Finite Group Databasebeta

← #160759

Subgroups — #160759

Order 512.

1422 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
1C4×#525C211yesmaximal, characteristic
2C4×#407C212yesmaximal, complemented
3#7634C214yesmaximal, complemented
4#7633C214yesmaximal, complemented
5#7564C212yesmaximal, complemented
6#7563C212yesmaximal, complemented
7C2×#525C411yescharacteristic
8C2×C4×#41C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#407—22no—
11C2×C4×M16—22no—
12C4×C22⋊Q8C412yes—
13C4×C22⋊Q8C2212yes—
14C4×#123—22no—
15C4×#121C412yes—
16C4×#121C2212yes—
17C4×#120—22no—
18#2297C2211yescharacteristic
19#2296C2211yescharacteristic
20#2290—22no—
21#2280C411yescharacteristic
22C42×D4—22no—
23C4×#117C411yescharacteristic
24#1749—22no—
25#1749C2214yes—
26C4×C22⋊C8—22no—
27C4×C22⋊C8C2214yes—
28#1728C2214yes—
29#1272C2214yescomplemented
30#1271C2218yescomplemented
31#1270C2214yescomplemented
32C22×#41C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4D412yes—
35C22×M16—24no—
36#525—24no—
37#525—24no—
38C2×#123—21no—
39C2×C22⋊Q8C2×C412yes—
40C2×#121C2×C412yes—
41C2×#120—21no—
42C4×#41—44no—
43C2×C4×Q8C2×C412yes—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4C2×C412yes—
49C2×#117—21no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C22×C42—21no—
54C22×C42—44no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62#407—216no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C4×M16—48no—
66C2×#97—24no—
67C2×C4×C8—24no—
68#391C2×C411yescharacteristic
69#391C2311yescharacteristic
70#390C2×C411yescharacteristic
71#390C2311yescharacteristic
72#389C2×C412yes—
73#389C2312yes—
74#388—22no—
75#388—22no—
76#387—22no—
77#387—22no—
78#386—22no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—22no—
84#381—21no—
85#380C2×C412yes—
86#380C2×C412yes—
87#379—21no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100#376—21no—
101C43—21no—
102#336—24no—
103#336C2318yes—
104C25C22⋊C411yescharacteristic
105C2×#41—22no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41M1614yescomplemented
109C22×Q8C22×C411yescharacteristic
110C22×D4—22no—
111C22×D4—21no—
112C22×D4C22×C411yescharacteristic
113C23×C4—28no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4C22⋊C412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C2×M16—216no—
133C2×M16—48no—
134C2×M16—48no—
135C22×C8—28no—
136#123—28no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139#121—24no—
140#121—24no—
141#120—28no—
142C4×Q8—44no—
143C4×D4—216no—
144C4×D4—44no—
145C4×D4—48no—
146C4×D4—48no—
147C4×D4—48no—
148#117—22no—
149#117—44no—
150#117—42no—
151#117—42no—
152#117M1614yescomplemented
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C42—21no—
172C2×C42—24no—
173C2×C42—28no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—42no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2411yescharacteristic
195C22⋊C8—216no—
196C22⋊C8—216no—
197#97—48no—
198C4×C8—48no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207C24—21no—
208C24—22no—
209C24—24no—
210C24—41no—
211C24—41no—
212C24—42no—
213C24C2×C22⋊C411yescharacteristic
214C24C2×C22⋊C412yes—
215#41—48no—
216#41—48no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C2×M1612yes—
221C2×D4—28no—
222C2×D4—24no—
223C2×D4—21no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4C2×M1612yes—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—432no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4#10418yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×M1612yes—
281C22×C4C2×M1612yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D412yes—
284M16—432no—
285C2×C8—216no—
286C2×C8—48no—
287C2×C8—48no—
288C4⋊C4—28no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4C2×M1614yes—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—24no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—22no—
311C42—24no—
312C42—24no—
313C42—42no—
314C42—41no—
315C42—42no—
316C42—41no—
317C42—44no—
318C42—44no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42#9914yes—
323C23—22no—
324C23—21no—
325C23—22no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—48no—
335C23—44no—
336C23—48no—
337C23—44no—
338C23—44no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—42no—
343C23#40712yes—
344C23C2×#10414yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—42no—
347Q8—42no—
348D4—42no—
349D4—42no—
350D4—416no—
351D4—48no—
352C2×C4—216no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—88no—
392C2×C4#40712yes—
393C2×C4C2×#9914yes—
394C2×C4C2×#9912yes—
395C2×C4#41312yes—
396C2×C4C2×#10414yes—
397C2×C4C2×#10414yes—
398C2×C4C22×M1611yescharacteristic
399C2×C4C22×M1611yescharacteristic
400C2×C4#57514yes—
401C8—416no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—22no—
406C22—24no—
407C22—416no—
408C22—42no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—41no—
415C22—44no—
416C22—42no—
417C22—81no—
418C22—82no—
419C22—81no—
420C22C2×#40711yescharacteristic
421C22C22×#9911yescharacteristic
422C22C2×#41311yescharacteristic
423C22#289612yes—
424C22C22×#10412yes—
425C4—24no—
426C4—24no—
427C4—24no—
428C4—24no—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4#146514yes—
438C2—44no—
439C2—42no—
440C2—82no—
441C2C2×#146511yescharacteristic
442C2#1744712yes—
443C2#1754212yes—
444C2#1763811yescharacteristic
445C2#1774011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. #7634C2semidirect
4. #7633C2semidirect
5. #7564C2semidirect
6. #7563C2semidirect
29. #1272C22semidirect
30. #1271C22semidirect
31. #1270C22semidirect
108. C2×#41M16semidirect
152. #117M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C4×#4072
#76342
#76332
#75642
#75632