Finite Group Databasebeta

← #194891

Subgroups — #194891

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2#34177C211yesmaximal, characteristic
3#21836C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9026C212yesmaximal, complemented
6#9025C212yesmaximal
7#9022C212yesmaximal, complemented
8#9021C212yesmaximal
9#8013C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#431C2212yes—
13#2611—24no—
14#2611C418yescomplemented
15C2×#400C2211yescharacteristic
16C2×#399C2211yescharacteristic
17C2×#398C2211yescharacteristic
18C2×#397C2211yescharacteristic
19#2343—24no—
20#2343—24no—
21#2315C2212yes—
22#2313—21no—
23#2313C2212yes—
24C2×#389—21no—
25C2×#388—21no—
26C2×#387—21no—
27C2×#385—21no—
28#2283—21no—
29#2283C2212yes—
30C2×#381—21no—
31#2273—21no—
32C2×#376C2211yescharacteristic
33#1991—24no—
34#1989—24no—
35#1824C2214yes—
36#1823C2214yes—
37#1458—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C24×C4D411yescharacteristic
43C2×#125—22no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—24no—
47C2×#125C2×C414yescomplemented
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—24no—
51C2×#117C2×C414yes—
52C2×#117D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69#431—24no—
70#431—24no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×C4×C8—22no—
74#400—22no—
75#400C2×C414yescomplemented
76#399—22no—
77#399C2×C414yes—
78#398—22no—
79#398C2×C414yescomplemented
80#397—22no—
81#397C2×C414yes—
82#389—28no—
83#388—28no—
84#387—28no—
85#387—28no—
86#387—28no—
87#385—28no—
88#385—28no—
89#383—22no—
90#383—22no—
91#382—22no—
92#382D414yes—
93#381—22no—
94#381—22no—
95#381—22no—
96#381—22no—
97#380—22no—
98#380C2314yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C4×C22⋊C4—28no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104#376—22no—
105#376—22no—
106#376—22no—
107#376C2×C414yescomplemented
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95C2311yescharacteristic
116C2×#95C2311yescharacteristic
117#340—28no—
118#340—28no—
119#324—24no—
120#323—24no—
121C25C2×D411yescharacteristic
122C23×C4—24no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—42no—
132C23×C4C22⋊C414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140#125—28no—
141#125—28no—
142#125—44no—
143#125—44no—
144#117—28no—
145#117—28no—
146#117—28no—
147#117—28no—
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—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D418yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C22⋊C414yes—
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—44no—
198C2×C42C22⋊C414yes—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42#4114yes—
204C4⋊C8—44no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C4×C8—44no—
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—24no—
218#95—24no—
219#95—24no—
220#95—21no—
221#95—21no—
222#95—24no—
223#95C22×C412yes—
224#95C22×C412yes—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24C2×C22⋊C412yes—
231C24#13514yes—
232C24C22×D411yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—28no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—48no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—48no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12012yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D412yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C2×C8—48no—
291C2×C8—44no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—84no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—28no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—48no—
314C42—24no—
315C42—24no—
316C42—48no—
317C42—84no—
318C42—84no—
319C42C4×D414yes—
320C42C4×D414yes—
321C42#12114yes—
322C42#12114yes—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—22no—
327C23—24no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—48no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—84no—
336C23—81no—
337C23#41614yes—
338C23C22×C22⋊C411yescharacteristic
339C23C2×#12011yescharacteristic
340C23C2×#12311yescharacteristic
341C23C2×#13512yes—
342C23#57512yes—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—168no—
392C2×C4#38612yes—
393C2×C4#38612yes—
394C2×C4#38614yes—
395C2×C4C2×C4×D412yes—
396C2×C4C2×#12112yes—
397C8—84no—
398C22—21no—
399C22—21no—
400C22—24no—
401C22—24no—
402C22—21no—
403C22—21no—
404C22—24no—
405C22—44no—
406C22—42no—
407C22—48no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22C2×#38611yescharacteristic
412C22C2×#41611yescharacteristic
413C22#289611yescharacteristic
414C22#300212yes—
415C22#301812yes—
416C22#302812yes—
417C22#309112yes—
418C4—22no—
419C4—22no—
420C4—24no—
421C4—22no—
422C4—22no—
423C4—44no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—88no—
431C2—21no—
432C2—21no—
433C2—22no—
434C2—44no—
435C2#1839314yes—
436C2#2090811yescharacteristic
437C2#2095211yescharacteristic
438C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×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
3. #21836C2semidirect
5. #9026C2semidirect
7. #9022C2semidirect
9. #8013C2semidirect
14. #2611C4semidirect
47. C2×#125C2×C4semidirect
75. #400C2×C4semidirect
79. #398C2×C4semidirect
107. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
#341772
#218362
C2×#14582
#90262
#90252
#90222
#90212
#80132