Finite Group Databasebeta

← #194834

Subgroups — #194834

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2638C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#21762C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9026C212yesmaximal, complemented
6#9025C212yesmaximal, complemented
7#9014C212yesmaximal, complemented
8#9013C212yesmaximal, complemented
9#8024C212yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2638—24no—
14#2638C418yescomplemented
15C2×#398C2211yescharacteristic
16C2×#397—21no—
17C2×#397C2211yescharacteristic
18C2×#395C2211yescharacteristic
19C2×#394—21no—
20C2×#393C2211yescharacteristic
21#2318—24no—
22#2315C2212yes—
23#2313C2212yes—
24#2312—21no—
25#2308—21no—
26#2290—24no—
27C2×#387—21no—
28C2×#385—21no—
29#2283—21no—
30#2280C2212yes—
31C2×#381C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1892—24no—
34#1873—24no—
35#1823C2214yes—
36#1787C2214yes—
37#1459—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8C2311yescharacteristic
43C23×D4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8C2×C414yescomplemented
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4C2×C414yes—
56C2×C4×D4D414yes—
57C2×#117D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42—22no—
71C22×C42D411yescharacteristic
72C2×#103—28no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2314yes—
77#398—22no—
78#398C2×C414yescomplemented
79#397—28no—
80#397—22no—
81#397C2×C414yes—
82#395—22no—
83#395C2×C414yescomplemented
84#394—28no—
85#393—22no—
86#393C2×C414yes—
87#388—28no—
88#387—28no—
89#387—28no—
90#386—28no—
91#385—28no—
92#385—28no—
93#383—22no—
94#382—22no—
95#381—22no—
96#381—22no—
97#381—22no—
98#381C2×C414yescomplemented
99#380—22no—
100#380D414yes—
101#380C2314yes—
102#379—22no—
103C4×C4⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109#376—22no—
110C2×#95—22no—
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#323—24no—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—82no—
127C22×Q8C22×C412yes—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—42no—
132C22×D4C22⋊C414yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C22⋊C414yes—
144C23×C4C22×C412yes—
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
154C22×C8—24no—
155C22×C8—44no—
156C22×C8—84no—
157C22⋊Q8—28no—
158C22⋊Q8—28no—
159C22⋊Q8—44no—
160C22⋊Q8—44no—
161C4×D4—28no—
162C4×D4—28no—
163C4×D4—28no—
164C4×D4—28no—
165#117—84no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D418yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4C22⋊C414yes—
204C2×C22⋊C4C22×C412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—84no—
212C2×C42C22⋊C414yes—
213#103—88no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—44no—
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—24no—
227#95—21no—
228#95—21no—
229#95—24no—
230#95C22×C412yes—
231#95C22×C412yes—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C412yes—
243C24#12012yes—
244C24#12312yes—
245C24#13514yes—
246C24C22×D411yescharacteristic
247C2×Q8—24no—
248C2×Q8—48no—
249C2×Q8—42no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—84no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—44no—
256C2×D4—48no—
257C2×D4—48no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—28no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C4×D414yes—
308C22×C4C4×D414yes—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12114yes—
312C22×C4#12114yes—
313C22×C4#12312yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D412yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4112yes—
318C2×C8—48no—
319C2×C8—44no—
320C2×C8—88no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—48no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C42—24no—
344C42—24no—
345C42—48no—
346C42—84no—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—48no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—84no—
370C23—84no—
371C23—84no—
372C23—81no—
373C23#38614yes—
374C23#41614yes—
375C23C22×C22⋊C411yescharacteristic
376C23C2×#12011yescharacteristic
377C23C2×#12311yescharacteristic
378C23C2×#13512yes—
379C23#57512yes—
380Q8—48no—
381Q8—84no—
382D4—416no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—84no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—816no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#38612yes—
433C2×C4#38612yes—
434C2×C4C2×C4×D412yes—
435C2×C4C2×#12112yes—
436C8—84no—
437C22—24no—
438C22—24no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—24no—
447C22—44no—
448C22—48no—
449C22—48no—
450C22—42no—
451C22—42no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—84no—
456C22—84no—
457C22C2×#38611yescharacteristic
458C22C2×#41611yescharacteristic
459C22#289611yescharacteristic
460C22#300212yes—
461C22#302212yes—
462C22#302612yes—
463C22#309112yes—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—82no—
472C4—82no—
473C4—88no—
474C4—82no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—44no—
480C2—44no—
481C2#1829114yes—
482C2#2090811yescharacteristic
483C2#2095511yescharacteristic
484C2#2096011yescharacteristic
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. #21762C2semidirect
5. #9026C2semidirect
6. #9025C2semidirect
7. #9014C2semidirect
8. #9013C2semidirect
9. #8024C2semidirect
14. #2638C4semidirect
51. C2×C22⋊Q8C2×C4semidirect
78. #398C2×C4semidirect
83. #395C2×C4semidirect
98. #381C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26382
#340572
#217622
C2×#14592
#90262
#90252
#90142
#90132
#80242