Finite Group Databasebeta

← #194925

Subgroups — #194925

Order 512.

1578 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2631C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#21808C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic, complemented
5#9020C214yesmaximal, complemented
6#9016C214yesmaximal, complemented
7#8018C212yesmaximal, complemented
8C22×#124C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2631—24no—
12#2631C418yescomplemented
13C2×#397—21no—
14C2×#393C2212yes—
15C2×#392—21no—
16C2×#392C2212yes—
17#2357—24no—
18#2354—24no—
19#2312C2214yes—
20#2309—21no—
21C2×#386—22no—
22#2285C2212yes—
23C2×#383C2211yescharacteristic
24#2280—22no—
25#2274—21no—
26#1983—24no—
27#1959—24no—
28#1787C2214yes—
29#1786C2214yes—
30#1466—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C23×D4—22no—
35C23×D4D411yescharacteristic
36C23×D4C2311yescharacteristic
37C24×C4D412yes—
38C2×#124—22no—
39C2×#124—22no—
40C2×#124—22no—
41C2×#124—24no—
42C2×#124C2×C414yes—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4D414yes—
48C2×#117—22no—
49C2×#117—24no—
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—24no—
59C2×C4⋊C8—22no—
60C2×#103—28no—
61C2×#102—28no—
62C2×C22⋊C8C2314yes—
63C2×#97—22no—
64#397—28no—
65#393—24no—
66#393C2×C418yes—
67#392—28no—
68#392—24no—
69#392C2×C418yes—
70#390—28no—
71#388—216no—
72#386—216no—
73#384D414yescomplemented
74#383—22no—
75#383C2×C414yes—
76#382—22no—
77#381—24no—
78#380—24no—
79#380—22no—
80#380C2314yes—
81#379—24no—
82C4×C4⋊C4—28no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yescomplemented
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95C2312yes—
91#323—28no—
92C25—21no—
93C25C2×D412yes—
94C22×Q8—21no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—82no—
98C22×Q8C22×C412yes—
99C22×D4—28no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—21no—
104C22×D4—22no—
105C22×D4—44no—
106C22×D4—48no—
107C22×D4—82no—
108C22×D4—84no—
109C22×D4—84no—
110C22×D4C22⋊C414yes—
111C22×D4C22×C412yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—84no—
118C23×C4C22⋊C418yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—24no—
124#124—28no—
125#124—28no—
126#124—44no—
127#124—44no—
128C4×D4—28no—
129C4×D4—28no—
130C4×D4—28no—
131C4×D4—28no—
132#117—48no—
133#117—88no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4C22⋊C414yes—
139C2×C4⋊C4C2×D418yes—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—84no—
155C2×C22⋊C4—84no—
156C2×C22⋊C4C22×C414yes—
157C2×C22⋊C4C2×D414yes—
158C2×C42—24no—
159C2×C42—28no—
160C2×C42C22×C412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42#4114yes—
165C4⋊C8—44no—
166#103—88no—
167#102—88no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170#97—44no—
171#95—28no—
172#95—24no—
173#95—24no—
174#95—28no—
175#95—24no—
176#95—22no—
177#95—24no—
178#95C22×C414yes—
179C24—22no—
180C24—24no—
181C24—22no—
182C24—24no—
183C24—22no—
184C24—44no—
185C24—44no—
186C24—42no—
187C24—42no—
188C24—44no—
189C24—41no—
190C24—41no—
191C24—44no—
192C24C2×C22⋊C414yes—
193C24C22×D411yescharacteristic
194C2×Q8—24no—
195C2×Q8—48no—
196C2×Q8—42no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—84no—
200C2×D4—24no—
201C2×D4—28no—
202C2×D4—48no—
203C2×D4—48no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—48no—
207C2×D4—416no—
208C2×D4—42no—
209C2×D4—82no—
210C2×D4—84no—
211C2×D4—82no—
212C2×D4—88no—
213C2×D4—88no—
214C2×D4—832no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—28no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—44no—
233C22×C4—84no—
234C22×C4—88no—
235C22×C4—88no—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C414yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4#12011yescharacteristic
240C22×C4#12012yes—
241C22×C4#12011yescharacteristic
242C22×C4#12312yes—
243C22×C4#12312yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D412yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4112yes—
248C2×C8—48no—
249C2×C8—44no—
250C4⋊C4—28no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—48no—
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—88no—
261C22⋊C4—88no—
262C22⋊C4—88no—
263C22⋊C4—88no—
264C42—48no—
265C42—88no—
266C42C4×D414yes—
267C42C4×D414yes—
268C42#12114yes—
269C42#12114yes—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—44no—
276C23—44no—
277C23—416no—
278C23—42no—
279C23—42no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23—42no—
287C23—82no—
288C23—82no—
289C23—88no—
290C23—88no—
291C23—82no—
292C23—88no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23C22×C22⋊C411yescharacteristic
299C23C2×#12011yescharacteristic
300C23C2×#12311yescharacteristic
301C23#57514yes—
302Q8—48no—
303Q8—84no—
304D4—416no—
305D4—48no—
306D4—84no—
307D4—832no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—416no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—42no—
332C2×C4—88no—
333C2×C4—82no—
334C2×C4—88no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#38614yes—
338C2×C4#38612yes—
339C2×C4#38612yes—
340C2×C4C2×C4×D412yes—
341C2×C4C2×#12112yes—
342C8—84no—
343C22—28no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—48no—
349C22—44no—
350C22—44no—
351C22—44no—
352C22—84no—
353C22—88no—
354C22—84no—
355C22—84no—
356C22—88no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—164no—
361C22—164no—
362C22C2×#38611yescharacteristic
363C22#289612yes—
364C22#300512yes—
365C22#301912yes—
366C22#302712yes—
367C22#309312yes—
368C4—24no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—44no—
374C4—84no—
375C4—88no—
376C4—82no—
377C2—22no—
378C2—22no—
379C2—84no—
380C2—162no—
381C2#1837014yes—
382C2#2091011yescharacteristic
383C2#2095311yescharacteristic
384C2#2096111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2631C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #21808C2semidirect
4. C2×#1466C2semidirect
5. #9020C2semidirect
6. #9016C2semidirect
7. #8018C2semidirect
12. #2631C4semidirect
73. #384D4semidirect
84. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26312
#341862
#218082
C2×#14662
#90202
#90162
#80182