Finite Group Databasebeta

← #363388

Subgroups — #363388

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic, complemented
3C42×C22⋊C4C211yesmaximal, characteristic, complemented
4#11201C212yesmaximal, complemented
5#10971C212yesmaximal
6#10923C212yesmaximal, complemented
7#10878C212yesmaximal, complemented
8#10808C212yesmaximal, complemented
9#10790C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×#104C2214yes—
14C2×#407C2212yes—
15C4×C22⋊Q8C412yescomplemented
16C4×C22⋊Q8C412yescomplemented
17C4×#123—22no—
18C4×#121C412yescomplemented
19C4×#121C412yescomplemented
20C4×#120—22no—
21#2297C2211yescharacteristic
22#2296C2211yescharacteristic
23#2290—22no—
24#2280C2211yescharacteristic
25C42×D4—22no—
26C4×#117C2211yescharacteristic
27C2×C4×C22⋊C4—22no—
28C2×C4×C22⋊C4C2212yes—
29C2×C43C2212yes—
30#1817—22no—
31#1817—22no—
32C2×#339C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1728C2212yes—
36#1728C2212yes—
37C4×#95—22no—
38C4×#95C2212yes—
39C22×#41C2311yescharacteristic
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C22×M16C2314yes—
43#525—24no—
44#525—24no—
45C2×#123—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×#121C2×C412yes—
48C2×#120—21no—
49C4×#41—44no—
50C2×C4×Q8C2×C412yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×#117—21no—
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C42—24no—
62C22×C42—21no—
63C22×C42—44no—
64C22×C42C2×C412yes—
65C22×C42C2×C412yes—
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42C2314yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#104—28no—
73C2×#104—28no—
74C2×#104—44no—
75#407—44no—
76C2×C22⋊C8—24no—
77#391C2×C411yescharacteristic
78#391C2×C411yescharacteristic
79#390C2×C411yescharacteristic
80#390C2×C411yescharacteristic
81#389C2×C412yes—
82#389C2×C412yes—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#386—22no—
88#386—22no—
89#386—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#381—21no—
94#380C2×C412yes—
95#380C2×C412yes—
96#379—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C22⋊C4—216no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—28no—
111#376—21no—
112C2×#95—22no—
113C2×#95—24no—
114C43—21no—
115C43—44no—
116#339—28no—
117#339—28no—
118#339—44no—
119#336—22no—
120#336—22no—
121#336—24no—
122C25C2×D411yescharacteristic
123C2×#41—22no—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41#4114yes—
127C22×Q8C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C22×C411yescharacteristic
131C23×C4—24no—
132C23×C4—28no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—24no—
157C22×C8—24no—
158#123—28no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161#121—24no—
162#121—24no—
163#120—28no—
164C4×Q8—44no—
165C4×D4—216no—
166C4×D4—44no—
167C4×D4—48no—
168C4×D4—48no—
169C4×D4—48no—
170#117—22no—
171#117—24no—
172#117—24no—
173#117—44no—
174#117#4114yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C22⋊C4—216no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—21no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—216no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—24no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—48no—
216C2×C42—48no—
217C2×C42C22⋊C412yes—
218C2×C42C22⋊C412yes—
219C2×C42C22⋊C412yes—
220C2×C42C22⋊C412yes—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42#4118yes—
228#104—416no—
229C22⋊C8—48no—
230#95—216no—
231#95—28no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240C24—22no—
241C24—24no—
242C24—21no—
243C24—22no—
244C24—22no—
245C24—42no—
246C24C2×C22⋊C412yes—
247C24C22×D411yescharacteristic
248#41—28no—
249#41—28no—
250#41—44no—
251#41—44no—
252C2×Q8—21no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8#11712yes—
256C2×D4—28no—
257C2×D4—24no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4#11712yes—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—216no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—432no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4#10418yescomplemented
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C4C2×#4114yes—
325C22×C4H(5,𝔽2)14yes—
326M16—44no—
327M16—44no—
328M16—44no—
329M16—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C4⋊C4—28no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4#11714yes—
342C22⋊C4—232no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—24no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C42—22no—
354C42—22no—
355C42—24no—
356C42—24no—
357C42—22no—
358C42—24no—
359C42—22no—
360C42—28no—
361C42—28no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—432no—
367C42—44no—
368C42—44no—
369C42—42no—
370C42—42no—
371C42—44no—
372C42—44no—
373C42—44no—
374C23—22no—
375C23—22no—
376C23—28no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—48no—
385C23—48no—
386C23—44no—
387C23—44no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23C2×#10414yes—
393C23C22×C22⋊C411yescharacteristic
394C23#52512yes—
395C23#52512yes—
396C23#53412yes—
397Q8—24no—
398Q8—42no—
399D4—24no—
400D4—416no—
401D4—48no—
402D4—42no—
403C2×C4—216no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—216no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—416no—
425C2×C4—416no—
426C2×C4—48no—
427C2×C4—41no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—88no—
445C2×C4C2×#10414yes—
446C2×C4C2×#10414yes—
447C2×C4#42114yes—
448C2×C4C2×#11711yescharacteristic
449C2×C4C2×#11711yescharacteristic
450C2×C4C4×#4112yes—
451C2×C4#51812yes—
452C2×C4#52512yes—
453C2×C4#53212yes—
454C2×C4#53412yes—
455C2×C4#55114yes—
456C2×C4#55314yes—
457C2×C4#55314yes—
458C2×C4#55914yes—
459C8—84no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—22no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—416no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—44no—
473C22—42no—
474C22—82no—
475C22—81no—
476C22—81no—
477C22#228011yescharacteristic
478C22#229711yescharacteristic
479C22#230811yescharacteristic
480C22#260014yes—
481C22C22×#10412yes—
482C22C2×#42112yes—
483C4—24no—
484C4—24no—
485C4—24no—
486C4—416no—
487C4—44no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—82no—
492C4—81no—
493C4—81no—
494C4#235312yes—
495C4#235712yes—
496C4#260214yes—
497C2—24no—
498C2—44no—
499C2—82no—
500C2#1394011yescharacteristic
501C2#2074712yes—
502C2#2141511yescharacteristic
503C2#2141811yescharacteristic
504C2#2177312yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20682C2semidirect
3. C42×C22⋊C4C2semidirect
4. #11201C2semidirect
6. #10923C2semidirect
7. #10878C2semidirect
8. #10808C2semidirect
9. #10790C2semidirect
15. C4×C22⋊Q8C4semidirect
16. C4×C22⋊Q8C4semidirect
18. C4×#121C4semidirect
19. C4×#121C4semidirect
314. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#206822
C42×C22⋊C42
#112012
#109712
#109232
#108782
#108082
#107902