2×Q8C2×M1614yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4#9912yes—
254C22×C4#9912yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D412yes—
265M16—88no—
266C2×C8—48no—
267C4⋊C4—216no—
268C4⋊C4—216no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C42—24no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—44no—
296C42—44no—
297C42—42no—
298C42C2×M1614yes—
299C23—21no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23C2×#9911yescharacteristic
304C23C2×#9911yescharacteristic
305C23C22×C22⋊C411yescharacteristic
306Q8—44no—
307Q8—48no—
308Q8—44no—
309Q8—44no—
310Q8—44no—
311Q8—48no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#40714yes—
359C2×C4C2×#9912yes—
360C2×C4C2×#9912yes—
361C2×C4#41312yes—
362C2×C4C22×M1612yes—
363C8—88no—
364C22—24no—
365C22—24no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22#212912yes—
373C22#213312yes—
374C22C2×#40711yescharacteristic
375C22C22×#9911yescharacteristic
376C22C2×#41311yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—24no—
382C4—44no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—44no—
389C4—82no—
390C4—82no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—22no—
395C2#1736211yescharacteristic
396C2C2×#212911yescharacteristic
397C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11536C2semidirect
6. #11482C2semidirect
125. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
#173652
#115362
#115112
#115062
#114822