Finite Group Databasebeta

← #482139

Subgroups — #482139

Order 512.

985 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#17353C211yesmaximal, characteristic
4#11902C214yesmaximal, complemented
5#11898C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11715C212yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2982—21no—
11C2×#410—21no—
12#2628C2211yescharacteristic
13#2627C412yescomplemented
14#2581C411yescharacteristic, complemented
15#2567C411yescharacteristic, complemented
16#2564C2212yes—
17#2451—21no—
18#2449—22no—
19#2440—21no—
20#2438—22no—
21#2409C2212yes—
22#2396C2211yescharacteristic
23C2×#395C412yescomplemented
24#2349C411yescharacteristic, complemented
25D4×C4⋊C4—21no—
26#2339—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1460—21no—
32#1460C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1351C2218yes—
38C22×#41D411yescharacteristic
39C23×Q8C2×C412yes—
40D4×Q8—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4C2312yes—
56C2×#117—22no—
57C2×#117D412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4D411yescharacteristic
62C8×Q8—44no—
63#431—24no—
64#431—22no—
65#410—44no—
66C2×#99—24no—
67#400—21no—
68#400C2×C412yes—
69#400C2×C411yescharacteristic
70#399—21no—
71#399C2311yescharacteristic
72#398—22no—
73#398—22no—
74#398—21no—
75#396—21no—
76#395—22no—
77#395—22no—
78#395C2×C412yes—
79#395C2×C412yes—
80#395C2×C412yes—
81#395C2×C412yes—
82#395C2×C411yescharacteristic
83#393C2312yes—
84#393C2311yescharacteristic
85#391—21no—
86#391—21no—
87#391—22no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—21no—
92#389—44no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#384—21no—
102#382C2×C411yescharacteristic
103#381—22no—
104#380—21no—
105#380—22no—
106#380—24no—
107#380C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C2×#95—22no—
112C2×#95C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—42no—
119C2×#41—42no—
120C2×#41SD1614yes—
121C2×#41Q1614yes—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—84no—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22×C411yescharacteristic
127C22×Q8C22×C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C2×D412yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C4⋊Q8—42no—
138C4⋊Q8—44no—
139C22⋊Q8—48no—
140C22⋊Q8—44no—
141C22⋊Q8—84no—
142C4×Q8—24no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×D4—24no—
146C4×D4—42no—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—44no—
150C4×D4—44no—
151#117—42no—
152#117—44no—
153#117—44no—
154#117—44no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C2×D411yescharacteristic
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2411yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42C22⋊C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C4⋊C8—44no—
199C4⋊C8—42no—
200#99—48no—
201C22⋊C8—24no—
202C4×C8—44no—
203C4×C8—42no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—44no—
218#95C22×C411yescharacteristic
219#95C22×C412yes—
220#95C22×C411yescharacteristic
221C24C2×C22⋊C412yes—
222C24C22×D411yescharacteristic
223#41—48no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8#10314yes—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4#10314yes—
242C2×D4C2×SD1614yes—
243C2×D4C2×Q1614yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#10318yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×SD1612yes—
276C22×C4C2×Q1612yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—44no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—84no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—164no—
302C4⋊C4#9914yes—
303C22⋊C4—24no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—82no—
314C42—24no—
315C42—22no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—84no—
321C23—22no—
322C23—21no—
323C23—22no—
324C23—21no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23—41no—
329C23C2×#10314yes—
330C23C22×C22⋊C411yescharacteristic
331C23C22×SD1611yescharacteristic
332C23C22×Q1611yescharacteristic
333Q8—28no—
334Q8—42no—
335Q8—42no—
336Q8—88no—
337Q8—84no—
338D4—28no—
339D4—44no—
340D4—44no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—88no—
380C2×C4—88no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—162no—
385C2×C4C2×#9914yes—
386C2×C4C2×#9912yes—
387C2×C4C2×#10312yes—
388C2×C4C2×#10314yes—
389C2×C4C2×#10312yes—
390C2×C4#42112yes—
391C8—84no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—24no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—42no—
403C22—41no—
404C22—42no—
405C22—81no—
406C22C22×#9911yescharacteristic
407C22C22×#10311yescharacteristic
408C22C2×#42111yescharacteristic
409C4—24no—
410C4—22no—
411C4—24no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—82no—
421C4—82no—
422C4—81no—
423C4—164no—
424C4#152014yes—
425C2—21no—
426C2—21no—
427C2—41no—
428C2—44no—
429C2—41no—
430C2C2×#152011yescharacteristic
431C2#3334811yescharacteristic
432C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
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
4. #11902C2semidirect
5. #11898C2semidirect
6. #11717C2semidirect
7. #11715C2semidirect
8. C2×D4×Q8C4semidirect
13. #2627C4semidirect
14. #2581C4semidirect
15. #2567C4semidirect
23. C2×#395C4semidirect
24. #2349C4semidirect

Maximal subgroups

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

subgroupindex
#357392
#206342
#173532
#119022
#118982
#117172
#117152