Finite Group Databasebeta

← #481906

Subgroups — #481906

Order 512.

880 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35846C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#11826C214yesmaximal, complemented
5#11822C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11694C212yesmaximal, complemented
8C2×#556C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2983—21no—
11C2×#410—21no—
12#2638C2212yes—
13#2621C2211yescharacteristic
14#2583C412yescomplemented
15#2581C411yescharacteristic, complemented
16#2546C411yescharacteristic, complemented
17#2484C412yescomplemented
18#2478—22no—
19#2466—22no—
20#2447—21no—
21#2432—21no—
22#2396C2211yescharacteristic
23#2395C2212yes—
24#2365—21no—
25#2360—21no—
26#2357C411yescharacteristic, complemented
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1466C2214yes—
32#1464—22no—
33#1460—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1344C2218yes—
38C22×#41D411yescharacteristic
39#556—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#126—22no—
43C2×#124—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2312yes—
55C2×#117—22no—
56C2×#117C2×C412yes—
57C2×#117D412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4D411yescharacteristic
61#446—44no—
62#432—24no—
63#431—22no—
64#410—44no—
65C2×#99—24no—
66#400C2×C412yes—
67#400C2×C411yescharacteristic
68#399C2311yescharacteristic
69#398—21no—
70#398—21no—
71#398C2×C412yes—
72#398C2×C412yes—
73#398C2×C412yes—
74#397—21no—
75#397—22no—
76#397—22no—
77#397C2312yes—
78#396—21no—
79#395C2×C412yes—
80#395C2×C411yescharacteristic
81#394—22no—
82#394—22no—
83#393C2311yescharacteristic
84#391—21no—
85#391—21no—
86#391—21no—
87#390—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—44no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#382C2×C411yescharacteristic
103#381—22no—
104#381C2×C412yes—
105#380—24no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—42no—
121C2×#41—42no—
122C22×Q8—21no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C411yescharacteristic
125C22×D4C22⋊C411yescharacteristic
126C22×D4C2×D412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C4⋊Q8—42no—
133C4⋊Q8—44no—
134#126—48no—
135#124—44no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C22⋊Q8—44no—
139C4×Q8—24no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—44no—
143C4×D4—24no—
144C4×D4—42no—
145C4×D4—44no—
146C4×D4—44no—
147#117—42no—
148#117—44no—
149#117—44no—
150#117—44no—
151#117—44no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
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⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C2×D411yescharacteristic
186C2×C22⋊C4C2×D412yes—
187C2×C22⋊C4C2411yescharacteristic
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42C22⋊C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C4⋊C8—42no—
198C4⋊C8—44no—
199#99—48no—
200C22⋊C8—24no—
201#97—44no—
202C4×C8—42no—
203#95—22no—
204#95—21no—
205#95—21no—
206#95—22no—
207#95—21no—
208#95—22no—
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
220C24C2×C22⋊C412yes—
221C24C22×D411yescharacteristic
222#41—48no—
223#41—48no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—81no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—42no—
238C2×D4—42no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4AGL(1,ℤ/8)12yes—
265C22×C4#13712yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—44no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—84no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—164no—
290C4⋊C4#9914yes—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—82no—
304C22⋊C4—81no—
305C42—22no—
306C42—24no—
307C42—22no—
308C42—42no—
309C42—44no—
310C42—42no—
311C42—42no—
312C42—84no—
313C42—82no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23C22×C22⋊C411yescharacteristic
324C23C2×AGL(1,ℤ/8)11yescharacteristic
325C23C2×#13711yescharacteristic
326Q8—44no—
327Q8—82no—
328Q8—84no—
329D4—44no—
330D4—44no—
331D4—44no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—48no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—162no—
375C2×C4C2×#9914yes—
376C2×C4C2×#9912yes—
377C2×C4#41712yes—
378C2×C4#41712yes—
379C2×C4#42112yes—
380C8—84no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—81no—
393C22—82no—
394C22—82no—
395C22C22×#9911yescharacteristic
396C22C2×#41711yescharacteristic
397C22C2×#42111yescharacteristic
398C4—24no—
399C4—22no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—81no—
408C4—82no—
409C4—81no—
410C4—81no—
411C4—82no—
412C4—164no—
413C4#151314yes—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—82no—
419C2C2×#151311yescharacteristic
420C2#3335011yescharacteristic
421C2#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. #11826C2semidirect
5. #11822C2semidirect
6. #11715C2semidirect
7. #11694C2semidirect
8. C2×#556C4semidirect
14. #2583C4semidirect
15. #2581C4semidirect
16. #2546C4semidirect
17. #2484C4semidirect
26. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#358462
#206342
#173722
#118262
#118222
#117152
#116942