Finite Group Databasebeta

← #379841

Subgroups — #379841

Order 512.

2261 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2261 conjugacy classes of subgroups — too many to draw. Only its 298 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×#2622C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic, complemented
3C2×#1898C212yesmaximal
4#14341C211yesmaximal, characteristic, complemented
5#11203C214yesmaximal, complemented
6#11043C214yesmaximal, complemented
7#10735C212yesmaximal, complemented
8C22×#120C2211yescharacteristic
9C23×C22⋊C4C2212yes—
10C23×C22⋊C4C2211yescharacteristic
11C22×#99C2214yes—
12#2622—24no—
13#2622—24no—
14C2×#397C414yescomplemented
15C2×#394C414yescomplemented
16#2315—28no—
17C2×#386C2212yes—
18C2×#385—24no—
19C2×#385C2212yes—
20C2×#380C2214yes—
21C2×#379C2212yes—
22C2×#379C2211yescharacteristic
23#1898—28no—
24C2×#342C2214yes—
25C2×#342C2218yes—
26#1440—24no—
27C22⋊#95C2214yes—
28C26C2311yescharacteristic
29C23×D4C2×C412yes—
30C23×D4C2311yescharacteristic
31C24×C4—22no—
32C24×C4C2312yes—
33C2×#120—22no—
34C2×#120—22no—
35C2×#120—24no—
36C2×#120—24no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4C2×C414yescomplemented
39C22×C22⋊C4—24no—
40C22×C22⋊C4—24no—
41C22×C22⋊C4—24no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4—48no—
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C414yes—
47C22×C22⋊C4C2×C414yes—
48C22×C22⋊C4C2×C414yescomplemented
49C22×C22⋊C4D414yes—
50C22×C22⋊C4D414yes—
51C22×C22⋊C4D418yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2314yes—
56C2×#99—28no—
57C2×#99—48no—
58#397—216no—
59#394—216no—
60#386—24no—
61#386—24no—
62#385—232no—
63#385—24no—
64#385—24no—
65#380—216no—
66#379—22no—
67#379—22no—
68#379—24no—
69#379—24no—
70C2×#95—24no—
71C2×#95—28no—
72C2×#95C2×C414yescomplemented
73C2×#95C2×C414yescomplemented
74#342—28no—
75#342—48no—
76C25—24no—
77C25—24no—
78C25—84no—
79C25C22×C411yescharacteristic
80C25C22×C412yes—
81C25C2×D412yes—
82C25C2×D412yes—
83C25C2×D414yes—
84C25#4114yes—
85C25C2411yescharacteristic
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—24no—
91C22×D4—24no—
92C22×D4—24no—
93C22×D4—28no—
94C22×D4—84no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—28no—
99C23×C4—24no—
100C23×C4—48no—
101C23×C4—84no—
102C23×C4—88no—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C414yes—
105C23×C4C22×C412yes—
106C23×C4C22×C414yes—
107C23×C4C2×D414yes—
108C23×C4C2×D414yes—
109#120—44no—
110#120—48no—
111#120—44no—
112C2×C4⋊C4—216no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—88no—
115C2×C4⋊C4—84no—
116C2×C22⋊C4—216no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—216no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—416no—
141C2×C22⋊C4—48no—
142C2×C22⋊C4—416no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—88no—
145C2×C22⋊C4—84no—
146C2×C22⋊C4—84no—
147C2×C22⋊C4C22⋊C4116yes—
148C2×C22⋊C4C2×D418yes—
149#99—416no—
150#95—216no—
151#95—28no—
152#95—28no—
153#95—88no—
154C24—24no—
155C24—24no—
156C24—28no—
157C24—28no—
158C24—28no—
159C24—28no—
160C24—216no—
161C24—21no—
162C24—22no—
163C24—24no—
164C24—48no—
165C24—44no—
166C24—44no—
167C24—48no—
168C24—44no—
169C24—44no—
170C24—88no—
171C24—88no—
172C24—816no—
173C24—816no—
174C24C2×C22⋊C414yes—
175C24C4×D418yes—
176C24#12014yes—
177C24#12118yes—
178C24#12314yes—
179C24C23×C411yescharacteristic
180C24C22×D411yescharacteristic
181C24C22×D411yescharacteristic
182C24C22×D412yes—
183C24C2×#4112yes—
184C24H(5,𝔽2)14yes—
185C242−1+414yes—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—44no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—48no—
195C2×D4—48no—
196C2×D4—88no—
197C2×D4—816no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—28no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—416no—
215C22×C4—416no—
216C22×C4—48no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—416no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—88no—
224C22×C4—88no—
225C22×C4—816no—
226C22×C4—832no—
227C22×C4C2×C22⋊C418yes—
228C4⋊C4—88no—
229C4⋊C4—816no—
230C22⋊C4—232no—
231C22⋊C4—28no—
232C22⋊C4—28no—
233C22⋊C4—416no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—48no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—48no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—48no—
245C22⋊C4—88no—
246C22⋊C4—88no—
247C22⋊C4—816no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—28no—
252C23—24no—
253C23—24no—
254C23—28no—
255C23—28no—
256C23—28no—
257C23—216no—
258C23—216no—
259C23—48no—
260C23—44no—
261C23—44no—
262C23—48no—
263C23—48no—
264C23—48no—
265C23—48no—
266C23—48no—
267C23—48no—
268C23—48no—
269C23—416no—
270C23—416no—
271C23—42no—
272C23—44no—
273C23—41no—
274C23—44no—
275C23—816no—
276C23—816no—
277C23—84no—
278C23—88no—
279C23—816no—
280C23—816no—
281C23—832no—
282C23#38618yes—
283C23C22×C22⋊C411yescharacteristic
284C23C2×C4×D412yes—
285C23#52014yes—
286C23C2×#12011yescharacteristic
287C23C2×#12112yes—
288C23C2×#12311yescharacteristic
289C23#53414yes—
290C23#53512yes—
291C23#53612yes—
292C23#53614yes—
293C23#53714yes—
294C23#53918yes—
295D4—816no—
296D4—81no—
297D4—81no—
298D4—84no—
299D4—82no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—88no—
316C2×C4—88no—
317C2×C4—816no—
318C2×C4—832no—
319C2×C4—88no—
320C2×C4—84no—
321C2×C4—88no—
322C2×C4—816no—
323C2×C4—84no—
324C22—24no—
325C22—28no—
326C22—28no—
327C22—24no—
328C22—22no—
329C22—22no—
330C22—24no—
331C22—24no—
332C22—28no—
333C22—28no—
334C22—28no—
335C22—44no—
336C22—44no—
337C22—48no—
338C22—48no—
339C22—48no—
340C22—416no—
341C22—416no—
342C22—816no—
343C22—816no—
344C22—84no—
345C22—81no—
346C22—82no—
347C22—164no—
348C22C2×#38611yescharacteristic
349C22#231112yes—
350C22#232214yes—
351C22#261311yescharacteristic
352C22#261411yescharacteristic
353C22#261814yes—
354C22#262312yes—
355C22#288314yes—
356C4—48no—
357C4—88no—
358C4—84no—
359C4—81no—
360C4—82no—
361C4—164no—
362C2—21no—
363C2—24no—
364C2—21no—
365C2—22no—
366C2—24no—
367C2—48no—
368C2—161no—
369C2#1425611yescharacteristic
370C2#2098412yes—
371C2#2187614yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2315C2semidirect
4. #14341C2semidirect
5. #11203C2semidirect
6. #11043C2semidirect
7. #10735C2semidirect
14. C2×#397C4semidirect
15. C2×#394C4semidirect
38. C22×C4⋊C4C2×C4semidirect
48. C22×C22⋊C4C2×C4semidirect
72. C2×#95C2×C4semidirect
73. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26222
C2×#23152
C2×#18982
#143412
#112032
#110432
#107352