Finite Group Databasebeta

← #481936

Subgroups — #481936

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36019C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11825C214yesmaximal, complemented
5#11823C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#556C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2975—21no—
11C2×#410—21no—
12#2631C2212yes—
13#2628C2211yescharacteristic
14#2610C412yescomplemented
15#2599C412yescomplemented
16#2563C411yescharacteristic, complemented
17#2535C411yescharacteristic, complemented
18#2481—22no—
19#2467—22no—
20#2421—21no—
21#2407—21no—
22#2398C2212yes—
23#2392C2211yescharacteristic
24#2363—21no—
25#2356C411yescharacteristic, complemented
26#2352—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1465C2214yes—
32#1463—22no—
33#1461—21no—
34#1458C2212yes—
35#1351C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39#556—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#126—22no—
42C2×#124C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×#121—21no—
45#520—44no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2312yes—
55C2×#117—21no—
56C2×#117—22no—
57C2×#117C2×C412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D412yes—
61#436—44no—
62#431—22no—
63#410—44no—
64C2×#99—24no—
65#407—24no—
66#400—21no—
67#400—22no—
68#399—21no—
69#398C2×C411yescharacteristic
70#397—22no—
71#397C2×C412yes—
72#397C2311yescharacteristic
73#396—21no—
74#395C2×C412yes—
75#394—22no—
76#394—21no—
77#394C2×C411yescharacteristic
78#393C2×C412yes—
79#393C2×C412yes—
80#393C2312yes—
81#392—22no—
82#392C2×C412yes—
83#392C2311yescharacteristic
84#391—21no—
85#390—21no—
86#390—22no—
87#389—21no—
88#389—44no—
89#388—22no—
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#385—22no—
100#383C2×C412yes—
101#382—21no—
102#382C2×C411yescharacteristic
103#380—22no—
104#380—21no—
105#380—21no—
106#380—24no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—42no—
121C2×#41—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C412yes—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22×C411yescharacteristic
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C2×M16—44no—
136C4⋊Q8—42no—
137#126—48no—
138#124—44no—
139C22⋊Q8—44no—
140#121—44no—
141C4×Q8—24no—
142C4×Q8—42no—
143C4×Q8—44no—
144C4×D4—24no—
145C4×D4—42no—
146C4×D4—44no—
147C4×D4—42no—
148C4×D4—44no—
149C4×D4—42no—
150C4×D4—42no—
151#117—42no—
152#117—44no—
153#117—44no—
154#117—44no—
155#117—44no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C22⋊C4C2411yescharacteristic
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42C22⋊C412yes—
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C4⋊C8—42no—
201#99—48no—
202C22⋊C8—24no—
203C22⋊C8—44no—
204C4×C8—42no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—21no—
215#95—22no—
216#95—44no—
217#95C22×C411yescharacteristic
218#95C22×C412yes—
219#95C22×C411yescharacteristic
220C24—21no—
221C24C2×C22⋊C412yes—
222C24C22×D411yescharacteristic
223#41—48no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—44no—
241C2×D4—81no—
242C2×D4—81no—
243C2×D4—81no—
244C2×D4—82no—
245C2×D4—82no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4#13712yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277M16—164no—
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—84no—
292C4⋊C4—82no—
293C4⋊C4—81no—
294C4⋊C4—164no—
295C4⋊C4#9914yes—
296C22⋊C4—24no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—81no—
310C22⋊C4—82no—
311C22⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—42no—
319C42—44no—
320C42—42no—
321C42—84no—
322C42—82no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23C22×C22⋊C411yescharacteristic
340C23C2×AGL(1,ℤ/8)11yescharacteristic
341C23C2×#13711yescharacteristic
342Q8—44no—
343Q8—84no—
344D4—44no—
345D4—44no—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—164no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—48no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—162no—
392C2×C4C2×#9914yes—
393C2×C4C2×#9912yes—
394C2×C4#41712yes—
395C2×C4#41712yes—
396C2×C4#42112yes—
397C8—84no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—44no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—81no—
411C22—82no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22C22×#9911yescharacteristic
420C22C2×#41711yescharacteristic
421C22C2×#42111yescharacteristic
422C4—24no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—42no—
428C4—41no—
429C4—42no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4—82no—
434C4—81no—
435C4—164no—
436C4#151314yes—
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—82no—
442C2—161no—
443C2C2×#151311yescharacteristic
444C2#3335111yescharacteristic
445C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36019C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17370C2semidirect
4. #11825C2semidirect
5. #11823C2semidirect
6. #11717C2semidirect
7. #11691C2semidirect
8. C2×#556C4semidirect
14. #2610C4semidirect
15. #2599C4semidirect
16. #2563C4semidirect
17. #2535C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#360192
#206342
#173702
#118252
#118232
#117172
#116912