Finite Group Databasebeta

← #196925

Subgroups — #196925

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic
3#21788C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic, complemented
5C2×#1503C211yesmaximal, characteristic, complemented
6#9146C214yesmaximal, complemented
7#9143C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2848C2214yes—
16#2633—22no—
17#2547C412yescomplemented
18#2542C412yescomplemented
19#2536C414yescomplemented
20#2502—21no—
21#2482—22no—
22#2435—21no—
23#2326—22no—
24#2283C2211yescharacteristic
25#2275—24no—
26#2275—24no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—22no—
30C2×C4×C22⋊C4C2212yes—
31C2×#376C2211yescharacteristic
32C22×#95C2212yes—
33#1955—22no—
34#1954—22no—
35#1779C2214yes—
36#1506—24no—
37#1503—24no—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×D4D411yescharacteristic
45C24×C4—22no—
46C24×C4C2312yes—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×C4⋊Q8C2×C411yescharacteristic, complemented
53C2×#125—21no—
54C2×#124—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C418yescomplemented
57C2×C22⋊Q8C2312yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yescomplemented
60C2×C4×D4—216no—
61C2×C4×D4—28no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#431—22no—
82#431—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C22⋊C8—28no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88C2×#97—21no—
89C2×C4×C8—21no—
90#399—24no—
91#398—24no—
92#397—24no—
93#394—24no—
94#391—22no—
95#390—22no—
96#388—28no—
97#385—28no—
98#385—24no—
99#384—22no—
100#383—21no—
101#382C2×C412yescomplemented
102#381—21no—
103#380C2×C418yescomplemented
104#380C2312yes—
105#379—24no—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C412yescomplemented
109C4×C4⋊C4D414yes—
110C4×C22⋊C4—216no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—24no—
114#376—21no—
115#376—22no—
116#376D414yes—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2311yescharacteristic
124#328—28no—
125#328—24no—
126#328—24no—
127C25C2×D412yes—
128C22×Q8—21no—
129C22×Q8C22×C412yes—
130C22×D4—28no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C22⋊C412yes—
134C22×D4C22⋊C412yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—216no—
139C23×C4—28no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—84no—
145C23×C4—84no—
146C23×C4—84no—
147C23×C4C22×C414yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D414yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159#125—42no—
160#125—42no—
161#124—48no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—88no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×D4—264no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C414yes—
198C2×C22⋊C4—216no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4C22⋊C414yes—
207C2×C22⋊C4C22⋊C414yes—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—28no—
213C2×C42—24no—
214C2×C42—82no—
215C2×C42—84no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—88no—
231#97—42no—
232C4×C8—42no—
233#95—28no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—28no—
238#95—24no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95C2×D412yes—
244#95C2×D414yes—
245#95C2×D412yes—
246C24—22no—
247C24—22no—
248C24—28no—
249C24—24no—
250C24—24no—
251C24—44no—
252C24C2×C22⋊C414yes—
253C24C22×D411yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—44no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×D4—216no—
260C2×D4—216no—
261C2×D4—28no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4#10418yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—232no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4—88no—
300C22×C4—832no—
301C22×C4—88no—
302C22×C4C2×C22⋊C414yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4#12012yes—
308C22×C4#12012yes—
309C22×C4#12312yes—
310C22×C4#12312yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—216no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—88no—
334C22⋊C4—232no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—416no—
342C22⋊C4—88no—
343C42—216no—
344C42—44no—
345C42—44no—
346C42—84no—
347C42—88no—
348C42C4×D412yes—
349C42C4×D412yes—
350C42C4×D412yes—
351C42C4×D412yes—
352C42#12112yes—
353C42#12112yes—
354C42#12112yes—
355C42#12112yes—
356C23—22no—
357C23—22no—
358C23—28no—
359C23—28no—
360C23—28no—
361C23—28no—
362C23—24no—
363C23—24no—
364C23—416no—
365C23—416no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23C2×#10418yes—
372C23C22×C22⋊C411yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×#12311yescharacteristic
375C23#57712yes—
376C23#57812yes—
377Q8—82no—
378Q8—82no—
379Q8—84no—
380D4—216no—
381D4—216no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—416no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—832no—
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4#38612yes—
427C2×C4#38612yes—
428C2×C4C2×#10414yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C8—82no—
434C8—82no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—216no—
442C22—28no—
443C22—48no—
444C22—48no—
445C22—416no—
446C22—48no—
447C22—44no—
448C22—44no—
449C22C2×#38611yescharacteristic
450C22#288312yes—
451C22#289812yes—
452C22C22×#10412yes—
453C22#300911yescharacteristic
454C22#301111yescharacteristic
455C22#309712yes—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—48no—
465C4—44no—
466C4—44no—
467C4—84no—
468C4—82no—
469C4—82no—
470C4#189614yes—
471C2—21no—
472C2—21no—
473C2—22no—
474C2—48no—
475C2#1754814yes—
476C2#2091311yescharacteristic
477C2C2×#189611yescharacteristic
478C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1506C2semidirect
5. C2×#1503C2semidirect
6. #9146C2semidirect
7. #9143C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
17. #2547C4semidirect
18. #2542C4semidirect
19. #2536C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
56. C2×C22⋊Q8C2×C4semidirect
59. C2×C4×Q8C2×C4semidirect
101. #382C2×C4semidirect
103. #380C2×C4semidirect
108. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×#22752
#217882
C2×#15062
C2×#15032
#91462
#91432
#80212
#80132