Finite Group Databasebeta

← #149430

Subgroups — #149430

Order 512.

1124 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 150 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
2C2×#1393C212yesmaximal
3C2×#1393C212yesmaximal
4#10761C212yesmaximal
5C22×#125C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7C22×#99C2214yes—
8#2566—24no—
9#2566—24no—
10C2×#396—22no—
11C2×#390C2211yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#385—22no—
14C2×#384C411yescharacteristic, complemented
15C2×#382C411yescharacteristic, complemented
16C2×C4×C4⋊C4C411yescharacteristic, complemented
17C2×#342—22no—
18C22⋊#95—22no—
19#1393—24no—
20#1393—24no—
21C23×D4C2311yescharacteristic
22C24×C4D412yes—
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—42no—
26C2×#125—42no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—42no—
30C2×C4×D4—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—24no—
41C22×C22⋊C4D412yes—
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C2×#99—24no—
46C2×#99—24no—
47C2×#99—24no—
48C2×#99—24no—
49C2×#99—24no—
50C2×#99Q818yes—
51#396—216no—
52#390—22no—
53#390—22no—
54#390D414yes—
55#390D414yes—
56#385—216no—
57#384—22no—
58#384—22no—
59#382—22no—
60#382—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C2×#95—22no—
64C2×#95—21no—
65C2×#95—44no—
66#342—44no—
67C25C2×D412yes—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—42no—
71C22×D4C2×D412yes—
72C22×D4C2×D412yes—
73C22×D4C2×Q814yes—
74C22×D4#4114yes—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—24no—
80C23×C4—24no—
81C23×C4—42no—
82C23×C4—44no—
83C23×C4—84no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88#125—48no—
89#125—48no—
90C4×D4—48no—
91C4×D4—48no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—84no—
112C2×C4⋊C4C22⋊C412yes—
113C2×C4⋊C4C22⋊C412yes—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—48no—
121C2×C22⋊C4—44no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—84no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137#99—28no—
138#99—28no—
139#99—44no—
140#99—44no—
141#95—24no—
142#95—28no—
143#95—88no—
144C24—22no—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—44no—
149C24—44no—
150C24C2×C22⋊C411yescharacteristic
151C24#12014yes—
152C24C22⋊Q814yes—
153C24#12412yes—
154C24C4⋊Q812yes—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4C22⋊Q814yes—
164C2×D4C22⋊Q814yes—
165C2×D4#12412yes—
166C2×D4#12412yes—
167C2×D4C4⋊Q812yes—
168C2×D4C4⋊Q812yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—48no—
186C22×C4—48no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—48no—
201C22×C4—84no—
202C22×C4—88no—
203C22×C4—88no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—88no—
207C22×C4—88no—
208C22×C4#9912yes—
209C22×C4#9912yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C4×Q814yes—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—48no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—48no—
221C4⋊C4—48no—
222C4⋊C4—88no—
223C22⋊C4—48no—
224C22⋊C4—416no—
225C22⋊C4—48no—
226C22⋊C4—48no—
227C22⋊C4—88no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C42—24no—
231C42—24no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—42no—
236C23—22no—
237C23—22no—
238C23—24no—
239C23—22no—
240C23—22no—
241C23—24no—
242C23—24no—
243C23—22no—
244C23—22no—
245C23—24no—
246C23—44no—
247C23—44no—
248C23—48no—
249C23—44no—
250C23—44no—
251C23—44no—
252C23—44no—
253C23—82no—
254C23—82no—
255C23—84no—
256C23—84no—
257C23#37911yescharacteristic
258C23#39112yes—
259C23#39314yes—
260C23#39514yes—
261C23C2×#9911yescharacteristic
262C23C2×#9911yescharacteristic
263C23#41012yes—
264D4—44no—
265D4—44no—
266D4—42no—
267D4—42no—
268D4—42no—
269D4—42no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—48no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—816no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—88no—
306C2×C4—816no—
307C2×C4—164no—
308C2×C4—164no—
309C2×C4—168no—
310C2×C4#39112yes—
311C2×C4#39112yes—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—24no—
317C22—24no—
318C22—24no—
319C22—24no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—48no—
325C22—44no—
326C22—44no—
327C22—84no—
328C22—82no—
329C22—84no—
330C22—82no—
331C22#144611yescharacteristic
332C22#178111yescharacteristic
333C22#178211yescharacteristic
334C22#199612yes—
335C22#199612yes—
336C22#212612yes—
337C22#212712yes—
338C4—22no—
339C4—22no—
340C4—44no—
341C4—42no—
342C4—42no—
343C4—82no—
344C4—82no—
345C4—82no—
346C4—82no—
347C4—168no—
348C2—21no—
349C2—21no—
350C2—22no—
351C2—44no—
352C2—44no—
353C2#1138111yescharacteristic
354C2#1269511yescharacteristic
355C2#1269611yescharacteristic
356C2#1330014yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×#125C4semidirect
14. C2×#384C4semidirect
15. C2×#382C4semidirect
16. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#13932
#107612