Finite Group Databasebeta

← #128177

Subgroups — #128177

Order 512.

902 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 138 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
1#17362C212yesmaximal
2#13847C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#407—22no—
6C2×C4×C4⋊C4C411yescharacteristic
7C2×C4×C22⋊C4—22no—
8C2×#376C411yescharacteristic
9#1465—22no—
10#1465C414yescomplemented
11#1463C2214yes—
12#1462—22no—
13#1459—22no—
14#1459C414yescomplemented
15#1445C2212yes—
16#1439—22no—
17#1438C412yes—
18C23×D4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C2×C4×D4—41no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4—41no—
25C2×C4×D4C2×C412yes—
26C2×C4×D4C2×C412yes—
27C2×C4×D4D411yescharacteristic
28C2×C4×D4D411yescharacteristic
29C2×C4×D4Q811yescharacteristic
30C2×C4×D4Q811yescharacteristic
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2311yescharacteristic
33C22×C22⋊C4—24no—
34C22×C22⋊C4D412yes—
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38#407—24no—
39#407—24no—
40#407—42no—
41#407—42no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—42no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C4×C22⋊C4—48no—
47#376—41no—
48#376—41no—
49C2×#95—21no—
50C2×#95—24no—
51C2×#95C2×C412yes—
52C2×#95C2×C412yes—
53#324—24no—
54#324C2×C418yes—
55#323—24no—
56#323—24no—
57C25C22⋊C412yes—
58C22×D4—41no—
59C22×D4—41no—
60C22×D4—44no—
61C22×D4C4212yes—
62C22×D4C4⋊C411yescharacteristic
63C22×D4C4⋊C411yescharacteristic
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—44no—
71C23×C4—44no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C4⋊C412yes—
77C23×C4C4⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C2×M16—44no—
81C2×M16—44no—
82C2×M16—84no—
83C2×M16—84no—
84C22×C8—44no—
85C4×D4—42no—
86C4×D4—42no—
87C4×D4—42no—
88C4×D4—42no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—41no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C22×C412yes—
102C2×C4⋊C4C2×D411yescharacteristic
103C2×C4⋊C4C2×Q811yescharacteristic
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—48no—
108C2×C22⋊C4—48no—
109C2×C22⋊C4C22⋊C412yes—
110C2×C22⋊C4C22⋊C412yes—
111C2×C22⋊C4C4⋊C412yes—
112C2×C22⋊C4C4⋊C412yes—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—24no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—44no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42C4211yescharacteristic
125C2×C42C4211yescharacteristic
126C2×C42C4⋊C411yescharacteristic
127C2×C42C4⋊C411yescharacteristic
128C22⋊C8—28no—
129C22⋊C8—44no—
130C22⋊C8—44no—
131C22⋊C8—44no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—48no—
136C24—24no—
137C24—22no—
138C24—22no—
139C24—42no—
140C24—42no—
141C24—44no—
142C24—42no—
143C24#9512yes—
144C24#9512yes—
145C24C2×C22⋊C411yescharacteristic
146C2×D4—24no—
147C2×D4—24no—
148C2×D4—42no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—48no—
154C2×D4—48no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4#9511yescharacteristic
196C22×C4#9511yescharacteristic
197C22×C4#9511yescharacteristic
198C22×C4#9511yescharacteristic
199C22×C4#9911yescharacteristic
200C22×C4#9911yescharacteristic
201C22×C4#9911yescharacteristic
202C22×C4#9911yescharacteristic
203C22×C4C2×C4211yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C4⋊C411yescharacteristic
206C22×C4C2×C4⋊C412yes—
207M16—84no—
208M16—84no—
209M16—88no—
210C2×C8—48no—
211C2×C8—88no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—48no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—432no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—44no—
230C42—44no—
231C42—82no—
232C42—82no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—44no—
243C23—44no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—48no—
249C23—48no—
250C23—42no—
251C23—42no—
252C23—44no—
253C23—82no—
254C23—82no—
255C23—84no—
256C23—84no—
257C23#34414yes—
258C23C2×#9511yescharacteristic
259C23C2×#9911yescharacteristic
260C23C2×#9911yescharacteristic
261D4—44no—
262D4—48no—
263D4—44no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#34211yescharacteristic
308C2×C4#34211yescharacteristic
309C2×C4#34211yescharacteristic
310C2×C4#34211yescharacteristic
311C2×C4#35312yes—
312C2×C4#35412yes—
313C8—88no—
314C22—24no—
315C22—24no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—24no—
323C22—44no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—82no—
331C22—84no—
332C22—88no—
333C22—84no—
334C22—82no—
335C22C2×#34211yescharacteristic
336C22C2×#34412yes—
337C22C2×#35311yescharacteristic
338C22C2×#35411yescharacteristic
339C22#213212yes—
340C4—21no—
341C4—21no—
342C4—42no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—84no—
349C4—84no—
350C4#139312yes—
351C4#139312yes—
352C2—21no—
353C2—22no—
354C2—22no—
355C2—21no—
356C2—44no—
357C2—82no—
358C2#811211yescharacteristic
359C2C2×#139311yescharacteristic
360C2#1258511yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect
10. #1465C4semidirect
14. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#173622
#138472
#73262