Finite Group Databasebeta

← #213617

Subgroups — #213617

Order 512.

1360 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 218 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×#2850C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#9124C218yesmaximal, complemented
4#8718C212yesmaximal, complemented
5#8334C212yesmaximal, complemented
6C22×#124C2211yescharacteristic
7C22×C22⋊Q8C411yescharacteristic
8C22×#121C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432C2214yescomplemented
11#2850—22no—
12#2850—22no—
13#2850C414yes—
14#2850C2214yes—
15C2×#391C411yescharacteristic
16C2×#390C411yescharacteristic
17C2×#388—22no—
18C2×#386—22no—
19C2×#382C2211yescharacteristic
20#1817—22no—
21#1797—22no—
22#1797—22no—
23#1779—22no—
24#1749C2214yes—
25#1749C2214yescomplemented
26C2×#329C2218yescomplemented
27#1295C2218yescomplemented
28C23×Q8C2×C411yescharacteristic
29C23×D4—21no—
30C23×D4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4D412yes—
33C2×#124—22no—
34C2×#124—44no—
35C2×#124C2×C412yes—
36C2×#124C2312yes—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—44no—
39C2×C22⋊Q8C2×C414yes—
40C2×#121—22no—
41C2×#121—44no—
42C2×#121C2×C414yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—22no—
50C22×C42—44no—
51C22×C42C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42D412yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58#432—48no—
59C2×C4⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×#97C2318yes—
64#391—21no—
65#391—21no—
66#391C2×C414yes—
67#390—21no—
68#390—21no—
69#390C2×C414yes—
70#388—24no—
71#388—24no—
72#388—24no—
73#388—24no—
74#386—24no—
75#386—24no—
76#386—24no—
77#386—24no—
78#382—22no—
79#382C2×C412yes—
80#382C2312yes—
81C2×#95—22no—
82C2×#95—22no—
83#336—24no—
84#336—24no—
85#329—48no—
86C25—21no—
87C25C22⋊C411yescharacteristic
88C22×Q8—21no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8C22×C412yes—
93C22×D4—24no—
94C22×D4—21no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4—42no—
98C22×D4—48no—
99C22×D4—44no—
100C22×D4C22×C412yes—
101C23×C4—24no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—44no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C22×C8—28no—
121#124—28no—
122#124—44no—
123#124—48no—
124C22⋊Q8—48no—
125C22⋊Q8—48no—
126#121—48no—
127#121—48no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C22×C412yes—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—48no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—44no—
146C2×C42—24no—
147C2×C42—28no—
148C2×C42—42no—
149C2×C42—44no—
150C2×C42—44no—
151C2×C42—44no—
152C2×C42—44no—
153C2×C42C22⋊C412yes—
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—48no—
164C22⋊C8—48no—
165C22⋊C8—48no—
166C22⋊C8—48no—
167#97—216no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172C24—24no—
173C24—22no—
174C24—41no—
175C24—41no—
176C24—42no—
177C24—42no—
178C24—44no—
179C24—42no—
180C24—42no—
181C24—44no—
182C24#10014yes—
183C24C2×C22⋊C412yes—
184C24C2×C22⋊C411yescharacteristic
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—88no—
192C2×D4—24no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—48no—
198C2×D4—48no—
199C2×D4—48no—
200C2×D4—42no—
201C2×D4—88no—
202C2×D4—816no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—416no—
237C22×C4—432no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D412yes—
252C2×C8—48no—
253C2×C8—48no—
254C2×C8—48no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4—48no—
266C42—24no—
267C42—22no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—41no—
272C42—42no—
273C42—41no—
274C42—42no—
275C42H(5,𝔽2)14yes—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—28no—
286C23—44no—
287C23—44no—
288C23—48no—
289C23—48no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—44no—
294C23—44no—
295C23—84no—
296C23—88no—
297C23—84no—
298C23—88no—
299C23—84no—
300C23C2×#10014yes—
301C23C2×#10012yes—
302C23#42114yes—
303C23C22×C22⋊C411yescharacteristic
304Q8—84no—
305Q8—84no—
306D4—84no—
307D4—84no—
308D4—816no—
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4#41312yes—
347C2×C4#42114yes—
348C2×C4#51812yes—
349C2×C4#53414yes—
350C8—88no—
351C22—24no—
352C22—22no—
353C22—24no—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—416no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—84no—
364C22—82no—
365C22—84no—
366C22—82no—
367C22—82no—
368C22—168no—
369C22#153314yes—
370C22#230811yescharacteristic
371C22C22×#10011yescharacteristic
372C22C2×#41311yescharacteristic
373C22C2×#42112yes—
374C22C2×#42112yes—
375C4—24no—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—82no—
383C4—82no—
384C4—82no—
385C2—24no—
386C2—44no—
387C2—162no—
388C2C2×#153311yescharacteristic
389C2#1767012yes—
390C2#1767511yescharacteristic
391C2#2175211yescharacteristic
392C2#2177412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #9124C2semidirect
4. #8718C2semidirect
5. #8334C2semidirect
10. C2×#432C22semidirect
25. #1749C22semidirect
26. C2×#329C22semidirect
27. #1295C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#207152
#91242
#87182
#83342