Finite Group Databasebeta

← #480729

Subgroups — #480729

Order 512.

1088 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2#17365C212yesmaximal, complemented
3#11565C218yesmaximal, complemented
4#11506C212yesmaximal, complemented
5#11505C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×#124C411yescharacteristic
8C22×#124C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—22no—
11#2545—22no—
12#2545—22no—
13#2545C414yes—
14#2545C2214yes—
15C2×#396—22no—
16C2×#391C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#388—22no—
19C2×C4×C4⋊C4C411yescharacteristic
20#1467—22no—
21#1467C2214yescomplemented
22#1463—22no—
23#1463C2214yes—
24#1461—22no—
25#1461C2214yescomplemented
26#1319C2218yescomplemented
27#1318C2218yes—
28C23×Q8C2×C411yescharacteristic
29C23×Q8C2×C411yescharacteristic
30C23×D4C2×C411yescharacteristic
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—44no—
34C2×C4⋊Q8C2×C414yes—
35C2×#124—21no—
36C2×#124—21no—
37C2×#124—22no—
38C2×#124—42no—
39C2×#124—41no—
40C2×#124—41no—
41C2×#124—44no—
42C2×#124C2×C414yes—
43C2×#124C2×C414yes—
44C2×C4×Q8—41no—
45C2×C4×Q8—41no—
46C2×C4×Q8—42no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8D412yes—
50C2×C4×Q8C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D412yes—
63#432—24no—
64#432—42no—
65#431—24no—
66#431—42no—
67#423—42no—
68#407—24no—
69C4×M16—42no—
70#396—24no—
71#396—24no—
72#396—24no—
73#396—24no—
74#391—22no—
75#391C2×C412yes—
76#391C2312yes—
77#390—22no—
78#390C2×C412yes—
79#390C2312yes—
80#388—24no—
81#388—24no—
82#388—24no—
83#388—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2×C414yes—
86C2×#95—21no—
87C2×#95—22no—
88#324—24no—
89#324—24no—
90#324—24no—
91#324C2318yes—
92C25—21no—
93C22×Q8—24no—
94C22×Q8—21no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8C22⋊C411yescharacteristic
103C22×Q8C22⋊C411yescharacteristic
104C22×Q8C22×C412yes—
105C22×Q8C22×C412yes—
106C22×D4—21no—
107C22×D4—42no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—42no—
111C22×D4C22×C412yes—
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—44no—
122C4⋊Q8—44no—
123C4⋊Q8—48no—
124#124—44no—
125#124—44no—
126#124—44no—
127#124—44no—
128#124—44no—
129#124—44no—
130#124—48no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2411yescharacteristic
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C412yes—
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C4⋊C8—44no—
188C4⋊C8—44no—
189C22⋊C8—28no—
190C22⋊C8—44no—
191#97—44no—
192C4×C8—44no—
193#95—24no—
194#95—24no—
195#95—24no—
196C24—22no—
197C24—42no—
198C24—44no—
199C24—42no—
200C24—44no—
201C24—41no—
202C24—41no—
203C24C2×C22⋊C411yescharacteristic
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—48no—
214C2×Q8—48no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—84no—
221C2×Q8—84no—
222C2×Q8#10014yes—
223C2×D4—24no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—84no—
230C2×D4—84no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4#9912yes—
260C22×C4#9911yescharacteristic
261C22×C4#9911yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D412yes—
270M16—88no—
271C2×C8—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—48no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—82no—
315C23—82no—
316C23—88no—
317C23—88no—
318C23—82no—
319C23—82no—
320C23—82no—
321C23—82no—
322C23C2×#9911yescharacteristic
323C23C2×#9911yescharacteristic
324C23C22×C22⋊C411yescharacteristic
325Q8—44no—
326Q8—48no—
327Q8—44no—
328Q8—84no—
329Q8—84no—
330D4—84no—
331D4—84no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—48no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4C2×#9912yes—
373C2×C4C2×#9912yes—
374C2×C4C2×#10012yes—
375C2×C4C2×#10014yes—
376C2×C4#56812yes—
377C8—88no—
378C22—24no—
379C22—24no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—84no—
387C22—84no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—164no—
392C22—164no—
393C22#212612yes—
394C22#213312yes—
395C22#287911yescharacteristic
396C22C22×#9911yescharacteristic
397C22C22×#10011yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—44no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—82no—
409C4—82no—
410C2—21no—
411C2—21no—
412C2—22no—
413C2—22no—
414C2—162no—
415C2#1739611yescharacteristic
416C2C2×#212611yescharacteristic
417C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×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
2. #17365C2semidirect
3. #11565C2semidirect
4. #11506C2semidirect
21. #1467C22semidirect
25. #1461C22semidirect
26. #1319C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
#173652
#115652
#115062
#115052