Finite Group Databasebeta

← #163965

Subgroups — #163965

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34258C211yesmaximal, characteristic
2#22532C212yesmaximal, complemented
3#7640C212yesmaximal, complemented
4#7626C212yesmaximal, complemented
5#7612C214yesmaximal, complemented
6#7597C211yesmaximal, characteristic, complemented
7#7572C212yesmaximal, complemented
8#7563C211yesmaximal, characteristic, complemented
9C22×H(5,𝔽2)C411yescharacteristic
10C2×#520C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#446—22no—
13#2341—22no—
14D4×C22⋊C4—22no—
15D4×C22⋊C4—24no—
16#2311C412yes—
17#2311C411yescharacteristic
18#2309C412yes—
19#2309C411yescharacteristic
20#2297C2211yescharacteristic
21#2297C2212yes—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24#1955—24no—
25#1954—22no—
26#1797C2212yescomplemented
27#1797C2214yescomplemented
28#1749C2214yescomplemented
29C4×C22⋊C8C2212yescomplemented
30#1272C2212yescomplemented
31#1272C2212yescomplemented
32#1270C2211yescharacteristic, complemented
33#1270C2211yescharacteristic, complemented
34#1270C2214yescomplemented
35#1270C2214yescomplemented
36#1270C2212yescomplemented
37C2×H(5,𝔽2)—24no—
38C2×H(5,𝔽2)—24no—
39C2×H(5,𝔽2)—44no—
40C2×H(5,𝔽2)—44no—
41C22×#41—21no—
42C22×#41C2×C412yes—
43C22×#41C2×C411yescharacteristic
44C22×#41C2×C411yescharacteristic
45C23×D4—22no—
46C23×D4—21no—
47C23×D4C2×C412yes—
48C23×D4C2×C411yescharacteristic
49#520—44no—
50C4×#41—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8D412yes—
53C2×C4×D4—22no—
54C2×C4×D4—21no—
55C2×C4×D4D412yes—
56C2×C4×D4D414yes—
57C2×#117C2×C412yes—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C412yes—
60C2×#117C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—24no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67#446—48no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—22no—
74C2×#97—24no—
75C2×C4×C8—22no—
76#391C2×C411yescharacteristic
77#391C2×C412yes—
78#391C2311yescharacteristic
79#390—24no—
80#390—22no—
81#390—22no—
82#390—24no—
83#390C2×C411yescharacteristic
84#390C2×C412yes—
85#390C2×C412yes—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2312yes—
89#390C2311yescharacteristic
90#386—22no—
91#386—24no—
92#386—22no—
93#386—24no—
94#380—24no—
95#380—22no—
96#379—22no—
97#379—24no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—24no—
100#336C2312yes—
101#336C2314yes—
102#336C2312yes—
103C25—22no—
104C25—21no—
105H(5,𝔽2)—432no—
106H(5,𝔽2)—432no—
107C2×#41—28no—
108C2×#41—24no—
109C2×#41—24no—
110C2×#41—24no—
111C2×#41—24no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—44no—
116C2×#41—44no—
117C2×#41—42no—
118C2×#41—42no—
119C2×#41—44no—
120C2×#41—44no—
121C2×#41M1614yes—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C411yescharacteristic
124C22×D4—216no—
125C22×D4—28no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—48no—
138C22×D4—48no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4C22⋊C412yes—
142C22×D4C22⋊C411yescharacteristic
143C22×D4M1614yes—
144C22×D4C22×C411yescharacteristic
145C22×D4C22×C412yes—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D414yes—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—24no—
157C4×Q8—28no—
158C4×Q8—44no—
159C4×D4—44no—
160C4×D4—48no—
161C4×D4—48no—
162C4×D4—44no—
163#117—44no—
164#117—42no—
165#117—44no—
166#117—42no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—84no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—84no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C414yes—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42—44no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D414yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—44no—
201C4⋊C8—48no—
202C22⋊C8—48no—
203C22⋊C8—44no—
204C22⋊C8—84no—
205C22⋊C8—82no—
206#97—48no—
207C4×C8—44no—
208#95—22no—
209#95—24no—
210#95—22no—
211C24—28no—
212C24—22no—
213C24—21no—
214C24—24no—
215C24—42no—
216C24—44no—
217C24—42no—
218C24—44no—
219C24—82no—
220C24—84no—
221C24—82no—
222C24—84no—
223C24C2×C22⋊C411yescharacteristic
224C24C2×C22⋊C412yes—
225#41—216no—
226#41—216no—
227#41—416no—
228#41—416no—
229#41—48no—
230#41—48no—
231#41—432no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8#9914yes—
238C2×Q8C2×M1612yes—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—28no—
242C2×D4—28no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—416no—
256C2×D4—416no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—464no—
260C2×D4—432no—
261C2×D4—42no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4#10014yes—
269C2×D4C2×M1612yes—
270C2×D4C2×M1614yes—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—84no—
298C22×C4—82no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C414yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×M1612yes—
304C22×C4C2×M1612yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D412yes—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—44no—
314C2×C8—42no—
315C2×C8—44no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—82no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C42—24no—
337C42—28no—
338C42—41no—
339C42—41no—
340C42—42no—
341C42—44no—
342C42—42no—
343C23—22no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—22no—
351C23—24no—
352C23—22no—
353C23—48no—
354C23—416no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23—44no—
360C23—42no—
361C23—44no—
362C23—42no—
363C23—88no—
364C23—816no—
365C23—164no—
366C23—164no—
367C23—162no—
368C23—162no—
369C23—164no—
370C23—168no—
371C23#40712yes—
372C23C2×#10014yes—
373C23C22×C22⋊C411yescharacteristic
374Q8—28no—
375Q8—42no—
376Q8—42no—
377D4—216no—
378D4—28no—
379D4—42no—
380D4—44no—
381D4—44no—
382D4—42no—
383D4—416no—
384D4—48no—
385D4—416no—
386D4—48no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—416no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—48no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4—164no—
419C2×C4—162no—
420C2×C4#40712yes—
421C2×C4C2×#9914yes—
422C2×C4C2×#9912yes—
423C2×C4C2×#10012yes—
424C2×C4C22×M1611yescharacteristic
425C2×C4C22×M1611yescharacteristic
426C2×C4#57612yes—
427C2×C4#57712yes—
428C8—44no—
429C8—82no—
430C8—84no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22—22no—
435C22—24no—
436C22—41no—
437C22—44no—
438C22—41no—
439C22—44no—
440C22—42no—
441C22—48no—
442C22—44no—
443C22—44no—
444C22—42no—
445C22—81no—
446C22—82no—
447C22—81no—
448C22—82no—
449C22—84no—
450C22—82no—
451C22—168no—
452C22—1616no—
453C22C2×#40711yescharacteristic
454C22C22×#9911yescharacteristic
455C22C22×#10011yescharacteristic
456C22#289712yes—
457C22#290112yes—
458C4—21no—
459C4—21no—
460C4—24no—
461C4—22no—
462C4—22no—
463C4—42no—
464C4—42no—
465C4—44no—
466C4—81no—
467C4—81no—
468C4—82no—
469C4#146214yes—
470C2—44no—
471C2—42no—
472C2—84no—
473C2—82no—
474C2C2×#146211yescharacteristic
475C2#1745312yes—
476C2#1761112yes—
477C2#1768511yescharacteristic
478C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×H(5,𝔽2)C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22532C2semidirect
3. #7640C2semidirect
4. #7626C2semidirect
5. #7612C2semidirect
6. #7597C2semidirect
7. #7572C2semidirect
8. #7563C2semidirect
26. #1797C22semidirect
27. #1797C22semidirect
28. #1749C22semidirect
29. C4×C22⋊C8C22semidirect
30. #1272C22semidirect
31. #1272C22semidirect
32. #1270C22semidirect
33. #1270C22semidirect
34. #1270C22semidirect
35. #1270C22semidirect
36. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#342582
#225322
#76402
#76262
#76122
#75972
#75722
#75632