Finite Group Databasebeta

← #201451

Subgroups — #201451

Order 512.

1801 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34963C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic, complemented
3#20747C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic, complemented
5#9477C214yesmaximal, complemented
6#9475C214yesmaximal, complemented
7#7940C212yesmaximal, complemented
8C2×#518—21no—
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2920C2212yes—
12#2598—22no—
13#2580C412yescomplemented
14#2535C412yescomplemented
15#2487—21no—
16#2482C2212yes—
17#2408—21no—
18#2404C2212yes—
19#2398C412yescomplemented
20#2397C412yescomplemented
21C4×#121—22no—
22C4×#120—22no—
23#2283C2211yescharacteristic
24#2275—24no—
25#2274—21no—
26#2273C2214yes—
27C42×D4—24no—
28C4×#117C2212yes—
29C2×C4×C22⋊C4—22no—
30#1826C2214yes—
31#1806—24no—
32#1789C2214yes—
33C4×#104—24no—
34#1466—22no—
35#1462C2214yes—
36#1460—22no—
37#1284C2218yes—
38C22×#41C2311yescharacteristic
39C23×D4D411yescharacteristic
40C24×C4D412yes—
41C2×#121—22no—
42C2×#120—22no—
43#518—216no—
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8C2×C411yescharacteristic, complemented
49C2×C4×Q8C2×C411yescharacteristic, complemented
50C2×C4×D4—28no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—24no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic, complemented
61C2×C4×D4C2×C411yescharacteristic, complemented
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×#117—22no—
65C2×#117—21no—
66C2×#117C2311yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—24no—
69C22×C22⋊C4C2312yes—
70C22×C42—24no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#431—22no—
76#431—22no—
77#423—22no—
78C2×#104—28no—
79#407C2314yes—
80C4×M16—22no—
81#400—22no—
82#399—22no—
83#399C2×C414yescomplemented
84#397C2×C414yescomplemented
85#394—22no—
86#393C2×C414yescomplemented
87#392—22no—
88#392C2×C414yescomplemented
89#390—22no—
90#389—22no—
91#388—24no—
92#386—24no—
93#386—22no—
94#386—24no—
95#385—24no—
96#385—22no—
97#383—21no—
98#382C2×C412yescomplemented
99#381C2312yes—
100#380—21no—
101#380—22no—
102#379—22no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D414yescomplemented
107C4×C22⋊C4—24no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—216no—
114C4×C22⋊C4C2314yes—
115#376—22no—
116#376C2×C412yescomplemented
117C2×#95—24no—
118C43D414yescomplemented
119#339—28no—
120#323—28no—
121C25C2×D412yes—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—44no—
125C22×Q8C22×C411yescharacteristic
126C22×D4—28no—
127C22×D4—21no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C22⋊C412yes—
132C22×D4C22⋊C412yes—
133C22×D4C22×C411yescharacteristic
134C23×C4—28no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—84no—
143C23×C4C22⋊C414yes—
144C23×C4C22⋊C414yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C2×M16—24no—
150#121—48no—
151#120—48no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×D4—264no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—42no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—42no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—88no—
176#117—216no—
177#117—22no—
178#117—22no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—28no—
193C2×C22⋊C4—232no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4C22×C414yes—
206C2×C22⋊C4C2×D414yes—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—24no—
212C2×C42—21no—
213C2×C42—28no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—88no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D418yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4112yes—
225C2×C42#4112yes—
226C4⋊C8—42no—
227C4⋊C8—42no—
228#104—416no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231#97—42no—
232C4×C8—42no—
233#95—28no—
234#95—22no—
235#95—24no—
236#95—22no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95C22×C414yes—
242C24—22no—
243C24—22no—
244C24—28no—
245C24—21no—
246C24—44no—
247C24—82no—
248C24C2×C22⋊C414yes—
249C24C22×D411yescharacteristic
250#41—28no—
251#41—44no—
252#41—44no—
253#41—44no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×D4—216no—
257C2×D4—216no—
258C2×D4—28no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—48no—
269C2×D4—48no—
270C2×D4—84no—
271C2×D4#10418yescomplemented
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—22no—
276C22×C4—232no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—44no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4—82no—
300C22×C4—84no—
301C22×C4—88no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C414yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4#12012yes—
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12312yes—
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D412yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4112yes—
319M16—48no—
320C2×C8—44no—
321C4⋊C4—216no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—44no—
330C4⋊C4—84no—
331C22⋊C4—216no—
332C22⋊C4—232no—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C22⋊C4—48no—
336C22⋊C4—44no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—48no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C42—216no—
346C42—48no—
347C42—48no—
348C42—44no—
349C42—42no—
350C42—41no—
351C42—42no—
352C42—41no—
353C42—88no—
354C42C4×D412yes—
355C42C4×D412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42#12112yes—
359C42#12112yes—
360C42#12112yes—
361C42#12112yes—
362C23—28no—
363C23—28no—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—22no—
369C23—41no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—41no—
375C23—44no—
376C23—42no—
377C23—41no—
378C23—82no—
379C23—82no—
380C23—816no—
381C23C2×#10418yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#12011yescharacteristic
384C23C2×#12311yescharacteristic
385Q8—24no—
386Q8—42no—
387D4—216no—
388D4—24no—
389D4—216no—
390D4—42no—
391D4—44no—
392D4—44no—
393D4—816no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—216no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—48no—
416C2×C4—41no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—832no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4#38611yescharacteristic
434C2×C4#38612yes—
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#38612yes—
438C2×C4#38611yescharacteristic
439C2×C4C2×#10414yes—
440C2×C4C2×C4×D412yes—
441C2×C4C2×#12112yes—
442C8—84no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—28no—
451C22—44no—
452C22—48no—
453C22—416no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—84no—
459C22—82no—
460C22—88no—
461C22—81no—
462C22—81no—
463C22C2×#38611yescharacteristic
464C22C22×#10412yes—
465C22#302512yes—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—416no—
473C4—44no—
474C4—42no—
475C4—84no—
476C4—81no—
477C4—81no—
478C4#189614yes—
479C4#191112yes—
480C2—22no—
481C2—22no—
482C2—41no—
483C2—48no—
484C2—82no—
485C2#2095911yescharacteristic
486C2C2×#189611yescharacteristic
487C2C2×#191111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radical#34963C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#518C2semidirect
3. #20747C2semidirect
4. #17364C2semidirect
5. #9477C2semidirect
6. #9475C2semidirect
7. #7940C2semidirect
13. #2580C4semidirect
14. #2535C4semidirect
19. #2398C4semidirect
20. #2397C4semidirect
48. C2×C4×Q8C2×C4semidirect
49. C2×C4×Q8C2×C4semidirect
60. C2×C4×D4C2×C4semidirect
61. C2×C4×D4C2×C4semidirect
83. #399C2×C4semidirect
84. #397C2×C4semidirect
86. #393C2×C4semidirect
88. #392C2×C4semidirect
98. #382C2×C4semidirect
106. C4×C4⋊C4D4semidirect
116. #376C2×C4semidirect
118. C43D4semidirect
271. C2×D4#104semidirect

Maximal subgroups

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

subgroupindex
#349632
C4×#5182
#207472
#173642
#94772
#94752
#79402