Finite Group Databasebeta

← #194853

Subgroups — #194853

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2634C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#21836C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9027C212yesmaximal
6#9026C212yesmaximal
7#9023C212yesmaximal
8#9022C212yesmaximal
9#8025C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2212yes—
13#2634—24no—
14#2634C418yescomplemented
15C2×#400—21no—
16C2×#400C2211yescharacteristic
17C2×#400C2211yescharacteristic
18C2×#398C2211yescharacteristic
19C2×#398C2211yescharacteristic
20C2×#395—21no—
21#2320—24no—
22#2316C2212yes—
23#2313—21no—
24#2313C2212yes—
25#2310—21no—
26#2295—24no—
27C2×#387—21no—
28C2×#387—21no—
29#2283—21no—
30#2283C2212yes—
31C2×#383C2211yescharacteristic
32C4×#117—21no—
33#1991—24no—
34#1989—24no—
35#1824C2214yes—
36#1823C2214yes—
37#1461—24no—
38C2×#324C2212yes—
39C2×#324C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8D411yescharacteristic
43C2×#125—22no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—24no—
47C2×#125C2×C414yes—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—24no—
51C2×C4×Q8C2×C414yes—
52C2×C4×Q8D414yes—
53C2×#117—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66#431—24no—
67#431—24no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8C2314yes—
70C2×C4×C8—22no—
71#400—28no—
72#400—22no—
73#400—22no—
74#400C2×C414yes—
75#400C2×C414yes—
76#398—22no—
77#398—22no—
78#398C2×C414yes—
79#398C2×C414yes—
80#395—28no—
81#391—28no—
82#389—28no—
83#387—28no—
84#387—28no—
85#387—28no—
86#387—28no—
87#383—22no—
88#383—22no—
89#383—22no—
90#383C2×C414yes—
91#382—22no—
92#382D414yes—
93#381—22no—
94#381—22no—
95#380—22no—
96#380C2314yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103#376—22no—
104#376—22no—
105#376—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95C2311yescharacteristic
115C2×#95C2311yescharacteristic
116C43—22no—
117#340—28no—
118#340—28no—
119#324—24no—
120#324—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8—42no—
125C22×Q8C22⋊C414yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—24no—
138#125—28no—
139#125—28no—
140#125—44no—
141#125—44no—
142C4×Q8—28no—
143C4×Q8—28no—
144C4×Q8—28no—
145C4×Q8—28no—
146#117—44no—
147#117—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C414yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D418yes—
181C2×C22⋊C4—24no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42C22⋊C414yes—
192C2×C42C22⋊C414yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42#4114yes—
198C4⋊C8—44no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C4×C8—44no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—21no—
218#95—21no—
219#95—24no—
220#95C22×C412yes—
221#95C22×C412yes—
222C24C22×D411yescharacteristic
223C2×Q8—28no—
224C2×Q8—28no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—48no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4#12011yescharacteristic
270C22×C4#12012yes—
271C22×C4#12011yescharacteristic
272C22×C4#12312yes—
273C22×C4#12312yes—
274C22×C4#13514yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D412yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4112yes—
279C2×C8—48no—
280C2×C8—44no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—28no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—84no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—48no—
305C42—48no—
306C42—84no—
307C42C4×D414yes—
308C42C4×D414yes—
309C42#12114yes—
310C42#12114yes—
311C23—22no—
312C23—22no—
313C23C22×C22⋊C411yescharacteristic
314C23C2×#12011yescharacteristic
315C23C2×#12311yescharacteristic
316C23C2×#13512yes—
317C23#57512yes—
318Q8—416no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#38612yes—
369C2×C4#38612yes—
370C2×C4#38614yes—
371C2×C4#41614yes—
372C2×C4C2×C4×D412yes—
373C2×C4C2×#12112yes—
374C8—84no—
375C22—24no—
376C22—24no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22C2×#38611yescharacteristic
382C22C2×#41611yescharacteristic
383C22#289611yescharacteristic
384C22#300212yes—
385C22#302012yes—
386C22#302812yes—
387C22#309112yes—
388C4—24no—
389C4—24no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—82no—
398C4—82no—
399C4—88no—
400C4—82no—
401C4—82no—
402C2—21no—
403C2—21no—
404C2—22no—
405C2#1831514yes—
406C2#2090811yescharacteristic
407C2#2095411yescharacteristic
408C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2634C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26342
#340592
#218362
C2×#14612
#90272
#90262
#90232
#90222
#80252