Finite Group Databasebeta

← #194759

Subgroups — #194759

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#20699C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9027C212yesmaximal, complemented
6#9026C212yesmaximal, complemented
7#9023C212yesmaximal, complemented
8#9022C212yesmaximal, complemented
9#8024C212yesmaximal
10C22×#117C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2484—24no—
14#2484C418yescomplemented
15C2×#400C2211yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#398C2211yescharacteristic
18C2×#398C2211yescharacteristic
19#2345—24no—
20#2344—24no—
21#2316C2212yes—
22#2313—21no—
23#2313C2212yes—
24C2×#387—21no—
25C2×#387—21no—
26C2×#385—21no—
27#2280—21no—
28#2280—21no—
29#2280C2212yes—
30C2×#381—21no—
31C2×#381C2211yescharacteristic
32#2274—21no—
33#1825—24no—
34#1824C2214yes—
35#1823C2214yes—
36#1735—24no—
37C2×#324—24no—
38C2×#324C2212yes—
39C2×#324C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—24no—
52C2×#117C2×C414yes—
53C2×#117D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—21no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8C2314yes—
74#400—22no—
75#400—22no—
76#400C2×C414yes—
77#400C2×C414yes—
78#398—22no—
79#398—22no—
80#398C2×C414yes—
81#398C2×C414yes—
82#389—28no—
83#387—28no—
84#387—28no—
85#387—28no—
86#387—28no—
87#385—28no—
88#381—22no—
89#381—28no—
90#381—22no—
91#381—22no—
92#381—22no—
93#381C2×C414yes—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380D414yes—
99#380C2314yes—
100#379—22no—
101C4×C4⋊C4—28no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
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#324—24no—
121C25C2×D411yescharacteristic
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4C22⋊C414yes—
137C23×C4C22⋊C414yes—
138C23×C4C22×C412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150C22×C8—48no—
151C22×C8—44no—
152#117—28no—
153#117—28no—
154#117—44no—
155#117—44no—
156#117—44no—
157#117—44no—
158#117—48no—
159#117—84no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D418yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—42no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—42no—
207C2×C42—42no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—44no—
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—24no—
221#95—24no—
222#95—24no—
223#95—21no—
224#95—21no—
225#95—24no—
226#95C22×C412yes—
227#95C22×C412yes—
228C24—22no—
229C24—22no—
230C24—24no—
231C24—42no—
232C24C2×C22⋊C412yes—
233C24#12011yescharacteristic
234C24#12011yescharacteristic
235C24#12312yes—
236C24C22×D411yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—28no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—28no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—84no—
287C22×C4—82no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C4×D414yes—
294C22×C4C4×D414yes—
295C22×C4#12012yes—
296C22×C4#12114yes—
297C22×C4#12114yes—
298C22×C4#12312yes—
299C22×C4#13514yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4112yes—
304C2×C8—48no—
305C2×C8—44no—
306C2×C8—88no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—84no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—48no—
329C42—48no—
330C42—48no—
331C42—84no—
332C42—84no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—48no—
348C23—48no—
349C23#38612yes—
350C23#38612yes—
351C23#45814yes—
352C23C22×C22⋊C411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12311yescharacteristic
355C23C2×#13512yes—
356C23#57512yes—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#38614yes—
410C2×C4#41614yes—
411C2×C4C2×C4×D412yes—
412C2×C4C2×#12112yes—
413C8—84no—
414C22—24no—
415C22—24no—
416C22—28no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—44no—
428C22—48no—
429C22—44no—
430C22#189014yes—
431C22C2×#38611yescharacteristic
432C22C2×#41611yescharacteristic
433C22#289611yescharacteristic
434C22#300212yes—
435C22#302012yes—
436C22C2×#45812yes—
437C22#309112yes—
438C4—24no—
439C4—22no—
440C4—22no—
441C4—24no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—82no—
446C4—82no—
447C4—88no—
448C4—82no—
449C4—82no—
450C2—21no—
451C2—21no—
452C2—22no—
453C2—24no—
454C2—42no—
455C2#1820114yes—
456C2#2090811yescharacteristic
457C2#2095411yescharacteristic
458C2C2×#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
7. #9023C2semidirect
8. #9022C2semidirect
14. #2484C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
#341782
#206992
C22×#3242
#90272
#90262
#90232
#90222
#80242