Finite Group Databasebeta

← #194836

Subgroups — #194836

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#21759C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9023C214yesmaximal, complemented
6#9015C214yesmaximal, complemented
7#8023C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117—21no—
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2627—24no—
13#2627C418yescomplemented
14C2×#400C2212yes—
15C2×#398—22no—
16C2×#395C2212yes—
17#2316—24no—
18#2316—22no—
19#2316C2214yes—
20C2×#389—24no—
21C2×#385—21no—
22C2×#381—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26#1892—24no—
27#1872—24no—
28#1824C2214yes—
29#1787C2214yes—
30C2×#324—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C2×C414yes—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—44no—
46C2×#117—44no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2×C414yes—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D414yes—
58C22×C4⋊C4D414yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42—22no—
65C2×#103—28no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70#400—24no—
71#400C2×C418yescomplemented
72#398—216no—
73#395—24no—
74#395C2×C418yescomplemented
75#389—216no—
76#389—216no—
77#385—28no—
78#381—28no—
79#380—24no—
80#380—22no—
81#380—22no—
82#380—24no—
83#380—22no—
84#380C2×C414yes—
85#380D414yes—
86#380C2314yes—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95C2312yes—
93#340—28no—
94#324—28no—
95C25C2×D411yescharacteristic
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—82no—
100C22×Q8C22×C412yes—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—82no—
114C23×C4—82no—
115C23×C4C22⋊C414yes—
116C23×C4C22⋊C414yes—
117C23×C4SD1614yes—
118C23×C4Q1614yes—
119C23×C4C22×C412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C22×C8—24no—
129C22×C8—44no—
130C22×C8—84no—
131C22⋊Q8—28no—
132C22⋊Q8—28no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135#117—48no—
136#117—416no—
137#117—48no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C22⋊C418yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D418yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C42—28no—
179C2×C42—48no—
180#103—88no—
181C22⋊C8—24no—
182C22⋊C8—24no—
183C22⋊C8—44no—
184#95—28no—
185#95—28no—
186#95—28no—
187#95—28no—
188#95—24no—
189#95—24no—
190#95—22no—
191#95C22×C414yes—
192C24—42no—
193C24—42no—
194C24C2×C22⋊C412yes—
195C24#12011yescharacteristic
196C24#12011yescharacteristic
197C24#12312yes—
198C24C2×SD1612yes—
199C24C2×Q1612yes—
200C24AGL(1,ℤ/8)12yes—
201C24#13712yes—
202C24C22×D411yescharacteristic
203C2×Q8—24no—
204C2×Q8—48no—
205C2×Q8—42no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—84no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—28no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—28no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—82no—
254C22×C4—84no—
255C22×C4#103116yescomplemented
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C4×D414yes—
260C22×C4C4×D414yes—
261C22×C4#12012yes—
262C22×C4#12114yes—
263C22×C4#12114yes—
264C22×C4#12312yes—
265C22×C4C2×SD1614yes—
266C22×C4C2×Q1614yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4112yes—
271C2×C8—48no—
272C2×C8—44no—
273C2×C8—88no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—216no—
277C4⋊C4—24no—
278C4⋊C4—48no—
279C4⋊C4—416no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4—88no—
285C4⋊C4—88no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C42—416no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23#38612yes—
308C23#38612yes—
309C23C2×#10314yes—
310C23#41714yes—
311C23#45012yes—
312C23#45012yes—
313C23#45112yes—
314C23#45112yes—
315C23#48314yes—
316C23#48414yes—
317C23C22×C22⋊C411yescharacteristic
318C23C2×#12011yescharacteristic
319C23C2×#12311yescharacteristic
320C23C22×SD1611yescharacteristic
321C23C22×Q1611yescharacteristic
322C23C2×AGL(1,ℤ/8)11yescharacteristic
323C23C2×#13711yescharacteristic
324Q8—48no—
325Q8—84no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—216no—
330C2×C4—216no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—816no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#38614yes—
369C2×C4C2×#10318yes—
370C2×C4C2×C4×D412yes—
371C2×C4C2×#12112yes—
372C8—84no—
373C22—24no—
374C22—22no—
375C22—22no—
376C22—24no—
377C22—22no—
378C22—22no—
379C22—24no—
380C22—24no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—22no—
387C22—21no—
388C22—22no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22#187214yes—
394C22#187214yes—
395C22C2×#38611yescharacteristic
396C22C22×#10311yescharacteristic
397C22C2×#41711yescharacteristic
398C22C2×#45011yescharacteristic
399C22C2×#45111yescharacteristic
400C22#302012yes—
401C22#302612yes—
402C22C2×#48311yescharacteristic
403C22C2×#48411yescharacteristic
404C4—28no—
405C4—22no—
406C4—22no—
407C4—24no—
408C4—44no—
409C4—84no—
410C4—88no—
411C4—82no—
412C4—82no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—24no—
418C2—24no—
419C2#1829312yes—
420C2#1829512yes—
421C2C2×#187211yescharacteristic
422C2#2095411yescharacteristic
423C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
5. #9023C2semidirect
6. #9015C2semidirect
13. #2627C4semidirect
71. #400C2×C4semidirect
74. #395C2×C4semidirect
255. C22×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#23162
#217592
C22×#3242
#90232
#90152
#80232