Finite Group Databasebeta

← #194763

Subgroups — #194763

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3C2×#1823C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9028C214yesmaximal, complemented
6#9026C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C22×#123—21no—
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2486—24no—
13#2486C418yescomplemented
14C2×#398C2212yes—
15C2×#397C2212yes—
16#2353—24no—
17#2352—24no—
18#2313C2212yes—
19#2312C2212yes—
20#2308—21no—
21C2×#389—21no—
22#2283—21no—
23#2280—21no—
24C2×#381—21no—
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28#2273—21no—
29#1823—24no—
30#1823C2218yes—
31C2×#340—24no—
32#1459—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×D4D411yescharacteristic
37C24×C4—21no—
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8—22no—
41C2×#123—28no—
42C2×#123—48no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4D414yes—
48C2×#117—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C414yes—
57C22×C4⋊C4D414yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42D411yescharacteristic
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70#398—24no—
71#398C2×C418yescomplemented
72#397—24no—
73#397C2×C418yescomplemented
74#389—28no—
75#388—28no—
76#387—28no—
77#386—28no—
78#383—22no—
79#382—22no—
80#381—22no—
81#381—22no—
82#381—28no—
83#380—22no—
84#380—28no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380C2×C414yes—
89#380D414yes—
90#380C2314yes—
91#379—22no—
92C4×C4⋊C4—22no—
93C4×C22⋊C4—28no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97#376—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95C2312yes—
102#340—216no—
103#324—24no—
104#323—24no—
105C25C2×D411yescharacteristic
106C25C2×D411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—24no—
110C22×D4—24no—
111C22×D4C22⋊C414yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4C22⋊C414yes—
126C23×C4C22×C412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4#4114yes—
135C23×C4C2411yescharacteristic
136C22×C8—24no—
137C22×C8—48no—
138C22×C8—44no—
139#123—432no—
140C4×D4—28no—
141C4×D4—28no—
142C4×D4—28no—
143C4×D4—28no—
144#117—84no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D418yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4C22⋊C414yes—
183C2×C22⋊C4C22×C412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42C22⋊C414yes—
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#95C22×C414yes—
204C24—21no—
205C24—22no—
206C24—21no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24C2×C22⋊C412yes—
213C24C2×C22⋊C412yes—
214C24#12012yes—
215C24#12312yes—
216C24#13514yes—
217C24AGL(1,ℤ/8)14yes—
218C24C22×D411yescharacteristic
219C2×D4—28no—
220C2×D4—28no—
221C2×D4—28no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—48no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—28no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—84no—
261C22×C4—82no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4—88no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C4×D414yes—
272C22×C4C4×D414yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12114yes—
276C22×C4#12114yes—
277C22×C4#12312yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C2×C8—48no—
283C2×C8—44no—
284C2×C8—88no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—28no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—416no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C22⋊C4—28no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—416no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—48no—
302C22⋊C4—48no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—24no—
306C42—24no—
307C42—84no—
308C42—84no—
309C23—22no—
310C23—24no—
311C23—22no—
312C23—24no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—48no—
321C23—48no—
322C23—42no—
323C23—42no—
324C23—48no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—84no—
332C23#38614yes—
333C23#41614yes—
334C23#41814yes—
335C23#44914yes—
336C23#44914yes—
337C23#48218yes—
338C23C22×C22⋊C411yescharacteristic
339C23C2×#12011yescharacteristic
340C23C2×#12311yescharacteristic
341C23C2×#13512yes—
342C23C2×AGL(1,ℤ/8)12yes—
343D4—416no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—48no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—84no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—88no—
380C2×C4—84no—
381C2×C4—816no—
382C2×C4—84no—
383C2×C4—168no—
384C2×C4#38612yes—
385C2×C4#38612yes—
386C2×C4#45814yes—
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12112yes—
389C8—84no—
390C22—24no—
391C22—28no—
392C22—24no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—24no—
397C22—21no—
398C22—21no—
399C22—24no—
400C22—44no—
401C22—44no—
402C22—48no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—82no—
408C22—84no—
409C22—84no—
410C22#187518yes—
411C22C2×#38611yescharacteristic
412C22C2×#41611yescharacteristic
413C22C2×#41811yescharacteristic
414C22C2×#44912yes—
415C22#302212yes—
416C22C2×#45812yes—
417C22C2×#48212yes—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—84no—
425C4—88no—
426C4—84no—
427C4#189014yes—
428C2—21no—
429C2—22no—
430C2—21no—
431C2—28no—
432C2—82no—
433C2#1820614yes—
434C2C2×#187511yescharacteristic
435C2#2095511yescharacteristic
436C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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. C2×#1823C2semidirect
5. #9028C2semidirect
6. #9026C2semidirect
7. #8023C2semidirect
13. #2486C4semidirect
71. #398C2×C4semidirect
73. #397C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#341852
C2×#18232
C2×#14592
#90282
#90262
#80232