Finite Group Databasebeta

← #481689

Subgroups — #481689

Order 512.

1055 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36041C211yesmaximal, characteristic
2#20355C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11758C214yesmaximal, complemented
5#11754C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11693C212yesmaximal
8C2×#559C411yescharacteristic, complemented
9C2×#519—21no—
10C2×#519C2211yescharacteristic
11#2983—21no—
12#2632C412yescomplemented
13#2619C2211yescharacteristic
14#2610C2212yes—
15#2598C2212yes—
16#2581C411yescharacteristic, complemented
17#2548C411yescharacteristic, complemented
18#2471—22no—
19#2458—22no—
20#2448—21no—
21#2436—21no—
22#2398C412yescomplemented
23#2396C2211yescharacteristic
24#2367—21no—
25#2363—21no—
26#2356C411yescharacteristic, complemented
27#1740—21no—
28#1740—22no—
29#1740C2212yes—
30#1740C2214yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1343C2218yes—
38C23×Q8D411yescharacteristic
39#559—44no—
40C2×#127C2×C411yescharacteristic
41C2×#124—21no—
42C2×#124—21no—
43C2×#124C2×C412yes—
44C2×#123—22no—
45C2×#121—21no—
46#519—28no—
47#519—44no—
48#519—44no—
49C2×C4×Q8—44no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×#117—24no—
60C2×#117—22no—
61C2×#117D411yescharacteristic
62C2×#117D412yes—
63C2×#117C2312yes—
64C2×#117C2311yescharacteristic
65#446—44no—
66#432—24no—
67#431—22no—
68#400C2×C412yes—
69#400C2×C411yescharacteristic
70#399—22no—
71#399C2311yescharacteristic
72#398—21no—
73#397—22no—
74#396—21no—
75#395C2×C412yes—
76#395C2×C411yescharacteristic
77#394—21no—
78#394C2×C412yes—
79#394C2×C412yes—
80#394C2×C412yes—
81#393—22no—
82#393—22no—
83#393C2311yescharacteristic
84#392—21no—
85#392C2312yes—
86#391—21no—
87#391—21no—
88#390—21no—
89#390—22no—
90#390—22no—
91#390—21no—
92#389—21no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#385—22no—
104#383C2×C412yes—
105#382C2×C411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112#376C2×C412yes—
113#342—24no—
114#342—22no—
115#342C2314yes—
116#324—22no—
117#324—24no—
118#323C2314yes—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22×C411yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4C22×C412yes—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134#127—42no—
135#124—44no—
136#124—44no—
137#124—44no—
138#123—48no—
139#121—42no—
140#121—42no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—44no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—44no—
147C4×Q8—42no—
148C4×Q8—432no—
149C4×D4—44no—
150C4×D4—44no—
151#117—216no—
152#117—28no—
153#117—42no—
154#117—44no—
155#117—42no—
156#117—44no—
157#117—48no—
158#117—44no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C22⋊C4C2411yescharacteristic
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42—42no—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C4⋊C8—42no—
205C4⋊C8—44no—
206C22⋊C8—24no—
207#97—44no—
208C4×C8—42no—
209#95—22no—
210#95—21no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24—22no—
224C24C22×D411yescharacteristic
225C2×Q8—216no—
226C2×Q8—216no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—81no—
236C2×Q8#9914yes—
237C2×Q8#10414yes—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4#13712yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—44no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—416no—
288C4⋊C4—48no—
289C4⋊C4—42no—
290C4⋊C4—81no—
291C4⋊C4—82no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C22⋊C4—28no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C42—22no—
312C42—24no—
313C42—24no—
314C42—22no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—416no—
319C42—48no—
320C42—42no—
321C42—42no—
322C42—82no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23C22×C22⋊C411yescharacteristic
335C23C2×AGL(1,ℤ/8)11yescharacteristic
336C23C2×#13711yescharacteristic
337Q8—216no—
338Q8—28no—
339Q8—28no—
340Q8—82no—
341D4—84no—
342D4—84no—
343D4—162no—
344D4—162no—
345D4—162no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—164no—
387C2×C4C2×#9914yes—
388C2×C4C2×#9912yes—
389C2×C4#41712yes—
390C2×C4#41712yes—
391C2×C4C2×#10414yes—
392C2×C4C2×#10412yes—
393C8—84no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22C22×#9911yescharacteristic
411C22C2×#41711yescharacteristic
412C22C22×#10411yescharacteristic
413C4—24no—
414C4—22no—
415C4—24no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—41no—
420C4—42no—
421C4—41no—
422C4—81no—
423C4—81no—
424C4—82no—
425C4—88no—
426C4—81no—
427C4—81no—
428C4#150414yes—
429C2—21no—
430C2—21no—
431C2—41no—
432C2—41no—
433C2—162no—
434C2C2×#150411yescharacteristic
435C2#3335111yescharacteristic
436C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#2619C22
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. #20355C2semidirect
3. #17371C2semidirect
4. #11758C2semidirect
5. #11754C2semidirect
6. #11715C2semidirect
8. C2×#559C4semidirect
12. #2632C4semidirect
16. #2581C4semidirect
17. #2548C4semidirect
22. #2398C4semidirect
26. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#360412
#203552
#173712
#117582
#117542
#117152
#116932