Finite Group Databasebeta

← #196843

Subgroups — #196843

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#21786C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic, complemented
5C2×#1503C211yesmaximal, characteristic, complemented
6#9151C214yesmaximal, complemented
7#9143C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2848C2214yes—
16#2626—22no—
17#2580—22no—
18#2566—22no—
19#2563C412yescomplemented
20#2542C412yescomplemented
21#2538C414yescomplemented
22#2349—24no—
23#2349—24no—
24#2317—21no—
25#2309—21no—
26C2×#389—21no—
27C2×#385—22no—
28#2285C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C22×#95C2212yes—
33#1954—22no—
34#1954—22no—
35#1779C2214yes—
36#1518—24no—
37#1503—24no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×D4D411yescharacteristic
45C24×C4—22no—
46C24×C4C2312yes—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×C4⋊Q8C2×C411yescharacteristic, complemented
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C418yescomplemented
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yescomplemented
59C2×C4×D4—28no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#431—22no—
81#431—22no—
82#431—22no—
83#431—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C22⋊C8—28no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#399—24no—
92#396—28no—
93#393—24no—
94#390—22no—
95#390—22no—
96#390—24no—
97#389—28no—
98#389—24no—
99#385—216no—
100#385—28no—
101#385—24no—
102#384—22no—
103#384C2×C412yescomplemented
104#382—22no—
105#382—21no—
106#382—22no—
107#382D414yes—
108#380—21no—
109#380—24no—
110#380—24no—
111#380C2×C418yescomplemented
112#380C2312yes—
113#379—24no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C412yescomplemented
117C4×C4⋊C4D414yes—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—24no—
121C2×#95—24no—
122C2×#95C2311yescharacteristic
123C2×#95C2312yes—
124C2×#95C2311yescharacteristic
125#328—24no—
126#328—24no—
127#328—28no—
128C25C2×D412yes—
129C22×Q8—21no—
130C22×Q8C22×C412yes—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22⋊C412yes—
136C22×D4C22⋊C412yes—
137C22×D4SD1614yes—
138C22×D4Q1614yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—28no—
147C23×C4—48no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—84no—
151C23×C4—84no—
152C23×C4C22×C414yes—
153C23×C4C2×D412yes—
154C23×C4C2×D414yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164#125—42no—
165#125—42no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C22⋊Q8—88no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—432no—
186#117—42no—
187#117—42no—
188#117—42no—
189#117—42no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—84no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C414yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4C22⋊C414yes—
221C2×C22⋊C4C22⋊C414yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—88no—
242C4×C8—42no—
243C4×C8—42no—
244#95—28no—
245#95—22no—
246#95—28no—
247#95—24no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95C2×D412yes—
253#95C2×D414yes—
254#95C2×D412yes—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—44no—
260C24—44no—
261C24C2×C22⋊C414yes—
262C24C2×SD1614yes—
263C24C2×Q1614yes—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—28no—
277C2×D4—28no—
278C2×D4#10318yes—
279C2×D4#10318yes—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—416no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—416no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—88no—
316C22×C4—816no—
317C22×C4—816no—
318C22×C4C2×C22⋊C414yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4#12012yes—
324C22×C4#12012yes—
325C22×C4#12312yes—
326C22×C4#12312yes—
327C22×C4C2×SD1612yes—
328C22×C4C2×Q1612yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4#13712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C4⋊C4—82no—
356C4⋊C4—88no—
357C4⋊C4—88no—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—416no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—88no—
366C22⋊C4—88no—
367C42—44no—
368C42—44no—
369C42—48no—
370C42C4×D412yes—
371C42C4×D412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42#12112yes—
375C42#12112yes—
376C42#12112yes—
377C42#12112yes—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—28no—
383C23—24no—
384C23—28no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—48no—
389C23—48no—
390C23—44no—
391C23—44no—
392C23—48no—
393C23—44no—
394C23—48no—
395C23—44no—
396C23C2×#10318yes—
397C23C22×C22⋊C411yescharacteristic
398C23C2×#12011yescharacteristic
399C23C2×#12311yescharacteristic
400C23C22×SD1611yescharacteristic
401C23C22×Q1611yescharacteristic
402C23C2×AGL(1,ℤ/8)11yescharacteristic
403C23C2×#13711yescharacteristic
404Q8—82no—
405Q8—82no—
406Q8—84no—
407D4—216no—
408D4—216no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—832no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4#38612yes—
451C2×C4#38612yes—
452C2×C4#38612yes—
453C2×C4#38612yes—
454C2×C4C2×#10312yes—
455C2×C4C2×#10312yes—
456C2×C4#41712yes—
457C2×C4#41712yes—
458C2×C4#45014yes—
459C2×C4#45114yes—
460C2×C4#48314yes—
461C2×C4#48414yes—
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C8—82no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—28no—
477C22—216no—
478C22—48no—
479C22—48no—
480C22—48no—
481C22—48no—
482C22—48no—
483C22—44no—
484C22—44no—
485C22C2×#38611yescharacteristic
486C22#288312yes—
487C22C22×#10311yescharacteristic
488C22C2×#41711yescharacteristic
489C22#290112yes—
490C22C2×#45011yescharacteristic
491C22C2×#45111yescharacteristic
492C22C2×#48311yescharacteristic
493C22C2×#48411yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—44no—
502C4—82no—
503C4—84no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4#187214yes—
508C4#187214yes—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—48no—
514C2#1761412yes—
515C2#1763012yes—
516C2C2×#187211yescharacteristic
517C2#2098011yescharacteristic
518C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1518C2semidirect
5. C2×#1503C2semidirect
6. #9151C2semidirect
7. #9143C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
19. #2563C4semidirect
20. #2542C4semidirect
21. #2538C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
55. C2×C22⋊Q8C2×C4semidirect
58. C2×C4×Q8C2×C4semidirect
103. #384C2×C4semidirect
111. #380C2×C4semidirect
116. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#23492
#217862
C2×#15182
C2×#15032
#91512
#91432
#80262
#80142