Finite Group Databasebeta

← #201370

Subgroups — #201370

Order 512.

1341 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic
2#34952C211yesmaximal, characteristic
3#20734C212yesmaximal, complemented
4#20632C211yesmaximal, characteristic
5#9469C214yesmaximal, complemented
6#9467C214yesmaximal, complemented
7#7981C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2988—24no—
11#2982—21no—
12#2972—21no—
13#2972C2214yes—
14C8×#41—24no—
15C2×#431—22no—
16C2×#431C2212yes—
17C2×#410—21no—
18#2582—21no—
19#2581C412yescomplemented
20#2580C412yescomplemented
21#2579—22no—
22#2538—21no—
23#2487C2212yes—
24#2409C2212yes—
25#2396C414yescomplemented
26C4×C22⋊Q8—22no—
27#2297—21no—
28#2296—21no—
29#2285C2212yes—
30#2285C2211yescharacteristic
31#1824C2214yes—
32#1803—24no—
33#1799—22no—
34#1799C2214yes—
35#1787C2214yes—
36C4×#103—24no—
37#1740—21no—
38#1289C2218yes—
39C22×#41D411yescharacteristic
40C23×Q8C2311yescharacteristic
41C2×C22⋊Q8—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2312yes—
60C2×#117—21no—
61C2×#117—22no—
62C2×#117C2×C412yes—
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C8×Q8—28no—
70C8×D4—28no—
71C8×D4—216no—
72#431—24no—
73#431—24no—
74#431—22no—
75#431—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8D414yes—
81C2×#103—28no—
82#410—44no—
83C2×#99—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8C2314yes—
86C2×C4×C8—24no—
87C2×C4×C8D414yes—
88#400C2×C414yes—
89#399C2×C414yes—
90#398—24no—
91#396—24no—
92#395C2×C414yes—
93#393C2×C414yes—
94#391—22no—
95#391—22no—
96#390—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#387—24no—
101#386—22no—
102#386—22no—
103#385—24no—
104#385—22no—
105#385—22no—
106#384—21no—
107#384—22no—
108#382—22no—
109#382C2×C412yes—
110#380—21no—
111#380—22no—
112#380—22no—
113#380C2314yes—
114#380C2312yes—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C22⋊C4—24no—
121#342—24no—
122#340—28no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—24no—
126C2×#41SD1614yes—
127C2×#41Q1614yes—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—44no—
132C22×Q8C22⋊C411yescharacteristic
133C22×Q8C22×C412yes—
134C22×D4C22⋊C411yescharacteristic
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—28no—
145C22×C8C2×D418yes—
146C22⋊Q8—48no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—42no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—42no—
160C4×Q8—41no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169#117—22no—
170#117—22no—
171#117—24no—
172#117—24no—
173#117—44no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—28no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227#103—416no—
228#99—48no—
229C22⋊C8—28no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C4×C8—28no—
233C4×C8—24no—
234C4×C8—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—42no—
242#95C22×C414yes—
243C24C2×C22⋊C412yes—
244C24C22×D411yescharacteristic
245#41—28no—
246#41—28no—
247#41—28no—
248#41—28no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—28no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8#10314yescomplemented
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4#10314yescomplemented
266C2×D4C2×SD1614yes—
267C2×D4C2×Q1614yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—81no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#10318yescomplemented
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4C2×SD1612yes—
300C22×C4C2×Q1612yes—
301C22×C4#13514yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4112yes—
306C2×C8—28no—
307C2×C8—48no—
308C2×C8—88no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—82no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—48no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—84no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—42no—
342C42—44no—
343C42—42no—
344C42—44no—
345C42—84no—
346C42C4×D412yes—
347C42C4×D412yes—
348C42C4×D412yes—
349C42C4×D412yes—
350C42#12012yes—
351C42#12011yescharacteristic
352C42#12011yescharacteristic
353C42#12311yescharacteristic
354C42#12312yes—
355C42#12311yescharacteristic
356C23—22no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23C2×#10314yes—
363C23C22×C22⋊C411yescharacteristic
364C23C2×#12112yes—
365C23C22×SD1611yescharacteristic
366C23C22×Q1611yescharacteristic
367C23C2×#13512yes—
368Q8—28no—
369Q8—28no—
370Q8—44no—
371Q8—44no—
372Q8—44no—
373D4—28no—
374D4—28no—
375D4—28no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4#38611yescharacteristic
411C2×C4#38611yescharacteristic
412C2×C4#38612yes—
413C2×C4#38611yescharacteristic
414C2×C4#38612yes—
415C2×C4#38611yescharacteristic
416C2×C4C2×#10314yes—
417C2×C4C2×#10312yes—
418C2×C4C2×#10312yes—
419C2×C4#41612yes—
420C2×C4#41612yes—
421C2×C4#46514yes—
422C2×C4#46514yes—
423C2×C4#46714yes—
424C2×C4#46714yes—
425C2×C4C2×C4×D412yes—
426C2×C4C2×#12011yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C8—48no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—22no—
433C22—24no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—44no—
438C22—42no—
439C22—42no—
440C22—82no—
441C22C2×#38611yescharacteristic
442C22C22×#10311yescharacteristic
443C22C2×#41611yescharacteristic
444C22C2×#46512yes—
445C22C2×#46712yes—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—21no—
450C4—22no—
451C4—24no—
452C4—42no—
453C4—44no—
454C4—42no—
455C4—84no—
456C4—82no—
457C4—164no—
458C4#189212yes—
459C4#189212yes—
460C4#189214yes—
461C4#302512yes—
462C2—21no—
463C2—22no—
464C2—21no—
465C2—22no—
466C2—44no—
467C2#2095911yescharacteristic
468C2C2×#189211yescharacteristic
469C2#2100911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20734C2semidirect
5. #9469C2semidirect
6. #9467C2semidirect
7. #7981C2semidirect
19. #2581C4semidirect
20. #2580C4semidirect
25. #2396C4semidirect
260. C2×Q8#103semidirect
265. C2×D4#103semidirect
287. C22×C4#103semidirect

Maximal subgroups

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

subgroupindex
#602762
#349522
#207342
#206322
#94692
#94672
#79812