Finite Group Databasebeta

← #194908

Subgroups — #194908

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#21768C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9023C212yesmaximal, complemented
6#9022C212yesmaximal, complemented
7#9015C212yesmaximal, complemented
8#9014C212yesmaximal, complemented
9#8016C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#407C2212yes—
13#2627—24no—
14#2627C418yescomplemented
15C2×#400C2211yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#398—21no—
18C2×#398—21no—
19C2×#395C2211yescharacteristic
20C2×#395C2211yescharacteristic
21#2345—24no—
22#2344—24no—
23#2316C2212yes—
24#2313—21no—
25#2313C2212yes—
26C2×#385—21no—
27#2280—21no—
28#2280—21no—
29#2280C2212yes—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32#2274—21no—
33#1894—24no—
34#1873—24no—
35#1824C2214yes—
36#1787C2214yes—
37#1463—24no—
38C2×#324C2212yes—
39C2×#324C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8C2311yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8C2×C414yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117D414yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D414yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C2×#103—28no—
72#407—24no—
73#407—24no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2314yes—
76#400—22no—
77#400—22no—
78#400C2×C414yes—
79#400C2×C414yes—
80#398—28no—
81#398—28no—
82#395—22no—
83#395—22no—
84#395C2×C414yes—
85#395C2×C414yes—
86#389—28no—
87#387—28no—
88#387—28no—
89#385—28no—
90#381—22no—
91#381—28no—
92#381—22no—
93#381—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380C2×C414yes—
100#380D414yes—
101#380C2314yes—
102#379—22no—
103C4×C4⋊C4—28no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2311yescharacteristic
117C2×#95C2311yescharacteristic
118#340—28no—
119#324—24no—
120#324—24no—
121C25C2×D411yescharacteristic
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—82no—
126C22×Q8C22×C412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4C22⋊C414yes—
142C23×C4C22⋊C414yes—
143C23×C4C22×C412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4#4114yes—
153C23×C4C2411yescharacteristic
154C2×M16—44no—
155C2×M16—84no—
156C22×C8—24no—
157C22⋊Q8—28no—
158C22⋊Q8—28no—
159C22⋊Q8—44no—
160C22⋊Q8—44no—
161#117—44no—
162#117—44no—
163#117—84no—
164#117—84no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C22⋊C414yes—
192C2×C4⋊C4C22⋊C414yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D418yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—42no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211#103—88no—
212C22⋊C8—24no—
213C22⋊C8—24no—
214C22⋊C8—44no—
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—24no—
224#95—24no—
225#95—24no—
226#95—21no—
227#95—21no—
228#95—24no—
229#95C22×C412yes—
230#95C22×C412yes—
231C24—22no—
232C24—22no—
233C24—42no—
234C24—42no—
235C24C2×C22⋊C412yes—
236C24#12011yescharacteristic
237C24#12011yescharacteristic
238C24#12312yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—48no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—84no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—82no—
296C22×C4—84no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C4×D414yes—
303C22×C4C4×D414yes—
304C22×C4#12012yes—
305C22×C4#12114yes—
306C22×C4#12114yes—
307C22×C4#12312yes—
308C22×C4#13714yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4112yes—
313M16—88no—
314C2×C8—48no—
315C2×C8—44no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—48no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C42—48no—
339C42—84no—
340C42—84no—
341C23—22no—
342C23—22no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—48no—
357C23—48no—
358C23#38612yes—
359C23#38612yes—
360C23C22×C22⋊C411yescharacteristic
361C23C2×#12011yescharacteristic
362C23C2×#12311yescharacteristic
363C23C2×#13712yes—
364C23#57512yes—
365Q8—48no—
366Q8—84no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#38614yes—
420C2×C4#41914yes—
421C2×C4C2×C4×D412yes—
422C2×C4C2×#12112yes—
423C8—84no—
424C22—24no—
425C22—24no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—44no—
437C22—48no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22C2×#38611yescharacteristic
442C22C2×#41911yescharacteristic
443C22#289611yescharacteristic
444C22#300712yes—
445C22#302012yes—
446C22#302612yes—
447C22#309512yes—
448C4—24no—
449C4—22no—
450C4—22no—
451C4—24no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—82no—
457C4—88no—
458C4—82no—
459C4—82no—
460C2—21no—
461C2—22no—
462C2—21no—
463C2—42no—
464C2—42no—
465C2#1834914yes—
466C2#2091111yescharacteristic
467C2#2095411yescharacteristic
468C2#2096011yescharacteristic
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. #9023C2semidirect
6. #9022C2semidirect
7. #9015C2semidirect
8. #9014C2semidirect
14. #2627C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
#341782
#217682
C2×#14632
#90232
#90222
#90152
#90142
#80162