Finite Group Databasebeta

← #497492

Subgroups — #497492

Order 512.

1291 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
1#36289C211yesmaximal, characteristic
2D4×#99C212yesmaximal
3#13299C212yesmaximal, complemented
4#13090C214yesmaximal, complemented
5#12856C212yesmaximal, complemented
6#12798C211yesmaximal, characteristic, complemented
7C4×#354C211yesmaximal, characteristic, complemented
8C2×#1393C212yesmaximal, complemented
9C2×#554C411yescharacteristic, complemented
10C2×D42C2211yescharacteristic
11C22×#99C2214yes—
12#2858C411yescharacteristic, complemented
13#2850C412yescomplemented
14#2563C2211yescharacteristic
15#2548C411yescharacteristic, complemented
16#2504C412yescomplemented
17#2490—22no—
18#2472—24no—
19#2435C2212yes—
20D4×C22⋊C4—22no—
21D4×C22⋊C4—22no—
22C4×#124C2211yescharacteristic
23C2×#390C2212yes—
24C42×D4C411yescharacteristic, complemented
25C2×#354C2212yes—
26C2×#354C2212yes—
27#1964—22no—
28#1898—24no—
29#1799C2212yes—
30#1781C2214yes—
31C4×#99C2212yes—
32#1393C2214yes—
33#1393C2214yes—
34#1393C2211yescharacteristic
35#1393C2212yes—
36#1393C2211yescharacteristic
37C23×D4D412yes—
38C23×D4C2312yes—
39#554—41no—
40#554—42no—
41#554—41no—
42D42—41no—
43D42—42no—
44D42—41no—
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×#127C2×C411yescharacteristic
47C2×#125—21no—
48C2×#124—22no—
49C2×#124C2×C412yescomplemented
50C2×#124C2311yescharacteristic
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yescomplemented
53C2×#121C2×C412yes—
54C2×#121D412yes—
55C2×#120D414yes—
56C2×C4×Q8C2×C411yescharacteristic, complemented
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—42no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C22×C4⋊C4C2×C412yescomplemented
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C42C2×C412yescomplemented
66C2×#99—28no—
67C2×#99—24no—
68C2×#99—24no—
69C2×#99—44no—
70C2×#99C2314yes—
71C2×#99C2312yes—
72C2×#99C2312yes—
73#397—24no—
74#394—22no—
75#394—24no—
76#393—22no—
77#391C2×C411yescharacteristic, complemented
78#390—22no—
79#390—22no—
80#390—24no—
81#390—42no—
82#390C2×C412yescomplemented
83#390C2×C412yescomplemented
84#390C2×C412yescomplemented
85#390C2312yes—
86#390C2311yescharacteristic
87#388—24no—
88#388—22no—
89#386—24no—
90#386—24no—
91#386—44no—
92#382C2×C412yescomplemented
93#382C2×C411yescharacteristic, complemented
94#380—22no—
95#380—22no—
96#379—24no—
97#379—24no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2×C411yescharacteristic, complemented
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4C2×C412yescomplemented
104C43C2×C411yescharacteristic, complemented
105#354—24no—
106#354—42no—
107#342—22no—
108#342—22no—
109C25C2×D414yes—
110C22×Q8—21no—
111C22×Q8C22×C411yescharacteristic
112C22×D4—22no—
113C22×D4—24no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—42no—
119C22×D4—42no—
120C22×D4—48no—
121C22×D4—82no—
122C22×D4C22⋊C414yes—
123C22×D4C22⋊C412yes—
124C22×D4C22⋊C412yes—
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22×C412yes—
127C22×D4C2×D414yes—
128C22×D4C2411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—44no—
132C23×C4C22×C412yes—
133C23×C4C2×D412yes—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136#127—41no—
137#127—41no—
138#125—42no—
139#125—42no—
140#124—22no—
141#124—21no—
142#124—21no—
143#124—84no—
144#124—82no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149#121—42no—
150#121—42no—
151#121—42no—
152#121—42no—
153#120—44no—
154#120—44no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×D4—22no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—41no—
161C4×D4—41no—
162C4×D4—44no—
163C4×D4—42no—
164C4×D4—41no—
165C4×D4—41no—
166C4×D4—88no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—88no—
196C2×C22⋊C4—84no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C414yes—
199C2×C22⋊C4C2×D412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—82no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C411yescharacteristic
210#99—22no—
211#99—22no—
212#99—24no—
213#99—24no—
214#99—22no—
215#99—48no—
216#99—84no—
217#99#4114yes—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—24no—
223#95—42no—
224#95—42no—
225C24—24no—
226C24—24no—
227C24—24no—
228C24—24no—
229C24—44no—
230C24—48no—
231C24—84no—
232C24C2×C22⋊C412yes—
233C24C2×C22⋊C414yes—
234C24C22×D412yes—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—84no—
261C2×D4—84no—
262C2×D4—84no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—88no—
269C2×D4#9912yes—
270C2×D4#9911yescharacteristic
271C2×D4#9911yescharacteristic
272C2×D4C2×#4112yes—
273C2×D4H(5,𝔽2)11yescharacteristic
274C2×D42−1+411yescharacteristic
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—82no—
297C22×C4—84no—
298C22×C4—88no—
299C22×C4—88no—
300C22×C4—164no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C23×C411yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—84no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—48no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C22⋊C4—84no—
339C22⋊C4—88no—
340C22⋊C4—88no—
341C22⋊C4—164no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—41no—
351C42—41no—
352C42—82no—
353C42—84no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—48no—
372C23—84no—
373C23—84no—
374C23—84no—
375C23—88no—
376C23—88no—
377C23—168no—
378C23C2×#9912yes—
379C23C2×#9912yes—
380C23C22×C22⋊C411yescharacteristic
381C23#54612yes—
382C23#54712yes—
383Q8—81no—
384Q8—81no—
385Q8—162no—
386D4—24no—
387D4—22no—
388D4—42no—
389D4—41no—
390D4—41no—
391D4—48no—
392D4—44no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4—84no—
397D4—84no—
398D4—84no—
399D4—162no—
400D4#35414yescomplemented
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—82no—
434C2×C4—88no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—162no—
444C2×C4—162no—
445C2×C4—1616no—
446C2×C4C2×#9911yescharacteristic
447C2×C4C2×#9911yescharacteristic
448C2×C4C4×#4111yescharacteristic
449C2×C4#52011yescharacteristic
450C22—21no—
451C22—21no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—44no—
460C22—44no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—82no—
469C22—88no—
470C22—88no—
471C22—84no—
472C22—84no—
473C22—164no—
474C22—164no—
475C22C2×#35414yes—
476C22#233911yescharacteristic
477C22C22×#9911yescharacteristic
478C22#288311yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—42no—
486C4—81no—
487C4—82no—
488C4—84no—
489C4—81no—
490C4—82no—
491C4—81no—
492C4—162no—
493C4—164no—
494C4C2×#35412yes—
495C2—21no—
496C2—21no—
497C2—24no—
498C2—42no—
499C2—42no—
500C2—42no—
501C2—84no—
502C2#2247611yescharacteristic
503C2C22×#35411yescharacteristic
504C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×D42 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13299C2semidirect
4. #13090C2semidirect
5. #12856C2semidirect
6. #12798C2semidirect
7. C4×#354C2semidirect
8. C2×#1393C2semidirect
9. C2×#554C4semidirect
12. #2858C4semidirect
13. #2850C4semidirect
15. #2548C4semidirect
16. #2504C4semidirect
24. C42×D4C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
49. C2×#124C2×C4semidirect
52. C2×C22⋊Q8C2×C4semidirect
56. C2×C4×Q8C2×C4semidirect
62. C22×C4⋊C4C2×C4semidirect
65. C22×C42C2×C4semidirect
77. #391C2×C4semidirect
82. #390C2×C4semidirect
83. #390C2×C4semidirect
84. #390C2×C4semidirect
92. #382C2×C4semidirect
93. #382C2×C4semidirect
99. C4×C4⋊C4C2×C4semidirect
103. C4×C22⋊C4C2×C4semidirect
104. C43C2×C4semidirect
400. D4#354semidirect

Maximal subgroups

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

subgroupindex
#362892
D4×#992
#132992
#130902
#128562
#127982
C4×#3542
C2×#13932