Finite Group Databasebeta

← #7503467

Subgroups — #7503467

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60517C212yesmaximal
2#60492C212yesmaximal, complemented
3#31115C214yesmaximal, complemented
4#31114C214yesmaximal, complemented
5#31112C214yesmaximal, complemented
6#31110C214yesmaximal, complemented
7#23675C212yesmaximal, complemented
8#23647C212yesmaximal, complemented
9#23560C211yesmaximal, characteristic, complemented
10#23498C211yesmaximal, characteristic
11#21099C212yesmaximal, complemented
12C2×#1859C212yesmaximal, complemented
13#20800C211yesmaximal, characteristic, complemented
14#3458C2211yescharacteristic
15#3449C2211yescharacteristic
16C2×#520C2212yes—
17#3099C2218yes—
18#3098C2218yes—
19#3097—24no—
20#3096—24no—
21#3093—24no—
22#3092—24no—
23#3084—24no—
24#3084C2218yescomplemented
25#3083—24no—
26#3083C2218yescomplemented
27#3078—24no—
28#3077—24no—
29#2911C2212yes—
30C2×#428C2212yes—
31C2×#428C2212yes—
32C2×#428C2212yes—
33#2905C2212yes—
34C22×#104C2212yes—
35#2896C2212yes—
36#2896C2212yes—
37C2×#417C2212yes—
38#2879C2212yes—
39C2×C4×M16C2211yescharacteristic
40#2285C2211yescharacteristic
41C2×#384C2211yescharacteristic
42#2090C2218yescomplemented
43#2089C2218yescomplemented
44#2079C2218yescomplemented
45#2078C2218yescomplemented
46#2061C2212yescomplemented
47#2061C2212yescomplemented
48#2060C2212yescomplemented
49#2060C2212yescomplemented
50#2040C2212yes—
51#2040C2212yes—
52#2039C2212yes—
53#2039C2212yes—
54#1938C2218yescomplemented
55#1937C2218yescomplemented
56#1859C2218yescomplemented
57#1859C2218yescomplemented
58#1847C2212yescomplemented
59#1847C2212yescomplemented
60#1845C2212yescomplemented
61#1844C2212yescomplemented
62#1742C2211yescharacteristic
63#1742C2211yescharacteristic
64C2×#343C2212yes—
65#1738C2212yes—
66C2×#339C2212yes—
67C22×#41C2312yes—
68C22×M16C2312yes—
69C22×M16C2312yes—
70C22×M16C2312yes—
71#544—22no—
72#544—22no—
73#544C2312yes—
74#544C2312yes—
75#543—41no—
76#537C2312yes—
77#537C2312yes—
78#536—22no—
79#535—22no—
80#534—41no—
81#533—24no—
82#533—41no—
83#532—41no—
84#525—24no—
85#520—24no—
86#520—24no—
87#520—42no—
88#520C2318yes—
89C2×C4×Q8—22no—
90C2×C4×D4—22no—
91C2×C4×D4—42no—
92C2×#117—22no—
93C2×#117C2312yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—22no—
97C22×C4⋊C4—41no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#485—28no—
101#485—28no—
102#484—28no—
103#483—28no—
104#482—28no—
105#482—28no—
106#481—28no—
107#480—28no—
108#479—28no—
109#479—28no—
110#478—28no—
111#478—28no—
112#477—28no—
113#476—28no—
114#475—28no—
115#474—28no—
116#428—24no—
117#428—24no—
118#428—24no—
119#428—24no—
120#428—24no—
121#428—22no—
122#428—22no—
123#428C2318yes—
124#428C2314yes—
125#428C2314yes—
126#427—24no—
127C2×#107—22no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—22no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133#424—24no—
134#424C2318yes—
135#423—24no—
136C2×#104—28no—
137C2×#104C2314yes—
138C2×#104C2314yes—
139#419—22no—
140#419D414yes—
141#418—22no—
142#418D414yes—
143#417—24no—
144#417D414yes—
145#417D414yes—
146#417C2314yes—
147#417C2314yes—
148#417C2318yes—
149#416—24no—
150#416C2318yes—
151C2×#103—22no—
152C2×#102—22no—
153#408—24no—
154#408C2318yes—
155#407—24no—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159#384—41no—
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#384C2312yes—
163#384C2312yes—
164#382C2311yescharacteristic
165#382C2311yescharacteristic
166#380—41no—
167C4×C4⋊C4—41no—
168#343C2312yes—
169#343C2312yes—
170#343C2318yes—
171#343C2312yes—
172#343C2312yes—
173#341C2318yes—
174#339C2314yes—
175#339C2314yes—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—24no—
179C2×#41—24no—
180C2×#41—24no—
181C2×#41—24no—
182C2×#41C2×D414yes—
183C2×#41C2414yes—
184C22×Q8C2×D412yes—
185C22×D4—22no—
186C22×D4C2×D412yes—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4C2411yescharacteristic
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16C2×D414yes—
201C2×M16C2×D414yes—
202C2×M16C2414yes—
203C2×M16C2414yes—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8C2414yes—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212#127—41no—
213#125—24no—
214#125—24no—
215#125—42no—
216#124—42no—
217#123—44no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—44no—
223C22⋊Q8—44no—
224#121—24no—
225#121—24no—
226#121—42no—
227#121—42no—
228#120—44no—
229C4×Q8—28no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—44no—
233C4×D4—28no—
234C4×D4—44no—
235C4×D4—48no—
236C4×D4—48no—
237C4×D4—42no—
238C4×D4—42no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—28no—
246#117—44no—
247#117C2×D414yes—
248#117C2414yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—84no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2411yescharacteristic
267C2×C4⋊C4C2411yescharacteristic
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—42no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—42no—
279C2×C42#4114yes—
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283#107—28no—
284#107—28no—
285#107—24no—
286#107—24no—
287#106—28no—
288#106—28no—
289#106—28no—
290#106—24no—
291#106—24no—
292C4⋊C8—28no—
293C4⋊C8—28no—
294#104—28no—
295#104—28no—
296#103—28no—
297#103—24no—
298#103—24no—
299#103C2×D418yes—
300#103C2×D418yes—
301#102—28no—
302#102—24no—
303#102—24no—
304#102C2×D418yes—
305#102C2×D418yes—
306C22⋊C8—28no—
307C22⋊C8—28no—
308#97—24no—
309C4×C8—24no—
310#95—41no—
311#95—41no—
312C24—22no—
313C24H(5,𝔽2)11yescharacteristic
314#41—28no—
315#41—28no—
316#41—24no—
317#41—24no—
318#41—24no—
319#41—24no—
320#41—44no—
321#41—44no—
322#41—44no—
323#41—44no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8C22×D414yes—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C22×D414yes—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—84no—
361C22×C4—164no—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+412yes—
367C22×C42−1+414yes—
368C22×C4C2511yescharacteristic
369M16—28no—
370M16—28no—
371M16—24no—
372M16—24no—
373M16—24no—
374M16—24no—
375M16—44no—
376M16—44no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8C22×D418yes—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4C22×D414yes—
406C4⋊C4C22×D414yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—48no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—21no—
418C42—21no—
419C42—48no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42C2×#4112yes—
425C42C2×#4112yes—
426C23—24no—
427C23—41no—
428C23—41no—
429C23—44no—
430C23—82no—
431C23—82no—
432C23#53912yes—
433C23D4212yes—
434C23#56312yes—
435C23C2×H(5,𝔽2)11yescharacteristic
436Q8—24no—
437Q8—24no—
438Q8—44no—
439Q8—162no—
440Q8—162no—
441D4—24no—
442D4—24no—
443D4—48no—
444D4—44no—
445D4—44no—
446D4—44no—
447D4—162no—
448D4—162no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—168no—
477C2×C4—162no—
478C2×C4#53614yes—
479C2×C4#53614yes—
480C2×C4#53912yes—
481C2×C4D4211yescharacteristic
482C2×C4D4211yescharacteristic
483C2×C4#54714yes—
484C2×C4#54714yes—
485C2×C4#56312yes—
486C2×C4C23×D412yes—
487C2×C4C22×#4111yescharacteristic
488C2×C4C2×2−1+411yescharacteristic
489C2×C4C2×2−1+412yes—
490C8—28no—
491C8—44no—
492C8—44no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—88no—
503C22#268014yes—
504C22C2×#53911yescharacteristic
505C22C2×D4211yescharacteristic
506C22C2×#56311yescharacteristic
507C4—24no—
508C4—42no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—84no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4#268012yes—
518C4#268012yes—
519C4C2×#53612yes—
520C4C2×#54712yes—
521C2—21no—
522C2—24no—
523C2—21no—
524C2—84no—
525C2C2×#268011yescharacteristic
526C2#6111011yescharacteristic
527C2#6111111yescharacteristic
subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC22×C4C25
prime radical#3449C22
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. #60492C2semidirect
3. #31115C2semidirect
4. #31114C2semidirect
5. #31112C2semidirect
6. #31110C2semidirect
7. #23675C2semidirect
8. #23647C2semidirect
9. #23560C2semidirect
11. #21099C2semidirect
12. C2×#1859C2semidirect
13. #20800C2semidirect
24. #3084C22semidirect
26. #3083C22semidirect
42. #2090C22semidirect
43. #2089C22semidirect
44. #2079C22semidirect
45. #2078C22semidirect
46. #2061C22semidirect
47. #2061C22semidirect
48. #2060C22semidirect
49. #2060C22semidirect
54. #1938C22semidirect
55. #1937C22semidirect
56. #1859C22semidirect
57. #1859C22semidirect
58. #1847C22semidirect
59. #1847C22semidirect
60. #1845C22semidirect
61. #1844C22semidirect

Maximal subgroups

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

subgroupindex
#605172
#604922
#311152
#311142
#311122
#311102
#236752
#236472
#235602
#234982
#210992
C2×#18592
#208002