Finite Group Databasebeta

← #481842

Subgroups — #481842

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36018C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11800C214yesmaximal, complemented
5#11798C214yesmaximal, complemented
6#11693C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#550C411yescharacteristic, complemented
9C2×#548—21no—
10C2×#519C2211yescharacteristic
11#2983—21no—
12#2619C2212yes—
13#2619C2211yescharacteristic
14#2548C411yescharacteristic, complemented
15#2548C412yescomplemented
16#2536C412yescomplemented
17#2536C411yescharacteristic, complemented
18#2426—21no—
19#2426—22no—
20#2414—22no—
21#2414—21no—
22#2392C2212yes—
23#2392C2211yescharacteristic
24#2351C411yescharacteristic, complemented
25#2341—21no—
26#2310—21no—
27#1820—22no—
28#1820C2214yes—
29#1740—21no—
30#1740C2212yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1344C2214yes—
36#1343C2218yes—
37#1343C2214yes—
38C22×#41—21no—
39C23×Q8D411yescharacteristic
40#550—44no—
41#548—48no—
42C2×#127C2×C412yes—
43C2×#127C2×C411yescharacteristic
44C2×#124—21no—
45C2×#124—22no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×#121—22no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#121—21no—
52#519—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×#117—22no—
65C2×#117D412yes—
66C2×#117D411yescharacteristic
67C2×#117C2312yes—
68C2×#117C2311yescharacteristic
69#446—44no—
70#432—24no—
71#431—22no—
72#398C2×C412yes—
73#398C2×C411yescharacteristic
74#397C2311yescharacteristic
75#396—22no—
76#396—21no—
77#394—22no—
78#394—21no—
79#394—22no—
80#394—21no—
81#394C2×C412yes—
82#394C2×C412yes—
83#394C2×C412yes—
84#394C2×C412yes—
85#394C2×C411yescharacteristic
86#392—22no—
87#392—21no—
88#392C2311yescharacteristic
89#392C2312yes—
90#391—44no—
91#390—22no—
92#390—21no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390—21no—
97#390—22no—
98#389—22no—
99#389—21no—
100#388—22no—
101#388—22no—
102#388—22no—
103#382C2×C412yes—
104#382C2×C411yescharacteristic
105#381—24no—
106#380—22no—
107#380—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112#342—22no—
113#342—24no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—44no—
119C2×#41—44no—
120C2×#41—44no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—42no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22⋊C412yes—
126C22×Q8C22⋊C411yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4C22×C412yes—
134C22×D4C22×C411yescharacteristic
135C23×C4—22no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139#127—44no—
140#127—42no—
141#124—48no—
142#124—44no—
143C22⋊Q8—44no—
144C22⋊Q8—48no—
145#121—48no—
146#121—44no—
147#121—48no—
148#121—44no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—44no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×D4—44no—
157C4×D4—42no—
158C4×D4—48no—
159C4×D4—44no—
160#117—42no—
161#117—44no—
162#117—42no—
163#117—44no—
164#117—44no—
165#117—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C22⋊C4C2×D412yes—
200C2×C22⋊C4C2411yescharacteristic
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C4⋊C8—42no—
212C4⋊C8—44no—
213C22⋊C8—24no—
214#97—44no—
215C4×C8—42no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—48no—
222#95C22×C411yescharacteristic
223#95C22×C412yes—
224#95C22×C411yescharacteristic
225C24—22no—
226C24—21no—
227C24C22×D411yescharacteristic
228#41—816no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8#9914yes—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—82no—
253C2×D4—81no—
254C2×D4—82no—
255C2×D4—81no—
256C2×D4—82no—
257C2×D4—81no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—82no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4AGL(1,ℤ/8)14yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—44no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4—81no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—81no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—81no—
324C22⋊C4—82no—
325C42—22no—
326C42—24no—
327C42—24no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—44no—
333C42—42no—
334C42—84no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—42no—
352C23—41no—
353C23C22×C22⋊C411yescharacteristic
354C23C2×AGL(1,ℤ/8)12yes—
355Q8—28no—
356Q8—48no—
357Q8—44no—
358D4—88no—
359D4—84no—
360D4—82no—
361D4—88no—
362D4—84no—
363D4—84no—
364D4—164no—
365D4—162no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—48no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—88no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4C2×#9914yes—
406C2×C4C2×#9912yes—
407C2×C4#41812yes—
408C2×C4#41812yes—
409C2×C4#42112yes—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—81no—
418C22—82no—
419C22—81no—
420C22—82no—
421C22—81no—
422C22—82no—
423C22—81no—
424C22—82no—
425C22—81no—
426C22—82no—
427C22—81no—
428C22—82no—
429C22—81no—
430C22—82no—
431C22—81no—
432C22C22×#9911yescharacteristic
433C22C2×#41811yescharacteristic
434C22C2×#42111yescharacteristic
435C4—22no—
436C4—24no—
437C4—41no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4—81no—
445C4—82no—
446C4—81no—
447C4—81no—
448C4—164no—
449C4#151014yes—
450C2—21no—
451C2—21no—
452C2—42no—
453C2—162no—
454C2—161no—
455C2C2×#151011yescharacteristic
456C2#3335111yescharacteristic
457C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36018C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20691C2semidirect
3. #17371C2semidirect
4. #11800C2semidirect
5. #11798C2semidirect
6. #11693C2semidirect
7. #11691C2semidirect
8. C2×#550C4semidirect
14. #2548C4semidirect
15. #2548C4semidirect
16. #2536C4semidirect
17. #2536C4semidirect
24. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#360182
#206912
#173712
#118002
#117982
#116932
#116912