Finite Group Databasebeta

← #215140

Subgroups — #215140

Order 512.

1392 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1392 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#9652C218yesmaximal, complemented
4#8308C212yesmaximal, complemented
5#8307C212yesmaximal
6C22×#124C411yescharacteristic
7C22×C22⋊Q8C2211yescharacteristic
8C22×#121C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432—22no—
11#2850—22no—
12#2850—22no—
13#2850—22no—
14#2850—22no—
15C2×#391C411yescharacteristic
16C2×#390C411yescharacteristic
17C2×#388C2214yes—
18C2×#386C414yes—
19C2×#382C411yescharacteristic
20#1817—22no—
21#1797—22no—
22#1797C2214yes—
23#1779C2214yes—
24#1749—22no—
25#1749C2214yes—
26C2×#324C2218yes—
27C2×#323C2218yes—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C23×D4C2×C412yes—
31C24×C4C2×C412yes—
32C24×C4C2311yescharacteristic
33C2×#124—21no—
34C2×#124—22no—
35C2×#124—21no—
36C2×#124—44no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—44no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—44no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2312yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C414yes—
48C22×C22⋊C4C2×C414yes—
49C22×C22⋊C4C2312yes—
50C22×C42—44no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42D412yes—
56C22×C42D412yes—
57C22×C42C2312yes—
58#432—48no—
59C2×C4⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8C2318yes—
63C2×#97—24no—
64#391—22no—
65#391—21no—
66#391—21no—
67#390—22no—
68#390—21no—
69#390—21no—
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—21no—
79#382—22no—
80#382—21no—
81C2×#95C2×C414yes—
82C2×#95C2×C414yes—
83#336—24no—
84#336—24no—
85#324—48no—
86#323—48no—
87C25C22×C411yescharacteristic
88C25C22×C412yes—
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8—42no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×D4—24no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—42no—
98C22×D4—44no—
99C22×D4—42no—
100C22×D4—42no—
101C22×D4—48no—
102C23×C4—24no—
103C23×C4—44no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—44no—
107C23×C4—44no—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22×C412yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C412yes—
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C22×C8—28no—
122#124—44no—
123#124—48no—
124#124—44no—
125C22⋊Q8—48no—
126C22⋊Q8—48no—
127#121—48no—
128#121—48no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—48no—
147C2×C42—24no—
148C2×C42—28no—
149C2×C42—42no—
150C2×C42—44no—
151C2×C42—44no—
152C2×C42—44no—
153C2×C42—44no—
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C412yes—
156C2×C42C22⋊C412yes—
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C4⋊C8—48no—
165C22⋊C8—216no—
166C22⋊C8—48no—
167C22⋊C8—48no—
168#97—48no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173C24—22no—
174C24—24no—
175C24—24no—
176C24—22no—
177C24—24no—
178C24—28no—
179C24—41no—
180C24—41no—
181C24—42no—
182C24—42no—
183C24—44no—
184C24C2×C22⋊C412yes—
185C24C23×C411yescharacteristic
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—88no—
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—42no—
202C2×D4—88no—
203C2×D4—816no—
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—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—28no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—416no—
241C22×C4—432no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C22×D412yes—
252C22×C4H(5,𝔽2)14yes—
253C2×C8—48no—
254C2×C8—48no—
255C2×C8—48no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4—48no—
266C22⋊C4—48no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—41no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42#9914yes—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—28no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—48no—
288C23—48no—
289C23—44no—
290C23—44no—
291C23—42no—
292C23—44no—
293C23—42no—
294C23—42no—
295C23—44no—
296C23—48no—
297C23—84no—
298C23—84no—
299C23—88no—
300C23—88no—
301C23#41314yes—
302C23C22×C22⋊C411yescharacteristic
303C23#51812yes—
304C23#56814yes—
305Q8—84no—
306Q8—84no—
307D4—84no—
308D4—84no—
309D4—816no—
310C2×C4—28no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
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—22no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4C2×#9914yes—
348C2×C4C2×#9912yes—
349C2×C4#41012yes—
350C2×C4#53514yes—
351C8—88no—
352C22—22no—
353C22—22no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—416no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—84no—
368C22—84no—
369C22—168no—
370C22#230911yescharacteristic
371C22#287912yes—
372C22C22×#9911yescharacteristic
373C22C2×#41011yescharacteristic
374C22C2×#41312yes—
375C22#298014yes—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—24no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—82no—
384C4—82no—
385C4—82no—
386C2—24no—
387C2—44no—
388C2—162no—
389C2#1739912yes—
390C2#1740012yes—
391C2#2179111yescharacteristic
392C2#2179311yescharacteristic
393C2#2179811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #9652C2semidirect
4. #8308C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#207152
#96522
#83082
#83072