Finite Group Databasebeta

← #481527

Subgroups — #481527

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35844C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11730C214yesmaximal, complemented
5#11726C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11693C212yesmaximal, complemented
8C2×#559C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2972—21no—
11C2×#410—21no—
12#2631C2212yes—
13#2619C2211yescharacteristic
14#2610C412yescomplemented
15#2598C412yescomplemented
16#2580C411yescharacteristic, complemented
17#2541C411yescharacteristic, complemented
18#2456—22no—
19#2448—21no—
20#2443—22no—
21#2431—21no—
22#2397C2212yes—
23#2396C2211yescharacteristic
24#2366—21no—
25#2361—21no—
26#2356C411yescharacteristic, complemented
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1352C2214yes—
36#1351C2214yes—
37#1343C2218yes—
38C22×#41D411yescharacteristic
39#559—44no—
40C2×#127—21no—
41C2×#127C2×C411yescharacteristic
42C2×#124—21no—
43C2×#124C2×C412yes—
44C2×#123—22no—
45C2×#121—21no—
46C2×#121—22no—
47#520—44no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4C2312yes—
59C2×#117—22no—
60C2×#117C2×C412yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C8×D4—44no—
65#431—22no—
66#410—28no—
67C2×#99—24no—
68C2×C22⋊C8—24no—
69#400C2×C411yescharacteristic
70#399—22no—
71#399C2×C412yes—
72#399C2311yescharacteristic
73#398—21no—
74#397—22no—
75#396—21no—
76#395C2×C411yescharacteristic
77#394—21no—
78#394C2×C412yes—
79#393C2×C412yes—
80#393C2311yescharacteristic
81#392—22no—
82#392—22no—
83#392—21no—
84#392C2×C412yes—
85#392C2×C412yes—
86#392C2312yes—
87#391—21no—
88#390—21no—
89#390—21no—
90#390—22no—
91#390—21no—
92#389—21no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#385—22no—
103#385—22no—
104#383C2×C412yes—
105#382C2×C411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112#376C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22×C412yes—
130C22×D4C2×D412yes—
131C23×C4—22no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C22×C8—44no—
138#127—42no—
139#127—44no—
140#124—44no—
141#124—44no—
142#123—48no—
143#121—44no—
144#121—84no—
145C4×Q8—24no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×D4—24no—
149C4×D4—42no—
150C4×D4—44no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—84no—
155C4×D4—84no—
156#117—28no—
157#117—42no—
158#117—44no—
159#117—44no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22×C412yes—
192C2×C22⋊C4C2×D411yescharacteristic
193C2×C22⋊C4C2411yescharacteristic
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—44no—
199C2×C42C22⋊C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C4⋊C8—42no—
204#99—216no—
205C22⋊C8—24no—
206C22⋊C8—44no—
207C4×C8—42no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95—44no—
218#95C22×C411yescharacteristic
219#95C22×C412yes—
220#95C22×C411yescharacteristic
221C24—22no—
222C24C2×C22⋊C412yes—
223C24C22×D411yescharacteristic
224#41—48no—
225#41—48no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—81no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—82no—
252C2×D4#10414yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#13514yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—44no—
284C2×C8—164no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—48no—
296C4⋊C4—42no—
297C4⋊C4—81no—
298C4⋊C4—82no—
299C4⋊C4#9914yes—
300C22⋊C4—24no—
301C22⋊C4—28no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—81no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—22no—
319C42—24no—
320C42—22no—
321C42—42no—
322C42—44no—
323C42—48no—
324C42—42no—
325C42—42no—
326C42—164no—
327C23—22no—
328C23—21no—
329C23—22no—
330C23—21no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23C2×#10414yes—
343C23C22×C22⋊C411yescharacteristic
344C23C2×#13512yes—
345Q8—44no—
346Q8—82no—
347D4—28no—
348D4—48no—
349D4—88no—
350D4—84no—
351D4—88no—
352D4—84no—
353D4—162no—
354D4—164no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—164no—
395C2×C4—162no—
396C2×C4C2×#9914yes—
397C2×C4C2×#9912yes—
398C2×C4#41612yes—
399C2×C4#41612yes—
400C2×C4C2×#10412yes—
401C8—84no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—24no—
406C22—42no—
407C22—44no—
408C22—41no—
409C22—42no—
410C22—41no—
411C22—42no—
412C22—81no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22C22×#9911yescharacteristic
422C22C2×#41611yescharacteristic
423C22C22×#10411yescharacteristic
424C4—24no—
425C4—22no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—42no—
431C4—41no—
432C4—42no—
433C4—82no—
434C4—81no—
435C4—88no—
436C4—81no—
437C4—81no—
438C4#150114yes—
439C2—21no—
440C2—21no—
441C2—42no—
442C2—44no—
443C2—162no—
444C2C2×#150111yescharacteristic
445C2#3335111yescharacteristic
446C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35844C2
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. #20393C2semidirect
3. #17351C2semidirect
4. #11730C2semidirect
5. #11726C2semidirect
6. #11715C2semidirect
7. #11693C2semidirect
8. C2×#559C4semidirect
14. #2610C4semidirect
15. #2598C4semidirect
16. #2580C4semidirect
17. #2541C4semidirect
26. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#358442
#203932
#173512
#117302
#117262
#117152
#116932