Finite Group Databasebeta

← #194785

Subgroups — #194785

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#20700C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9027C214yesmaximal, complemented
6#9026C214yesmaximal, complemented
7#8020C212yesmaximal
8C23×C4⋊C4—21no—
9C23×C4⋊C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#407C2212yes—
12C2×#398—24no—
13C2×#398—21no—
14C2×#398—21no—
15C2×#398C418yescomplemented
16C2×#398C2212yes—
17C2×#398C2212yes—
18#2345—24no—
19#2344—24no—
20#2316C2212yes—
21#2313—21no—
22#2313C2212yes—
23#2280—21no—
24#2280—21no—
25#2280C2212yes—
26#2274—21no—
27C22×#95—21no—
28C22×#95C2211yescharacteristic
29#1824—24no—
30#1823C2218yes—
31#1736—24no—
32#1463—24no—
33C2×#324C2212yes—
34C2×#324C2212yes—
35#1294C2218yes—
36C24×C4—21no—
37C24×C4—21no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117D414yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—28no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—48no—
57C22×C4⋊C4C2×C414yes—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D414yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65#407—24no—
66#407—24no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2314yes—
69#398—28no—
70#398—28no—
71#398—24no—
72#398—24no—
73#398C2×C418yes—
74#398C2×C418yes—
75#389—28no—
76#387—28no—
77#387—28no—
78#381—22no—
79#381—22no—
80#381—22no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380D414yes—
86#380C2314yes—
87#379—22no—
88C4×C4⋊C4—28no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C2×#95—28no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2×C414yes—
101C2×#95C2312yes—
102#340—216no—
103#324—24no—
104#324—24no—
105C25C2×D411yescharacteristic
106C23×C4—28no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4—42no—
128C23×C4C22⋊C414yes—
129C23×C4C22⋊C414yes—
130C23×C4C22×C412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4#4114yes—
140C23×C4C2411yescharacteristic
141C2×M16—48no—
142C2×M16—44no—
143C22×C8—24no—
144#117—44no—
145#117—44no—
146#117—48no—
147#117—84no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—432no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4C22⋊C414yes—
178C2×C4⋊C4C22⋊C414yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D418yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—44no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—28no—
199#95—28no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—24no—
204#95C22×C414yes—
205C24—22no—
206C24—22no—
207C24—24no—
208C24—24no—
209C24C2×C22⋊C412yes—
210C24#12011yescharacteristic
211C24#12011yescharacteristic
212C24#12312yes—
213C24C22×D411yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—28no—
218C22×C4—216no—
219C22×C4—28no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—432no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4—82no—
265C22×C4—88no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C4×D414yes—
274C22×C4C4×D414yes—
275C22×C4#12012yes—
276C22×C4#12114yes—
277C22×C4#12114yes—
278C22×C4#12312yes—
279C22×C4AGL(1,ℤ/8)14yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4112yes—
284M16—88no—
285C2×C8—48no—
286C2×C8—44no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—48no—
295C4⋊C4—416no—
296C4⋊C4—48no—
297C4⋊C4—44no—
298C4⋊C4—48no—
299C4⋊C4—416no—
300C4⋊C4—84no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—48no—
304C22⋊C4—48no—
305C42—48no—
306C42—84no—
307C42—84no—
308C23—216no—
309C23—216no—
310C23—24no—
311C23—22no—
312C23—22no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23#38612yes—
324C23#38612yes—
325C23#45614yes—
326C23#45814yes—
327C23C22×C22⋊C411yescharacteristic
328C23C2×#12011yescharacteristic
329C23C2×#12311yescharacteristic
330C23C2×AGL(1,ℤ/8)12yes—
331C23#57512yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—432no—
356C2×C4—44no—
357C2×C4—416no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—84no—
375C2×C4—88no—
376C2×C4—88no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4#38614yes—
380C2×C4#41814yes—
381C2×C4C2×C4×D412yes—
382C2×C4C2×#12112yes—
383C8—84no—
384C22—28no—
385C22—216no—
386C22—28no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—24no—
396C22—24no—
397C22—28no—
398C22—28no—
399C22#188614yes—
400C22#189014yes—
401C22C2×#38611yescharacteristic
402C22C2×#41811yescharacteristic
403C22#289611yescharacteristic
404C22#300612yes—
405C22C2×#45612yes—
406C22C2×#45812yes—
407C22#309412yes—
408C4—24no—
409C4—22no—
410C4—22no—
411C4—24no—
412C4—48no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—84no—
417C4—88no—
418C2—21no—
419C2—22no—
420C2—21no—
421C2—24no—
422C2—24no—
423C2#1817114yes—
424C2#2091211yescharacteristic
425C2C2×#188611yescharacteristic
426C2C2×#189011yescharacteristic
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. #9027C2semidirect
6. #9026C2semidirect
15. C2×#398C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
#341782
#207002
C2×#14632
#90272
#90262
#80202