Finite Group Databasebeta

← #194734

Subgroups — #194734

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#20700C212yesmaximal, complemented
4C2×#1465C211yesmaximal, characteristic
5#9022C214yesmaximal, complemented
6#9021C214yesmaximal, complemented
7#8016C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2487—24no—
12#2487C418yescomplemented
13C2×#400C2212yes—
14C2×#399C2212yes—
15#2318—24no—
16#2315C2212yes—
17#2313C2212yes—
18#2312—21no—
19#2308—21no—
20C2×#389—21no—
21#2290—24no—
22C2×#386—21no—
23C2×#385—21no—
24#2283—21no—
25#2280C2212yes—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1824C2218yes—
30#1823—24no—
31#1736—24no—
32#1465—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×D4D411yescharacteristic
37C24×C4—21no—
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C414yes—
45C2×C4×D4D414yes—
46C2×#117D414yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C414yescomplemented
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42—22no—
63C22×C42D411yescharacteristic
64#407—24no—
65#407—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#400—24no—
69#400C2×C418yescomplemented
70#399—24no—
71#399C2×C418yes—
72#389—28no—
73#388—28no—
74#387—28no—
75#386—28no—
76#386—28no—
77#385—28no—
78#385—28no—
79#383—22no—
80#382—22no—
81#381—22no—
82#381—22no—
83#380—28no—
84#380—22no—
85#380—22no—
86#380C2×C414yescomplemented
87#380D414yes—
88#380C2314yes—
89#379—22no—
90C4×C4⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96#376—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2312yes—
101#340—216no—
102#324—24no—
103#323—24no—
104C25C2×D411yescharacteristic
105C25C2×D411yescharacteristic
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—24no—
109C22×D4—42no—
110C22×D4C22⋊C414yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—84no—
123C23×C4C22⋊C414yes—
124C23×C4C22×C412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4#4114yes—
133C23×C4C2411yescharacteristic
134C2×M16—48no—
135C2×M16—44no—
136C22×C8—24no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4—28no—
141#117—48no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D418yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—84no—
178C2×C22⋊C4C22⋊C414yes—
179C2×C22⋊C4C22×C412yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—84no—
187C2×C42C22⋊C414yes—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—44no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—28no—
196#95—28no—
197#95—24no—
198#95—22no—
199#95—24no—
200#95C22×C414yes—
201C24—21no—
202C24—22no—
203C24—22no—
204C24—21no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24C2×C22⋊C412yes—
211C24C2×C22⋊C412yes—
212C24#12012yes—
213C24#12312yes—
214C24#13714yes—
215C24C22×D411yescharacteristic
216C2×D4—28no—
217C2×D4—28no—
218C2×D4—44no—
219C2×D4—48no—
220C2×D4—48no—
221C22×C4—24no—
222C22×C4—28no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—48no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—48no—
254C22×C4—84no—
255C22×C4—82no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—88no—
259C22×C4—88no—
260C22×C4—88no—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C4×D414yes—
266C22×C4C4×D414yes—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12114yes—
270C22×C4#12114yes—
271C22×C4#12312yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D412yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276M16—88no—
277C2×C8—48no—
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—28no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—84no—
288C4⋊C4—88no—
289C22⋊C4—28no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C22⋊C4—88no—
299C42—24no—
300C42—24no—
301C42—48no—
302C42—84no—
303C23—22no—
304C23—22no—
305C23—24no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—24no—
310C23—24no—
311C23—24no—
312C23—22no—
313C23—24no—
314C23—22no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—48no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—48no—
324C23—42no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23—84no—
329C23#38614yes—
330C23#41914yes—
331C23C22×C22⋊C411yescharacteristic
332C23C2×#12011yescharacteristic
333C23C2×#12311yescharacteristic
334C23C2×#13712yes—
335C23#57512yes—
336D4—416no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—88no—
373C2×C4—84no—
374C2×C4—816no—
375C2×C4—816no—
376C2×C4—84no—
377C2×C4#38612yes—
378C2×C4#38612yes—
379C2×C4#45814yes—
380C2×C4C2×C4×D412yes—
381C2×C4C2×#12112yes—
382C8—84no—
383C22—21no—
384C22—21no—
385C22—24no—
386C22—24no—
387C22—21no—
388C22—21no—
389C22—24no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22—48no—
397C22—42no—
398C22—48no—
399C22—48no—
400C22—42no—
401C22—82no—
402C22—84no—
403C22C2×#38611yescharacteristic
404C22C2×#41911yescharacteristic
405C22#289611yescharacteristic
406C22#300712yes—
407C22#302112yes—
408C22C2×#45812yes—
409C22#309512yes—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—84no—
417C4—84no—
418C4—88no—
419C4#189014yes—
420C2—21no—
421C2—21no—
422C2—22no—
423C2—44no—
424C2—44no—
425C2#1814114yes—
426C2#2091111yescharacteristic
427C2#2095611yescharacteristic
428C2C2×#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. #20700C2semidirect
5. #9022C2semidirect
6. #9021C2semidirect
7. #8016C2semidirect
12. #2487C4semidirect
54. C22×C4⋊C4C2×C4semidirect
69. #400C2×C4semidirect
86. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#340572
#207002
C2×#14652
#90222
#90212
#80162