Finite Group Databasebeta

← #7503469

Subgroups — #7503469

Order 512.

1782 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60520C212yesmaximal
2#60493C212yesmaximal, complemented
3#31113C218yesmaximal, complemented
4#31112C214yesmaximal, complemented
5#31110C214yesmaximal, complemented
6#23675C212yesmaximal, complemented
7#23647C212yesmaximal, complemented
8C2×#2060C211yesmaximal, characteristic, complemented
9C2×#2039C211yesmaximal, characteristic
10#21099C212yesmaximal, complemented
11#20863C212yesmaximal, complemented
12#20800C211yesmaximal, characteristic, complemented
13C2×#544C2211yescharacteristic
14C2×#537C2211yescharacteristic
15#3428C2212yes—
16#3099—24no—
17#3099C2218yes—
18#3098—24no—
19#3098C2218yes—
20#3095—24no—
21#3094—24no—
22#3084C2218yescomplemented
23#3083C2218yescomplemented
24#3082—24no—
25#3081—24no—
26#3080—24no—
27#3079—24no—
28#2911C2212yes—
29C2×#428C2212yes—
30C2×#428C2212yes—
31C2×#428C2212yes—
32#2905C2212yes—
33C22×#104C2212yes—
34#2896C2212yes—
35#2896C2212yes—
36C2×#417C2212yes—
37#2879C2212yes—
38C2×C4×M16C2211yescharacteristic
39#2285C2211yescharacteristic
40C2×#384C2211yescharacteristic
41#2090C2218yescomplemented
42#2089C2218yescomplemented
43#2079C2218yescomplemented
44#2078C2218yescomplemented
45#2060C2214yescomplemented
46#2060C2214yescomplemented
47#2039C2214yes—
48#2039C2214yes—
49#1938C2218yescomplemented
50#1937C2218yescomplemented
51#1860C2214yescomplemented
52#1860C2214yescomplemented
53#1859C2214yescomplemented
54#1859C2214yescomplemented
55#1847C2214yescomplemented
56#1845C2212yescomplemented
57#1844C2212yescomplemented
58#1742C2212yes—
59C2×#343C2211yescharacteristic
60C2×#343C2211yescharacteristic
61#1738C2212yes—
62C2×#339C2212yes—
63C22×#41C2312yes—
64C22×M16C2312yes—
65C22×M16C2312yes—
66C22×M16C2312yes—
67#544—24no—
68#544—41no—
69#544C2314yes—
70#537—24no—
71#537—41no—
72#537C2314yes—
73C2×C4⋊Q8—22no—
74C2×#125—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—41no—
77C2×#121—22no—
78C2×#121—41no—
79#520—22no—
80#520—22no—
81#520C2314yes—
82#520C2314yes—
83#519—22no—
84#518—22no—
85#518—42no—
86C4×#41—24no—
87C4×#41—42no—
88C2×#117—24no—
89C2×#117C2312yes—
90C2×#117C2312yes—
91C22×C4⋊C4—41no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#485—28no—
96#485—28no—
97#484—28no—
98#483—28no—
99#482—28no—
100#482—28no—
101#481—28no—
102#480—28no—
103#479—28no—
104#479—28no—
105#478—28no—
106#478—28no—
107#477—28no—
108#476—28no—
109#475—28no—
110#474—28no—
111#428—24no—
112#428—24no—
113#428—24no—
114#428—24no—
115#428—24no—
116#428—22no—
117#428—22no—
118#428C2314yes—
119#428C2318yes—
120#428C2314yes—
121#427—24no—
122C2×#107—22no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106—22no—
126C2×#106C2314yes—
127C2×#106C2314yes—
128#424—24no—
129#424C2318yes—
130#423—24no—
131C2×#104—28no—
132C2×#104C2314yes—
133C2×#104C2314yes—
134#419—22no—
135#419D414yes—
136#418—22no—
137#418D414yes—
138#417—24no—
139#417D414yes—
140#417D414yes—
141#417C2318yes—
142#417C2314yes—
143#417C2314yes—
144#416—24no—
145#416C2318yes—
146C2×#103—22no—
147C2×#102—22no—
148#408—24no—
149#408C2318yes—
150#407—24no—
151C4×M16—24no—
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154#384—41no—
155#384C2312yes—
156#384C2312yes—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#382C2312yes—
160#380—41no—
161C4×C4⋊C4—41no—
162#343C2314yes—
163#343C2314yes—
164#343C2314yes—
165#343C2314yes—
166#341C2318yes—
167#339C2314yes—
168#339C2314yes—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—24no—
172C2×#41—24no—
173C2×#41—24no—
174C2×#41—24no—
175C2×#41C2×D414yes—
176C2×#41C2414yes—
177C22×Q8—41no—
178C22×Q8C2×D412yes—
179C22×D4—22no—
180C22×D4—41no—
181C22×D4C2×D412yes—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C2411yescharacteristic
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16C2×D414yes—
196C2×M16C2×D414yes—
197C2×M16C2414yes—
198C2×M16C2414yes—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2414yes—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204#125—28no—
205#125—44no—
206C22⋊Q8—28no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209C22⋊Q8—44no—
210#121—28no—
211#121—44no—
212#121—44no—
213#121—44no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—44no—
225C4×D4—44no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—24no—
233#117—24no—
234#117—44no—
235#117—44no—
236#117C2×D414yes—
237#117C2414yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—84no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—44no—
269C2×C42#4114yes—
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—28no—
273#107—28no—
274#107—28no—
275#107—24no—
276#107—24no—
277#106—28no—
278#106—28no—
279#106—28no—
280#106—24no—
281#106—24no—
282C4⋊C8—28no—
283C4⋊C8—28no—
284#104—28no—
285#104—28no—
286#103—28no—
287#103—24no—
288#103—24no—
289#103C2×D418yes—
290#103C2×D418yes—
291#102—28no—
292#102—24no—
293#102—24no—
294#102C2×D418yes—
295#102C2×D418yes—
296C22⋊C8—28no—
297C22⋊C8—28no—
298#97—24no—
299C4×C8—24no—
300#95—42no—
301C24—22no—
302C24H(5,𝔽2)11yescharacteristic
303#41—28no—
304#41—28no—
305#41—24no—
306#41—24no—
307#41—24no—
308#41—24no—
309#41—44no—
310#41—44no—
311#41—44no—
312#41—44no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—44no—
317C2×Q8—162no—
318C2×Q8C22×D414yes—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—162no—
328C2×D4C22×D414yes—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C2×#4112yes—
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C42−1+412yes—
358C22×C42−1+414yes—
359C22×C4C2511yescharacteristic
360M16—28no—
361M16—28no—
362M16—24no—
363M16—24no—
364M16—24no—
365M16—24no—
366M16—44no—
367M16—44no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8C22×D418yes—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4C22×D414yes—
395C4⋊C4C22×D414yes—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—44no—
409C42—44no—
410C42—42no—
411C42—42no—
412C42C2×#4112yes—
413C42C2×#4112yes—
414C23—24no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—82no—
419C23—82no—
420C23#53912yes—
421C23D4211yescharacteristic
422C23D4211yescharacteristic
423C23#56312yes—
424C23C2×H(5,𝔽2)11yescharacteristic
425Q8—24no—
426Q8—24no—
427Q8—44no—
428Q8—164no—
429D4—24no—
430D4—24no—
431D4—48no—
432D4—44no—
433D4—44no—
434D4—44no—
435D4—164no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—48no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—164no—
463C2×C4—164no—
464C2×C4—161no—
465C2×C4—161no—
466C2×C4#53614yes—
467C2×C4#53614yes—
468C2×C4#53912yes—
469C2×C4D4212yes—
470C2×C4#54714yes—
471C2×C4#54714yes—
472C2×C4#56312yes—
473C2×C4C23×D412yes—
474C2×C4C22×#4111yescharacteristic
475C2×C4C2×2−1+411yescharacteristic
476C2×C4C2×2−1+412yes—
477C8—28no—
478C8—44no—
479C8—44no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—164no—
492C22#268012yes—
493C22#268012yes—
494C22C2×#53911yescharacteristic
495C22C2×D4211yescharacteristic
496C22C2×#56311yescharacteristic
497C4—24no—
498C4—42no—
499C4—44no—
500C4—44no—
501C4—44no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4#268014yes—
507C4C2×#53612yes—
508C4C2×#54712yes—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—21no—
513C2—84no—
514C2C2×#268011yescharacteristic
515C2#6111011yescharacteristic
516C2#6111111yescharacteristic
subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60493C2semidirect
3. #31113C2semidirect
4. #31112C2semidirect
5. #31110C2semidirect
6. #23675C2semidirect
7. #23647C2semidirect
8. C2×#2060C2semidirect
10. #21099C2semidirect
11. #20863C2semidirect
12. #20800C2semidirect
22. #3084C22semidirect
23. #3083C22semidirect
41. #2090C22semidirect
42. #2089C22semidirect
43. #2079C22semidirect
44. #2078C22semidirect
45. #2060C22semidirect
46. #2060C22semidirect
49. #1938C22semidirect
50. #1937C22semidirect
51. #1860C22semidirect
52. #1860C22semidirect
53. #1859C22semidirect
54. #1859C22semidirect
55. #1847C22semidirect
56. #1845C22semidirect
57. #1844C22semidirect

Maximal subgroups

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

subgroupindex
#605202
#604932
#311132
#311122
#311102
#236752
#236472
C2×#20602
C2×#20392
#210992
#208632
#208002