Finite Group Databasebeta

← #181538

Subgroups — #181538

Order 512.

2140 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×#2349C211yesmaximal, characteristic
4#8026C212yesmaximal, complemented
5#8023C214yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C2×#1294C214yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11#2972—24no—
12#2972C418yescomplemented
13C2×C8×D4—24no—
14C2×#431C2212yes—
15C22×C22⋊C8C2214yes—
16#2349—24no—
17#2349—24no—
18#2349C418yescomplemented
19D4×C4⋊C4C418yescomplemented
20#2316—21no—
21#2315—22no—
22C2×#389—21no—
23C2×#385—22no—
24#2285C2212yes—
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2214yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C22×#95C2212yes—
30#1294C2214yes—
31#1294C2218yes—
32#1294C2214yes—
33C23×D4D411yescharacteristic
34C24×C4—22no—
35C24×C4C2312yes—
36C23×C8—24no—
37#518—44no—
38#518—44no—
39C2×C4×D4—28no—
40C2×C4×D4—28no—
41C2×C4×D4—44no—
42C2×C4×D4—44no—
43C2×C4×D4C2×C414yes—
44C2×C4×D4C2×C414yes—
45C2×C4×D4C2×C414yes—
46C2×C4×D4D414yes—
47C2×C4×D4D414yes—
48C2×C4×D4Q814yes—
49C2×#117—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C418yescomplemented
54C22×C4⋊C4D411yescharacteristic
55C22×C22⋊C4—24no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D412yes—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C8×D4—232no—
61#431—24no—
62#431C2×C418yescomplemented
63C2×C4⋊C8—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—28no—
66C2×C22⋊C8C2×C4116yescomplemented
67C2×C22⋊C8C2314yes—
68C2×C4×C8—22no—
69#389—28no—
70#389—28no—
71#385—216no—
72#385—216no—
73#384C2×C414yescomplemented
74#382—22no—
75#382—22no—
76#382C2×C414yescomplemented
77#380—22no—
78#380—24no—
79#380—24no—
80#380—24no—
81#380C2×C418yescomplemented
82#380C2×C418yescomplemented
83#380C2314yes—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2×C414yescomplemented
87C4×C4⋊C4C2×C414yescomplemented
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95C2×C418yescomplemented
93C2×#95C2311yescharacteristic
94C2×#95C2312yes—
95C2×#95C2311yescharacteristic
96C25C2×D412yes—
97C22×D4C22⋊C414yes—
98C22×D4C22⋊C414yes—
99C22×D4C4⋊C414yes—
100C22×D4D814yes—
101C22×D4SD1614yes—
102C22×D4SD1614yes—
103C22×D4Q1614yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—28no—
108C23×C4—28no—
109C23×C4—48no—
110C23×C4—44no—
111C23×C4—44no—
112C23×C4—84no—
113C23×C4—84no—
114C23×C4C22×C414yes—
115C23×C4C22×C414yes—
116C23×C4C22×C414yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C23×C4C2×D414yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×Q814yes—
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C22×C8—216no—
126C22×C8—88no—
127C4×D4—432no—
128C4×D4—432no—
129C4×D4C42116yes—
130C4×D4C22⋊C4116yes—
131C4×D4C4⋊C4116yes—
132C4×D4C4⋊C4116yes—
133#117—44no—
134#117—44no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—88no—
145C2×C4⋊C4—84no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C4⋊C414yes—
149C2×C4⋊C4C22×C418yes—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—48no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—48no—
157C2×C22⋊C4C22⋊C418yes—
158C2×C22⋊C4C22⋊C418yes—
159C2×C22⋊C4C4⋊C418yes—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42C22×C412yes—
165C2×C42C22×C412yes—
166C2×C42C22×C412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×Q812yes—
170C4⋊C8—28no—
171C22⋊C8—216no—
172C22⋊C8—24no—
173C22⋊C8C22×C418yes—
174C4×C8—28no—
175#95—28no—
176#95—28no—
177#95—24no—
178#95—24no—
179#95—21no—
180#95—22no—
181#95—24no—
182#95—21no—
183#95C22×C412yes—
184#95C22×C414yes—
185#95C22×C412yes—
186C24—44no—
187C24—44no—
188C24C2×C22⋊C414yes—
189C24C2×C22⋊C414yes—
190C24C2×C4⋊C414yes—
191C24C2×D814yes—
192C24C2×SD1614yes—
193C24C2×SD1614yes—
194C24C2×Q1614yes—
195C24C22×D411yescharacteristic
196C2×D4#95116yes—
197C2×D4#102116yes—
198C2×D4#102116yes—
199C2×D4#103116yes—
200C2×D4#103116yes—
201C2×D4#106116yes—
202C2×D4#107116yes—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—28no—
208C22×C4—42no—
209C22×C4—48no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—42no—
213C22×C4—416no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—416no—
226C22×C4—416no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—88no—
232C22×C4—88no—
233C22×C4—88no—
234C22×C4—816no—
235C22×C4—816no—
236C22×C4C2×C4218yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C414yes—
239C22×C4C2×C22⋊C418yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C414yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C4⋊C418yes—
245C22×C4C2×C4⋊C418yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C414yes—
248C22×C4C2×C4⋊C412yes—
249C22×C4C2×D812yes—
250C22×C4C2×SD1612yes—
251C22×C4C2×SD1612yes—
252C22×C4C2×Q1612yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C23×C411yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C2×C8—28no—
260C2×C8—48no—
261C2×C8—816no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C4⋊C4—84no—
267C4⋊C4—88no—
268C4⋊C4—88no—
269C4⋊C4#95116yes—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—416no—
273C22⋊C4—416no—
274C22⋊C4#95132yes—
275C42—48no—
276C42—48no—
277C42C2×C4214yes—
278C42C2×C22⋊C414yes—
279C42C2×C4⋊C414yes—
280C42C2×C4⋊C414yes—
281C23—22no—
282C23—28no—
283C23—22no—
284C23—28no—
285C23—44no—
286C23—44no—
287C23—44no—
288C23—48no—
289C23—48no—
290C23—44no—
291C23—48no—
292C23—48no—
293C23C2×#9518yes—
294C23C2×#10218yes—
295C23C2×#10218yes—
296C23C2×#10318yes—
297C23C2×#10318yes—
298C23C2×#10618yes—
299C23C2×#10718yes—
300C23C22×C22⋊C411yescharacteristic
301C23C22×C22⋊C411yescharacteristic
302C23C22×C4⋊C411yescharacteristic
303C23C22×D811yescharacteristic
304C23C22×SD1611yescharacteristic
305C23C22×SD1611yescharacteristic
306C23C22×Q1611yescharacteristic
307D4#340164yes—
308C2×C4—22no—
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—832no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—1616no—
344C2×C4C2×#9514yes—
345C2×C4C2×#9518yes—
346C2×C4C2×#9514yes—
347C2×C4C2×#10214yes—
348C2×C4C2×#10214yes—
349C2×C4C2×#10314yes—
350C2×C4C2×#10314yes—
351C2×C4C2×#10614yes—
352C2×C4C2×#10714yes—
353C2×C4C22×C4211yescharacteristic
354C2×C4C22×C22⋊C411yescharacteristic
355C2×C4C22×C4⋊C411yescharacteristic
356C2×C4C22×C4⋊C411yescharacteristic
357C8—48no—
358C22—21no—
359C22—21no—
360C22—28no—
361C22—21no—
362C22—21no—
363C22—28no—
364C22—22no—
365C22—216no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—216no—
370C22—48no—
371C22—48no—
372C22—48no—
373C22—48no—
374C22C2×#340116yes—
375C22C22×#9511yescharacteristic
376C22#288312yes—
377C22C22×#10211yescharacteristic
378C22C22×#10211yescharacteristic
379C22C22×#10311yescharacteristic
380C22C22×#10311yescharacteristic
381C22#290112yes—
382C22C22×#10611yescharacteristic
383C22C22×#10711yescharacteristic
384C4—24no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—28no—
390C4—44no—
391C4—82no—
392C4—84no—
393C4—88no—
394C4C2×#34018yes—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—216no—
400C2#1761312yes—
401C2#1761412yes—
402C2#2034311yescharacteristic
403C2C22×#34011yescharacteristic
404C2#2035411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8026C2semidirect
5. #8023C2semidirect
6. #8014C2semidirect
7. C2×#1294C2semidirect
12. #2972C4semidirect
18. #2349C4semidirect
19. D4×C4⋊C4C4semidirect
53. C22×C4⋊C4C2×C4semidirect
62. #431C2×C4semidirect
66. C2×C22⋊C8C2×C4semidirect
73. #384C2×C4semidirect
76. #382C2×C4semidirect
81. #380C2×C4semidirect
82. #380C2×C4semidirect
86. C4×C4⋊C4C2×C4semidirect
87. C4×C4⋊C4C2×C4semidirect
92. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#341812
C2×#23492
#80262
#80232
#80142
C2×#12942