Finite Group Databasebeta

← #204641

Subgroups — #204641

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic, complemented
3C2×#1513C212yesmaximal, complemented
4C2×#1504C212yesmaximal, complemented
5#13966C211yesmaximal, characteristic, complemented
6#9684C212yesmaximal, complemented
7#9683C212yesmaximal, complemented
8#9681C214yesmaximal, complemented
9C2×#525C2212yes—
10C2×#407C2212yes—
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#2850C2212yes—
14#2617—22no—
15#2602—24no—
16#2500C412yescomplemented
17#2499C412yescomplemented
18#2432C412yescomplemented
19#2431C412yescomplemented
20#2325—22no—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C22×#95C2211yescharacteristic
24#1753—24no—
25#1749C2212yes—
26C4×C22⋊C8C2212yes—
27#1513—28no—
28#1504—28no—
29C2×#328C2214yes—
30C2×#328C2214yes—
31C2×#327C2214yes—
32C2×#327C2214yes—
33#1442—22no—
34#1434C2212yes—
35#1434C2212yes—
36C22⋊#95—22no—
37C22×#41C2×C412yes—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C2×C4⋊Q8C2×C411yescharacteristic, complemented
42C2×#127C2×C411yescharacteristic, complemented
43C2×#124C2×C412yescomplemented
44#525—24no—
45#525—24no—
46#525—24no—
47#525—24no—
48C2×#123—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2312yes—
51C2×#121—22no—
52C2×#121C2×C412yes—
53C2×#121C2312yes—
54C2×#120—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×#117C2×C412yes—
58C22×C4⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#407—24no—
64#407D418yescomplemented
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×#97—22no—
68C2×C4×C8—22no—
69#397—24no—
70#394—24no—
71#394—24no—
72#392—24no—
73#391C2×C412yescomplemented
74#391C2×C412yescomplemented
75#390—22no—
76#390C2×C412yescomplemented
77#390C2×C412yescomplemented
78#389—22no—
79#388—24no—
80#388—24no—
81#387—24no—
82#386—24no—
83#386—24no—
84#385—24no—
85#384C2×C411yescharacteristic, complemented
86#384C2×C411yescharacteristic, complemented
87#382C2×C412yescomplemented
88#380—22no—
89#380—24no—
90C4×C22⋊C4C2×C412yes—
91C4×C22⋊C4C2×C412yes—
92C4×C22⋊C4D414yes—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100C2×#95C2312yes—
101C2×#95C2312yes—
102#336—22no—
103#336—22no—
104#328—28no—
105#328—28no—
106#327—28no—
107#327—28no—
108C25C2×D411yescharacteristic
109C2×#41—24no—
110C2×#41—24no—
111C2×#41—82no—
112C2×#41—82no—
113C22×Q8C22×C412yes—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4C22×C412yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—44no—
123C23×C4—42no—
124C23×C4—44no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D414yes—
135C23×C4#4114yes—
136C23×C4C2411yescharacteristic
137C2×M16—28no—
138C2×M16—82no—
139C2×M16—82no—
140C22×C8—24no—
141C4⋊Q8—81no—
142#127—81no—
143#124—82no—
144#123—44no—
145#123—44no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152#121—24no—
153#121—24no—
154#121—22no—
155#121—22no—
156#121—22no—
157#121—22no—
158#121—44no—
159#121—44no—
160#120—44no—
161#120—44no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—44no—
166#117—24no—
167#117—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4C22⋊C414yes—
191C2×C22⋊C4C22⋊C414yes—
192C2×C42—22no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C22⋊C8—28no—
201C22⋊C8—24no—
202C22⋊C8C2×D418yes—
203#97—82no—
204C4×C8—82no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—44no—
211#95—44no—
212#95—42no—
213#95—82no—
214#95—82no—
215#95C22⋊C414yes—
216#95C22⋊C414yes—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—42no—
221C24—41no—
222C24—41no—
223C24C2×C22⋊C412yes—
224C24#12412yes—
225C24#12712yes—
226C24C22×D411yescharacteristic
227#41—88no—
228#41—88no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—84no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4—88no—
289C22×C4—88no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4C2×C22⋊C414yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C4×D414yes—
297C22×C4C4×D414yes—
298C22×C4#12114yes—
299C22×C4#12114yes—
300C22×C4#12412yes—
301C22×C4#12712yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D412yes—
305C22×C4C2×#4112yes—
306M16—88no—
307C2×C8—28no—
308C2×C8—44no—
309C2×C8—44no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—48no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—41no—
338C42—41no—
339C42—44no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23#39014yes—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#12411yescharacteristic
366C23C2×#12711yescharacteristic
367C23#57512yes—
368C23#57512yes—
369Q8—82no—
370Q8—82no—
371D4—82no—
372D4—82no—
373D4—88no—
374D4—88no—
375D4—84no—
376D4—82no—
377D4—82no—
378D4—164no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—88no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—816no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4#39014yes—
430C2×C4C2×C4×D412yes—
431C2×C4C2×#12112yes—
432C8—48no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—44no—
445C22—44no—
446C22—48no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—82no—
451C22—84no—
452C22—84no—
453C22—82no—
454C22C2×#39011yescharacteristic
455C22#288311yescharacteristic
456C22#288311yescharacteristic
457C22#289612yes—
458C22#290112yes—
459C22#314212yes—
460C22#315311yescharacteristic
461C22#315311yescharacteristic
462C4—22no—
463C4—24no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—44no—
468C4—84no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—162no—
473C2—21no—
474C2—21no—
475C2—22no—
476C2—44no—
477C2—162no—
478C2#1762312yes—
479C2#1762912yes—
480C2#2120111yescharacteristic
481C2#2121311yescharacteristic
482C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20524C2semidirect
3. C2×#1513C2semidirect
4. C2×#1504C2semidirect
5. #13966C2semidirect
6. #9684C2semidirect
7. #9683C2semidirect
8. #9681C2semidirect
16. #2500C4semidirect
17. #2499C4semidirect
18. #2432C4semidirect
19. #2431C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
42. C2×#127C2×C4semidirect
43. C2×#124C2×C4semidirect
64. #407D4semidirect
73. #391C2×C4semidirect
74. #391C2×C4semidirect
76. #390C2×C4semidirect
77. #390C2×C4semidirect
85. #384C2×C4semidirect
86. #384C2×C4semidirect
87. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350322
#205242
C2×#15132
C2×#15042
#139662
#96842
#96832
#96812