Finite Group Databasebeta

← #6477651

Subgroups — #6477651

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic
3#37453C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic
5#33119C214yesmaximal, complemented
6#29764C214yesmaximal, complemented
7#24579C214yesmaximal, complemented
8#24577C214yesmaximal, complemented
9#24482C214yesmaximal, complemented
10#24480C214yesmaximal, complemented
11#21931C211yesmaximal, characteristic
12#21762C211yesmaximal, characteristic
13#21762C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#484—28no—
18C2×#484C2214yes—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#483—28no—
22C2×#483C2214yes—
23C2×#483C2212yes—
24C2×#483C2212yes—
25C22×#107C2212yes—
26C22×#106C2212yes—
27C22×#103C2212yes—
28C22×#103C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2817—22no—
32#2627—22no—
33#2627—22no—
34#2621C2214yes—
35#2503C2212yes—
36#2503C2212yes—
37#2483—22no—
38#2434C2212yes—
39#2434C2212yes—
40#2345—22no—
41#2345C2214yes—
42#2344—22no—
43#2344C2214yes—
44#2316C2212yes—
45#2313—22no—
46#2280—22no—
47#2280C2211yescharacteristic
48#2280C2211yescharacteristic
49#2274C2212yes—
50#2085C2218yescomplemented
51#2068C2218yescomplemented
52#2066C2218yescomplemented
53#2043C2218yescomplemented
54#1981C2218yescomplemented
55#1949C2218yescomplemented
56#1928C2218yescomplemented
57#1927C2218yescomplemented
58#1892C2214yes—
59#1892C2214yes—
60#1873C2214yes—
61#1873C2214yes—
62#1850C2214yes—
63#1850C2214yes—
64#1823C2212yes—
65#1823C2212yes—
66#1798C2214yes—
67#1787C2212yes—
68#1787C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4—21no—
72C24×C4C2311yescharacteristic
73C23×C8C2312yes—
74C23×C8C2312yes—
75C2×C4⋊Q8C2314yes—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2312yes—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2312yes—
88C2×#117—24no—
89C2×#117C2312yes—
90C2×#117C2312yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106#484—216no—
107#484—28no—
108#484—28no—
109#483—216no—
110#483—28no—
111#483—28no—
112C2×#107—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106—28no—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2318yes—
119C2×#103—28no—
120C2×#103—28no—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2318yes—
126C2×#103C2318yes—
127C2×C22⋊C8—24no—
128C2×C22⋊C8—24no—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133#400—24no—
134#400—24no—
135#398—28no—
136#397—28no—
137#395—24no—
138#395—24no—
139#391C2314yes—
140#391C2314yes—
141#389—24no—
142#389C2318yes—
143#387—28no—
144#387—28no—
145#385—24no—
146#385—24no—
147#382C2314yes—
148#381—21no—
149#381—24no—
150#381—24no—
151#381—24no—
152#381D412yes—
153#380—24no—
154#380—24no—
155#380D412yes—
156#380D412yes—
157#380C2312yes—
158#380C2312yes—
159#379—21no—
160#379D412yes—
161C4×C4⋊C4—24no—
162C4×C4⋊C4C2318yes—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4C2312yes—
168C4×C22⋊C4C2312yes—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172C25C2×D411yescharacteristic
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8C2412yes—
182C22×Q8C2412yes—
183C23×C4—22no—
184C23×C4—28no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—44no—
193C23×C4—84no—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C22×C8—24no—
203C22×C8—24no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—48no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—24no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221#117—44no—
222#117—44no—
223#117—88no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×D414yes—
269C2×C22⋊C4C2×D414yes—
270C2×C22⋊C4C2×D414yes—
271C2×C42—22no—
272C2×C42—28no—
273C2×C42C2414yes—
274#107—28no—
275#107—28no—
276#106—28no—
277#106—28no—
278C4⋊C8—48no—
279#103—28no—
280#103—28no—
281#103—28no—
282#103—28no—
283#103—48no—
284#103—48no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289#95—28no—
290#95—28no—
291#95—22no—
292#95—22no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95C2×D414yes—
297#95C2×D414yes—
298C24—24no—
299C24—24no—
300C24—42no—
301C24#12114yes—
302C24C22×D411yescharacteristic
303C24C22×D411yescharacteristic
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8—84no—
318C2×Q8—84no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—48no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—816no—
346C22×C4—82no—
347C22×C4—84no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4#12114yes—
352C22×C4#12114yes—
353C22×C4#12114yes—
354C22×C4C22×D418yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2511yescharacteristic
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—28no—
368C2×C8—28no—
369C2×C8—48no—
370C2×C8—48no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—88no—
390C4⋊C4—88no—
391C4⋊C4C22×D414yes—
392C4⋊C4C22×D414yes—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C42—28no—
404C42—44no—
405C42—88no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—28no—
412C23—28no—
413C23—22no—
414C23—22no—
415C23—44no—
416C23—416no—
417C23C2×#12112yes—
418C23C2×#12112yes—
419C23C2×#12112yes—
420C23#57714yes—
421C23#57814yes—
422C23C23×D411yescharacteristic
423Q8—44no—
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—84no—
428Q8—84no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—416no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C2×#12112yes—
468C2×C4C2×#12112yes—
469C2×C4C2×#12112yes—
470C2×C4C23×D411yescharacteristic
471C2×C4C22×#4111yescharacteristic
472C2×C4#58514yes—
473C8—44no—
474C8—44no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—28no—
480C22—28no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—416no—
485C22—44no—
486C22#304214yes—
487C22#304414yes—
488C22C22×#12111yescharacteristic
489C22#348512yes—
490C22C2×#57712yes—
491C22C2×#57812yes—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—44no—
496C4—44no—
497C4—48no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—84no—
503C4#344912yes—
504C4#349112yes—
505C2—24no—
506C2—24no—
507C2—42no—
508C2#3508911yescharacteristic
509C2C2×#304211yescharacteristic
510C2C2×#304411yescharacteristic
511C2#6176714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #33119C2semidirect
6. #29764C2semidirect
7. #24579C2semidirect
8. #24577C2semidirect
9. #24482C2semidirect
10. #24480C2semidirect
50. #2085C22semidirect
51. #2068C22semidirect
52. #2066C22semidirect
53. #2043C22semidirect
54. #1981C22semidirect
55. #1949C22semidirect
56. #1928C22semidirect
57. #1927C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4832
#374532
#341782
#331192
#297642
#245792
#245772
#244822
#244802
#219312
#217622