Finite Group Databasebeta

← #194741

Subgroups — #194741

Order 512.

1758 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1758 conjugacy classes of subgroups — too many to draw. Only its 266 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×#2482C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3#20702C212yesmaximal, complemented
4C2×#1462C211yesmaximal, characteristic
5#9028C214yesmaximal, complemented
6#9024C214yesmaximal, complemented
7#8015C212yesmaximal, complemented
8C22×#117C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2482—24no—
12#2482C418yescomplemented
13C2×#399C2212yes—
14C2×#397C2212yes—
15#2312—24no—
16#2312—22no—
17#2312C2214yes—
18C2×#388—24no—
19C2×#388—22no—
20C2×#385—21no—
21C2×#381C2211yescharacteristic
22C2×#380C2212yes—
23C2×#379—21no—
24C2×#379—21no—
25C2×C4×C22⋊C4—21no—
26#1825—24no—
27#1824C2214yescomplemented
28#1823C2214yescomplemented
29#1736—24no—
30#1462—24no—
31C2×#323C2214yes—
32#1294C2218yescomplemented
33C26D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—24no—
41C2×#117C2×C414yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D414yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4C2×C414yes—
57C22×C22⋊C4D412yes—
58C22×C22⋊C4D414yes—
59C22×C22⋊C4C2312yes—
60C22×C42—21no—
61C22×C42—22no—
62#407—24no—
63#407—24no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8C2314yes—
66#399—24no—
67#399C2×C418yescomplemented
68#397—24no—
69#397C2×C418yescomplemented
70#388—216no—
71#388—216no—
72#388—216no—
73#385—28no—
74#381—22no—
75#381C2×C414yes—
76#380—22no—
77#380—22no—
78#380D414yes—
79#380C2314yes—
80#379—24no—
81#379—28no—
82#379—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95C2312yes—
90#340—28no—
91#340—28no—
92#323—28no—
93C25—21no—
94C25—21no—
95C25—22no—
96C25—44no—
97C25—44no—
98C25—82no—
99C25C22⋊C414yes—
100C25C2×D411yescharacteristic
101C25C2×D412yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—24no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—44no—
110C23×C4—82no—
111C23×C4C22⋊C414yes—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4#4114yes—
119C23×C4C2411yescharacteristic
120C2×M16—48no—
121C2×M16—44no—
122C22×C8—24no—
123#117—28no—
124#117—28no—
125#117—44no—
126#117—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—84no—
137C2×C4⋊C4C22×C412yes—
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C2×D418yes—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—48no—
158C2×C22⋊C4—48no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—84no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4C22⋊C418yes—
166C2×C22⋊C4C22×C414yes—
167C2×C22⋊C4C2×D414yes—
168C2×C42—24no—
169C2×C42—28no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—84no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—44no—
176#95—28no—
177#95—24no—
178#95—28no—
179#95—24no—
180#95—24no—
181#95—22no—
182#95C22×C414yes—
183C24—24no—
184C24—24no—
185C24—22no—
186C24—24no—
187C24—24no—
188C24—22no—
189C24—24no—
190C24—24no—
191C24—42no—
192C24—48no—
193C24—44no—
194C24—44no—
195C24—44no—
196C24—44no—
197C24—48no—
198C24—44no—
199C24—48no—
200C24—44no—
201C24—84no—
202C24—82no—
203C24—82no—
204C24—84no—
205C24—84no—
206C24—832no—
207C24C2×C22⋊C412yes—
208C24C2×C22⋊C414yes—
209C24#12011yescharacteristic
210C24#12011yescharacteristic
211C24#12312yes—
212C24C22×D411yescharacteristic
213C22×C4—28no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—28no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—48no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—84no—
243C22×C4—82no—
244C22×C4—84no—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C4×D414yes—
248C22×C4C4×D414yes—
249C22×C4#12012yes—
250C22×C4#12114yes—
251C22×C4#12114yes—
252C22×C4#12312yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D412yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4112yes—
257M16—88no—
258C2×C8—48no—
259C2×C8—44no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—48no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—84no—
266C4⋊C4—88no—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—216no—
270C22⋊C4—48no—
271C22⋊C4—416no—
272C22⋊C4—48no—
273C22⋊C4—48no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—88no—
277C22⋊C4—88no—
278C22⋊C4—88no—
279C42—48no—
280C42—88no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—24no—
294C23—24no—
295C23—42no—
296C23—42no—
297C23—416no—
298C23—44no—
299C23—44no—
300C23—48no—
301C23—48no—
302C23—48no—
303C23—416no—
304C23—44no—
305C23—48no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—82no—
310C23—82no—
311C23—84no—
312C23—84no—
313C23—88no—
314C23—816no—
315C23—832no—
316C23—832no—
317C23—816no—
318C23#38612yes—
319C23#38612yes—
320C23#45814yes—
321C23C22×C22⋊C411yescharacteristic
322C23C2×#12011yescharacteristic
323C23C2×#12311yescharacteristic
324C23#57514yes—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—88no—
353C2×C4—82no—
354C2×C4—816no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#38614yes—
359C2×C4C2×C4×D412yes—
360C2×C4C2×#12112yes—
361C8—84no—
362C22—22no—
363C22—21no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—28no—
369C22—28no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—48no—
374C22—48no—
375C22—48no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—84no—
381C22—88no—
382C22—816no—
383C22—816no—
384C22—88no—
385C22#189014yes—
386C22C2×#38611yescharacteristic
387C22#289612yes—
388C22#300512yes—
389C22#301912yes—
390C22C2×#45812yes—
391C22#309312yes—
392C4—22no—
393C4—22no—
394C4—24no—
395C4—44no—
396C4—84no—
397C4—82no—
398C4—82no—
399C4—88no—
400C2—22no—
401C2—22no—
402C2—24no—
403C2—42no—
404C2—84no—
405C2#1814914yes—
406C2#2091011yescharacteristic
407C2#2095311yescharacteristic
408C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20702C2semidirect
5. #9028C2semidirect
6. #9024C2semidirect
7. #8015C2semidirect
12. #2482C4semidirect
27. #1824C22semidirect
28. #1823C22semidirect
32. #1294C22semidirect
67. #399C2×C4semidirect
69. #397C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#23122
#207022
C2×#14622
#90282
#90242
#80152