Finite Group Databasebeta

← #124137

Subgroups — #124137

Order 512.

1304 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 106 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#13851C211yesmaximal, characteristic
2C4×#323C212yesmaximal, complemented
3#7131C212yesmaximal, complemented
4#7103C212yesmaximal, complemented
5C23×#41C411yescharacteristic
6C2×C4×C22⋊C4—22no—
7C2×C4×C22⋊C4C411yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9C4×C22⋊C8—22no—
10C2×#323—22no—
11#1446C411yescharacteristic
12#1446C2211yescharacteristic
13#1438—22no—
14#1438C411yescharacteristic
15#1438C2211yescharacteristic
16#1280C2214yes—
17C22×#41—22no—
18C22×#41—48no—
19C22×#41—41no—
20C22×#41—41no—
21C22×#41—42no—
22C22×#41C814yescomplemented
23C23×Q8C2×C411yescharacteristic
24C23×D4—21no—
25C23×D4C2×C411yescharacteristic
26C24×C4—21no—
27C24×C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—22no—
31C22×C22⋊C4D412yes—
32C22×C42—21no—
33C22×C42D412yes—
34C2×C22⋊C8—24no—
35C2×C4×C8—42no—
36C4×C22⋊C4—41no—
37C4×C22⋊C4—41no—
38C4×C22⋊C4—48no—
39C4×C22⋊C4—42no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95C2×C411yescharacteristic
43C2×#95C2×C411yescharacteristic
44C2×#95C2×C411yescharacteristic
45C2×#95C2311yescharacteristic
46#336—42no—
47#323—28no—
48C25—21no—
49C25C22⋊C411yescharacteristic
50C2×#41—416no—
51C2×#41—416no—
52C2×#41—44no—
53C2×#41—44no—
54C2×#41—44no—
55C2×#41—44no—
56C2×#41—48no—
57C2×#41—432no—
58C2×#41—48no—
59C2×#41—48no—
60C2×#41—48no—
61C22×Q8—21no—
62C22×Q8—44no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8C2×C812yes—
66C22×D4—24no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—44no—
71C22×D4—44no—
72C22×D4—42no—
73C22×D4—42no—
74C22×D4—44no—
75C22×D4C2×C812yes—
76C23×C4—28no—
77C23×C4—21no—
78C23×C4—24no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—44no—
85C23×C4—44no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—42no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C2×C812yes—
92C23×C4M1612yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C22×C8—44no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—44no—
111C2×C42—44no—
112C2×C42—42no—
113C22⋊C8—28no—
114C22⋊C8—44no—
115C4×C8—84no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—44no—
121#95—44no—
122C24—24no—
123C24—22no—
124C24—21no—
125C24—21no—
126C24—22no—
127C24—22no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—41no—
132C24—41no—
133C24—42no—
134C24—41no—
135C24—41no—
136C24—42no—
137C24—42no—
138C24C22⋊C812yes—
139C24C2×C22⋊C411yescharacteristic
140#41—416no—
141#41—416no—
142#41—432no—
143#41—416no—
144#41—432no—
145#41—416no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—48no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×D4—42no—
154C2×D4—48no—
155C2×D4—42no—
156C2×D4—48no—
157C2×D4—48no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—48no—
161C2×D4—416no—
162C2×D4—416no—
163C2×D4—44no—
164C2×D4—44no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—216no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—48no—
182C22×C4—44no—
183C22×C4—416no—
184C22×C4—48no—
185C22×C4—48no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4C22⋊C812yes—
209C22×C4#9911yescharacteristic
210C22×C4#9911yescharacteristic
211C22×C4#10012yes—
212C22×C4#10414yes—
213C22×C4C22×C811yescharacteristic
214C22×C4C2×M1611yescharacteristic
215C22×C4#13514yes—
216C2×C8—48no—
217C2×C8—84no—
218C2×C8—84no—
219C22⋊C4—48no—
220C22⋊C4—416no—
221C22⋊C4—416no—
222C22⋊C4—48no—
223C42—44no—
224C42—82no—
225C42—84no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—24no—
230C23—22no—
231C23—24no—
232C23—42no—
233C23—44no—
234C23—44no—
235C23—42no—
236C23—42no—
237C23—44no—
238C23—44no—
239C23—44no—
240C23—44no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—42no—
245C23—44no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—44no—
250C23—42no—
251C23—41no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23—82no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—82no—
265C23—81no—
266C23C2×C22⋊C811yescharacteristic
267C23C2×#9911yescharacteristic
268C23C2×#10011yescharacteristic
269C23#41612yes—
270C23C2×#10412yes—
271Q8—44no—
272Q8—44no—
273Q8—48no—
274D4—44no—
275D4—432no—
276D4—44no—
277D4—48no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—416no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—416no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#32312yes—
319C2×C4#32312yes—
320C2×C4AΣL(1,𝔽16)14yes—
321C8—88no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—48no—
331C22—42no—
332C22—42no—
333C22—44no—
334C22—42no—
335C22—84no—
336C22—82no—
337C22—84no—
338C22—82no—
339C22—84no—
340C22—84no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—82no—
346C22—84no—
347C22C2×#32311yescharacteristic
348C22#147612yes—
349C22#150112yes—
350C22#152511yescharacteristic
351C22#154311yescharacteristic
352C22C2×AΣL(1,𝔽16)12yes—
353C22#212611yescharacteristic
354C22#213211yescharacteristic
355C4—22no—
356C4—44no—
357C4—41no—
358C4—41no—
359C4—44no—
360C4—44no—
361C4—82no—
362C4—82no—
363C4#131614yes—
364C2—21no—
365C2—21no—
366C2—22no—
367C2—82no—
368C2—84no—
369C2#756311yescharacteristic
370C2C2×#131611yescharacteristic
371C2#1143111yescharacteristic
372C2#1172512yes—
373C2#1173312yes—
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC23×#41C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
2. C4×#323C2semidirect
3. #7131C2semidirect
4. #7103C2semidirect
22. C22×#41C8semidirect

Maximal subgroups

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

subgroupindex
#138512
C4×#3232
#71312
#71032