Finite Group Databasebeta

← #182748

Subgroups — #182748

Order 512.

1120 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34240C211yesmaximal, characteristic
2#20480C211yesmaximal, characteristic, complemented
3#20467C211yesmaximal, characteristic, complemented
4C2×#1533C211yesmaximal, characteristic
5C2×#1527C211yesmaximal, characteristic
6#8226C212yesmaximal, complemented
7#8226C212yesmaximal, complemented
8#8224C212yesmaximal, complemented
9C4×#329C212yesmaximal, complemented
10#8035C211yesmaximal, characteristic, complemented
11#8032C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2372—22no—
18#2367C412yes—
19#2367C412yes—
20#2342—22no—
21Q8×C22⋊C4C412yes—
22D4×C22⋊C4C412yes—
23#2337—21no—
24#2337—21no—
25#2312—21no—
26C4×#124C2212yes—
27C4×#124C2212yes—
28#2283C2211yescharacteristic
29#2273—21no—
30C4×#117C2211yescharacteristic
31#1770—21no—
32C8×C4⋊C4—21no—
33#1768—22no—
34#1753—21no—
35#1753—21no—
36#1750—22no—
37#1745C2211yescharacteristic, complemented
38#1745C2211yescharacteristic, complemented
39#1745C2212yescomplemented
40C4×#97C2211yescharacteristic, complemented
41C4×#97C2211yescharacteristic, complemented
42C4×#97C2212yescomplemented
43#1533C418yescomplemented
44#1527C418yescomplemented
45C2×#329C2212yes—
46C2×#329C2212yes—
47C2×#329C2212yes—
48C2×#329C2212yes—
49#1295C2212yescomplemented
50#1295C2212yescomplemented
51#1295C2212yescomplemented
52#1295C2212yescomplemented
53C23×Q8C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C22×M16—21no—
56C2×#124—21no—
57C2×#124—21no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124C2×C412yes—
61C2×#124C2×C414yes—
62C2×#124C2×C414yes—
63C2×#124C2312yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C412yes—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×#117D412yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#432—22no—
76#432C2×C414yescomplemented
77#432C2×C414yescomplemented
78#432C2×C414yescomplemented
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—22no—
86C4×M16C2×C414yescomplemented
87C2×#97—22no—
88C2×#97—21no—
89C2×#97—21no—
90C2×#97C2311yescharacteristic
91C2×#97C2311yescharacteristic
92C2×#97C2311yescharacteristic
93C2×#97C2312yes—
94C2×#97C2312yes—
95C2×#97C2311yescharacteristic
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#391—22no—
101#391—22no—
102#391C2×C412yes—
103#391C2×C412yes—
104#390C2×C412yes—
105#390C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#385—22no—
117#385—22no—
118#383—21no—
119#382C2312yes—
120#381—21no—
121#380—22no—
122#380C2×C412yes—
123#379—22no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D411yescharacteristic
139C4×C22⋊C4D411yescharacteristic
140#376—21no—
141#376D412yes—
142C43C2311yescharacteristic
143C43C2311yescharacteristic
144#336—24no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#329C2×C418yescomplemented
150#329C2×C418yescomplemented
151#329C2×C418yescomplemented
152#329C2×C418yescomplemented
153C25—21no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—82no—
157C22×Q8C4214yes—
158C22×Q8C22×C412yes—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—82no—
162C22×D4C4214yes—
163C22×D4C22×C412yes—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—24no—
171C2×M16—42no—
172C2×M16—42no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179#124—24no—
180#124—24no—
181#124—24no—
182#124—24no—
183#124—24no—
184#124—22no—
185#124—22no—
186#124C4218yes—
187#124C22×C414yes—
188#124C22×C414yes—
189C4×Q8—82no—
190C4×Q8—82no—
191C4×D4—82no—
192C4×D4—82no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42C2411yescharacteristic
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244#97—21no—
245#97—21no—
246#97—22no—
247#97—21no—
248#97—22no—
249#97—21no—
250#97—44no—
251#97—42no—
252#97—42no—
253#97C22×C412yes—
254#97C22×C412yes—
255#97C22×C414yes—
256#97C22×C412yes—
257#97C22×C414yes—
258#97C22×C412yes—
259C4×C8—24no—
260C4×C8—44no—
261C4×C8—42no—
262C4×C8—42no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276C24—22no—
277C24—24no—
278C24—44no—
279C24—82no—
280C24—81no—
281C24—81no—
282C24C2×C22⋊C411yescharacteristic
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—82no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8C2×C4214yes—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—82no—
296C2×D4—84no—
297C2×D4—84no—
298C2×D4C2×C4214yes—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4C2×C4212yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4#11712yes—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330M16—48no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—28no—
360C22⋊C4—28no—
361C22⋊C4—24no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—24no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—81no—
379C42—81no—
380C42—82no—
381C42—82no—
382C42C2×C4212yes—
383C42#11712yes—
384C42C4×D412yes—
385C42C4×D412yes—
386C42C4×D412yes—
387C42C4×D412yes—
388C42C23×C411yescharacteristic
389C42C23×C411yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C23—22no—
393C23—22no—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—48no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—84no—
405C23—84no—
406C23—164no—
407C23—164no—
408C23C4×C22⋊C412yes—
409C23C22×C22⋊C411yescharacteristic
410Q8—44no—
411Q8—44no—
412Q8—84no—
413D4—44no—
414D4—44no—
415D4—84no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4—164no—
452C2×C4—164no—
453C2×C4C4×C22⋊C412yes—
454C2×C4C4×C22⋊C412yes—
455C2×C4C4×C22⋊C412yes—
456C2×C4#42114yes—
457C2×C4C22×C4211yescharacteristic
458C2×C4C2×#11711yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C8—24no—
464C8—44no—
465C8—44no—
466C8—42no—
467C8—42no—
468C8—42no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—168no—
482C22C2×C4×C22⋊C411yescharacteristic
483C22#288812yes—
484C22C2×#42112yes—
485C22#290111yescharacteristic
486C22#290111yescharacteristic
487C4—24no—
488C4—21no—
489C4—22no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4—82no—
502C4—82no—
503C2—42no—
504C2—84no—
505C2#1767912yes—
506C2#1768112yes—
507C2#2039811yescharacteristic
508C2C4×#42111yescharacteristic
509C2#2040511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20480C2semidirect
3. #20467C2semidirect
6. #8226C2semidirect
7. #8226C2semidirect
8. #8224C2semidirect
9. C4×#329C2semidirect
10. #8035C2semidirect
11. #8032C2semidirect
37. #1745C22semidirect
38. #1745C22semidirect
39. #1745C22semidirect
40. C4×#97C22semidirect
41. C4×#97C22semidirect
42. C4×#97C22semidirect
43. #1533C4semidirect
44. #1527C4semidirect
49. #1295C22semidirect
50. #1295C22semidirect
51. #1295C22semidirect
52. #1295C22semidirect
76. #432C2×C4semidirect
77. #432C2×C4semidirect
78. #432C2×C4semidirect
86. C4×M16C2×C4semidirect
149. #329C2×C4semidirect
150. #329C2×C4semidirect
151. #329C2×C4semidirect
152. #329C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342402
#204802
#204672
C2×#15332
C2×#15272
#82262
#82242
C4×#3292
#80352
#80322