Finite Group Databasebeta

← #125592

Subgroups — #125592

Order 512.

578 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 86 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#15135C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7132C214yesmaximal, complemented
5C2×#399C412yes—
6C2×#389—21no—
7C2×#384C411yescharacteristic
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11#1779—21no—
12C2×#323—21no—
13#1445C2212yes—
14#1444—21no—
15#1439—21no—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—22no—
27C22×C22⋊C4C2×C411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C2×C4⋊C8—41no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—41no—
35C2×C22⋊C8—41no—
36#399—44no—
37#389—44no—
38#384—41no—
39#384—41no—
40#380—44no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95C2×C412yes—
47C2×#95D411yescharacteristic
48C2×#95C2311yescharacteristic
49#336—41no—
50#323—42no—
51C25C22⋊C411yescharacteristic
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—41no—
58C23×C4—42no—
59C23×C4—42no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C22×C8—42no—
67C22×C8—42no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—82no—
82C2×C4⋊C4—82no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—44no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—82no—
91C2×C22⋊C4SD1612yes—
92C2×C22⋊C4Q1612yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—42no—
98C2×C42—42no—
99C4⋊C8—82no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C22⋊C8—82no—
105C22⋊C8—82no—
106#95—41no—
107#95—41no—
108#95—42no—
109#95—42no—
110#95—44no—
111#95—44no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—41no—
119C24—41no—
120C24—41no—
121C24—41no—
122C24#10312yes—
123C24C2×C22⋊C411yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—48no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—84no—
170C22×C4#9911yescharacteristic
171C22×C4#9911yescharacteristic
172C22×C4#10012yes—
173C22×C4#10312yes—
174C22×C4#13112yes—
175C22×C4C2×SD1611yescharacteristic
176C22×C4C2×Q1611yescharacteristic
177C22×C4#13712yes—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C4⋊C4—48no—
185C4⋊C4—84no—
186C4⋊C4—82no—
187C4⋊C4—88no—
188C4⋊C4—84no—
189C4⋊C4—82no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—44no—
193C22⋊C4—88no—
194C42—42no—
195C42—42no—
196C42—42no—
197C42—81no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23#40811yescharacteristic
226C23C2×#9911yescharacteristic
227C23C2×#10011yescharacteristic
228C23C2×#10311yescharacteristic
229C23#41911yescharacteristic
230C23#42111yescharacteristic
231C23#42111yescharacteristic
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#35212yes—
282C2×C4#35312yes—
283C8—162no—
284C8—162no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#147111yescharacteristic
309C22#148111yescharacteristic
310C22#149711yescharacteristic
311C22#151111yescharacteristic
312C22#152011yescharacteristic
313C22#153611yescharacteristic
314C22#154811yescharacteristic
315C22C2×#35211yescharacteristic
316C22#212311yescharacteristic
317C22C2×#35311yescharacteristic
318C22#213411yescharacteristic
319C4—22no—
320C4—22no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—42no—
327C4—84no—
328C4—162no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—82no—
334C2#772711yescharacteristic
335C2#1156211yescharacteristic
336C2#1156311yescharacteristic
337C2#1180911yescharacteristic
338C2#1181811yescharacteristic
339C2#1190111yescharacteristic
340C2#1190311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7132C2semidirect

Maximal subgroups

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

subgroupindex
#151352
#87002
#85592
#71322