Finite Group Databasebeta

← #6477659

Subgroups — #6477659

Order 512.

1820 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
1C2×#3095C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3#37453C211yesmaximal, characteristic
4#34179C211yesmaximal, characteristic
5#33119C214yesmaximal, complemented
6#29764C214yesmaximal, complemented
7#24595C214yesmaximal, complemented
8#24593C214yesmaximal, complemented
9#24498C214yesmaximal, complemented
10#24496C214yesmaximal, complemented
11#21935C211yesmaximal, characteristic
12#21768C211yesmaximal, characteristic
13#21768C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3095—28no—
18#3095—28no—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#485C2214yes—
22C2×#485C2214yes—
23C2×#483C2212yes—
24C2×#483C2212yes—
25C2×#428C2212yes—
26C2×#428C2212yes—
27C2×#419C2212yes—
28C2×#419C2212yes—
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31#2817—22no—
32#2627—22no—
33#2627—22no—
34#2621C2214yes—
35#2503C2212yes—
36#2503C2212yes—
37#2483—22no—
38#2434C2212yes—
39#2434C2212yes—
40#2347—22no—
41#2347C2214yes—
42#2346—22no—
43#2346C2214yes—
44#2316C2212yes—
45#2312—22no—
46#2283—22no—
47#2280C2211yescharacteristic
48#2280C2211yescharacteristic
49#2274C2212yes—
50#2085C2218yescomplemented
51#2068C2218yescomplemented
52#2066C2218yescomplemented
53#2043C2218yescomplemented
54#1981C2218yescomplemented
55#1949C2218yescomplemented
56#1928C2218yescomplemented
57#1927C2218yescomplemented
58#1894C2214yes—
59#1894C2214yes—
60#1873C2214yes—
61#1873C2214yes—
62#1851C2214yes—
63#1851C2214yes—
64#1824C2212yes—
65#1824C2212yes—
66#1798C2214yes—
67#1787C2212yes—
68#1787C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C22×M16C2312yes—
73C22×M16C2312yes—
74C2×C4⋊Q8C2314yes—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8C2312yes—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2312yes—
86C2×C22⋊Q8C2312yes—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117—24no—
90C2×#117D418yes—
91C2×#117C2312yes—
92C2×#117C2312yes—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C42—21no—
105#485—216no—
106#485—216no—
107#484—28no—
108#484—28no—
109#483—28no—
110#483—28no—
111#428—28no—
112#428—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2318yes—
118#419—28no—
119#419—28no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2318yes—
125C2×#103C2318yes—
126#407—24no—
127#407—24no—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132#400—24no—
133#400—24no—
134#398—28no—
135#397—28no—
136#395—24no—
137#395—24no—
138#391C2314yes—
139#391C2314yes—
140#389—24no—
141#389C2318yes—
142#388—28no—
143#388—28no—
144#385—24no—
145#385—24no—
146#383—24no—
147#382—24no—
148#382C2314yes—
149#381—21no—
150#381—24no—
151#381—24no—
152#381D412yes—
153#380—24no—
154#380D412yes—
155#380D412yes—
156#380C2312yes—
157#380C2312yes—
158#379—21no—
159#379D412yes—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4C2318yes—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4C2312yes—
167C4×C22⋊C4C2312yes—
168#376—24no—
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—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—44no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D414yes—
194C23×C4#4114yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—44no—
201C2×M16—44no—
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—48no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—44no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216C22⋊Q8—44no—
217#117—28no—
218#117—28no—
219#117—24no—
220#117—24no—
221#117—44no—
222#117—44no—
223#117C2×D418yes—
224#117C2×D418yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C2×D414yes—
264C2×C22⋊C4C2×D414yes—
265C2×C22⋊C4C2×D414yes—
266C2×C42—28no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—28no—
271C2×C42C2414yes—
272#107—28no—
273#107—28no—
274#106—28no—
275#106—28no—
276C4⋊C8—48no—
277#103—28no—
278#103—28no—
279#103—28no—
280#103—28no—
281#103—48no—
282#103—48no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287#95—22no—
288#95—28no—
289#95—22no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95C2×D414yes—
295#95C2×D414yes—
296C24—42no—
297C24—42no—
298C24—42no—
299C24#12114yes—
300C24C22×D411yescharacteristic
301C24C22×D411yescharacteristic
302C24C22×D411yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—28no—
308C2×Q8—28no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—84no—
316C2×Q8—84no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4#12114yes—
345C22×C4#12114yes—
346C22×C4#12114yes—
347C22×C4C22×D418yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C2×#4112yes—
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2511yescharacteristic
358M16—48no—
359M16—48no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—28no—
363C2×C8—28no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—28no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—88no—
381C4⋊C4—88no—
382C4⋊C4C22×D414yes—
383C4⋊C4C22×D414yes—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C42—28no—
397C42—28no—
398C42—28no—
399C42—44no—
400C42—88no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—88no—
414C23C2×#12112yes—
415C23C2×#12112yes—
416C23C2×#12112yes—
417C23C23×D411yescharacteristic
418Q8—44no—
419Q8—44no—
420Q8—44no—
421Q8—44no—
422Q8—84no—
423Q8—84no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4C2×#12112yes—
461C2×C4C2×#12112yes—
462C2×C4C2×#12112yes—
463C2×C4C23×D411yescharacteristic
464C2×C4C22×#4111yescharacteristic
465C2×C4#58514yes—
466C8—44no—
467C8—44no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—88no—
481C22C22×#12111yescharacteristic
482C22#348512yes—
483C22#358612yes—
484C22#358612yes—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—44no—
489C4—44no—
490C4—48no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4—84no—
496C4#344912yes—
497C4#349112yes—
498C2—42no—
499C2—42no—
500C2—42no—
501C2#3508911yescharacteristic
502C2#6042711yescharacteristic
503C2#6042711yescharacteristic
504C2#6177914yes—
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. #24595C2semidirect
8. #24593C2semidirect
9. #24498C2semidirect
10. #24496C2semidirect
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
C2×#30952
#374532
#341792
#331192
#297642
#245952
#245932
#244982
#244962
#219352
#217682