Finite Group Databasebeta

← #480750

Subgroups — #480750

Order 512.

1080 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1080 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×#2566C211yesmaximal, characteristic
2#17362C212yesmaximal
3#11572C218yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11511C212yesmaximal, complemented
6C22×#125C411yescharacteristic
7C22×C4×D4C2211yescharacteristic
8C2×#407—22no—
9#2566—22no—
10#2566—22no—
11#2566C414yes—
12#2566C2214yes—
13C2×#396—22no—
14C2×#390C411yescharacteristic
15C2×#390C411yescharacteristic
16C2×#385—22no—
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C411yescharacteristic
20#1465—22no—
21#1465C2214yes—
22#1463C2214yes—
23#1462—22no—
24#1459—22no—
25#1459C2214yes—
26#1325C2218yescomplemented
27#1324C2218yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4D412yes—
30C22×M16—42no—
31C2×#125—22no—
32C2×#125—42no—
33C2×#125—41no—
34C2×#125—41no—
35C2×#125C2×C414yes—
36C2×C4×D4—42no—
37C2×C4×D4—41no—
38C2×C4×D4—41no—
39C2×C4×D4C2×C412yes—
40C2×C4×D4D412yes—
41C2×C4×D4D412yes—
42C2×C4×D4C2312yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D412yes—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56#407—24no—
57#407—24no—
58#407—42no—
59#407—42no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—42no—
62#396—24no—
63#396—24no—
64#396—24no—
65#396—24no—
66#390—21no—
67#390—21no—
68#390—21no—
69#390—21no—
70#390C2×C414yes—
71#390C2×C414yes—
72#385—24no—
73#385—24no—
74#385—24no—
75#385—24no—
76#384—22no—
77#384C2×C412yes—
78#384C2312yes—
79#382—22no—
80#382C2×C412yes—
81#382C2312yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4C2×C414yes—
84C2×#95—21no—
85C2×#95—22no—
86#324—24no—
87#324C2318yes—
88#323—24no—
89#323—24no—
90C25C22⋊C412yes—
91C22×D4—24no—
92C22×D4—42no—
93C22×D4—41no—
94C22×D4—41no—
95C22×D4C22⋊C411yescharacteristic
96C22×D4C22⋊C411yescharacteristic
97C22×D4C22×C412yes—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—84no—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C2×M16—44no—
116C2×M16—44no—
117C2×M16—88no—
118C22×C8—44no—
119#125—44no—
120#125—44no—
121#125—44no—
122#125—44no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—42no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—84no—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2411yescharacteristic
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C22⋊C412yes—
176C2×C42C22×C412yes—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C22⋊C8—28no—
180C22⋊C8—44no—
181C22⋊C8—44no—
182C22⋊C8—44no—
183#95—24no—
184#95—24no—
185#95—24no—
186C24—24no—
187C24—22no—
188C24—22no—
189C24—42no—
190C24—44no—
191C24—44no—
192C24C2×C22⋊C411yescharacteristic
193C24C2×C22⋊C412yes—
194C24C2×C22⋊C412yes—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—48no—
204C2×D4—48no—
205C2×D4#10114yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—48no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—84no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4#9912yes—
250C22×C4#9911yescharacteristic
251C22×C4#9911yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D412yes—
258M16—88no—
259M16—88no—
260C2×C8—48no—
261C2×C8—88no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—48no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—48no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—48no—
280C22⋊C4—48no—
281C22⋊C4—84no—
282C22⋊C4—84no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—24no—
288C42—24no—
289C42—42no—
290C42—42no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—416no—
307C23—44no—
308C23—44no—
309C23—82no—
310C23—82no—
311C23—88no—
312C23C2×#9911yescharacteristic
313C23C2×#9911yescharacteristic
314C23C2×#10114yes—
315C23C22×C22⋊C411yescharacteristic
316D4—44no—
317D4—48no—
318D4—44no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4C2×#9912yes—
364C2×C4C2×#9912yes—
365C2×C4C2×#10112yes—
366C2×C4#56812yes—
367C8—88no—
368C22—24no—
369C22—24no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—44no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—88no—
386C22—88no—
387C22—82no—
388C22#212712yes—
389C22#213212yes—
390C22#287911yescharacteristic
391C22C22×#9911yescharacteristic
392C22C22×#10111yescharacteristic
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—44no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—82no—
402C4—82no—
403C4—82no—
404C2—21no—
405C2—21no—
406C2—22no—
407C2—22no—
408C2—44no—
409C2—82no—
410C2#1739711yescharacteristic
411C2C2×#212711yescharacteristic
412C2C2×#213211yescharacteristic
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×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11572C2semidirect
4. #11512C2semidirect
5. #11511C2semidirect
26. #1325C22semidirect
27. #1324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
#173622
#115722
#115122
#115112